S Karkar
email: slim.karkar@cea.fr

A Gloaguen

Y Le Guen

M Pierre-Jean

C Dandine-Roulland

E Le Floch

C Philippe

A Tenenhaus

V Frouin

MULTIVARIATE HAPLOTYPE ANALYSIS OF 96 SULCI OPENING FOR 15,612 UK-BIOBANK SUJECTS

Keywords: Imaging genetics, haplotype, multivariate analysis

Imaging genetic studies of large control cohorts such as UK Biobank enable to assess the range of normal variations in brain structures. Previous studies by our group have shown that the width of several cortical sulci is associated with a variant in the upstream region of KCNK2 gene even if this effect is corrected with age. Here we propose to analyze in a multivariate setup the associations between sets of genetic variants and multiple sulci widths. The genetic variants we consider are sets of SNPs of known phase called haplotypes, taken from the upstream region of KCNK2 gene. To the best of our knowledge, multivariate analysis in imaging genetics has never been used in haplotype studies. Our method was able to recover the expected association signal and uncover new associations between imaging data and genetic variants.

I. INTRODUCTION

Grey matter thickness is known to shrink with aging in both diseased and normal brains [START_REF] Ge | Age-related total gray matter and white matter changes in normal adult brain. part ii: Quantitative magnetization transfer ratio histogram analysis[END_REF], [START_REF] Fjell | Structural brain changes in aging: Courses, causes and cognitive consequences[END_REF], [START_REF] Lockhart | Structural imaging measures of brain aging[END_REF]. A related effect is the cortical sulcus widening [START_REF] Kochunov | Age-related morphology trends of cortical sulci[END_REF], [START_REF] Shen | Variation in longitudinal trajectories of cortical sulci in normal elderly[END_REF]. The width of a sulcus can be estimated using a feature called opening [START_REF] Riviere | Brainvisa: an extensible software environment for sharing multimodal neuroimaging data and processing tools[END_REF] shown to be robustly related to grey matter thickness and which does not require spatial normalization nor regional atlas. Heritability studies pointed to a dozen sulci that appeared to be under strong genetic control [START_REF] Le Guen | Genetic influence on the sulcal pits: On the origin of the first cortical folds[END_REF]. Furthermore, in [START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF], GWAS has identified a reproducible genetic marker associated with the opening of the left, posterior, Calloso-Marginal sulcus. In this work, we used the Brainvisa cortical sulci recognition pipeline to automatically segment [START_REF] Fischer | Morphologist 2012: the new morphological pipeline of brainvisa[END_REF] and label [START_REF] Perrot | Cortical sulci recognition and spatial normalization[END_REF] nearly one hundred brain sulci. GWAS use a univariate approach and as such, suffer from several drawbacks, in particular the use of an unduly conservative multiple test correction and the fact that the correlation structure of the genome is not accounted for. In the context of complex traits, where individual variant effect size is expected to be small, only

Authors contributed equally

SNPs that are frequent in the population can significantly be associated with the phenotype. Moreover, univariate analyses are unable to model or predict the role of a genetic variant within the genomic region. Finally, univariate approaches are inadequate in situations where a set of variants are jointly associated to multiple phenotypes (pleiotropy). Using a multiple phenotype multivariate approach, we propose to alleviate these drawbacks by simultaneously analyzing one hundred related phenotypes and to model interactions between genetic variants within the same genomic segment.

II. MATERIAL AND METHODS

We obtained 20,060 T1-weighted MRI from the January 2018 release, under UK Biobank (UKB) data appl. #25251. UKB cohort is particularly suited to study aging, with a mean participants age of 57 years, and a standard deviation of 8.2. We retained 15,612 subjects after QC protocol, British ancestry selection, and additional filtering for high heterogeneity, high missingness, first-degree relatedness and sex mismatch.

II-A. Imaging

For each selected subject, the brain mask of the T1-weighted image is obtained using SPM8 (fil.ion.ucl.ac.uk/spm). Next, individual brain images were segmented into grey matter, white matter and cerebrospinal fluid (CSF) in BrainVisa. Finally, individual sulci were extracted using Morphologist, the sulcus identification pipeline of BrainVisa. We retained the 96 (out of 126) most sample-wide identified sulci : a sulci was retained if it was missing in less than 1000 individuals (94.6% presence rate, see [START_REF] Borne | A patch-based segmentation approach with high level representation of the data for cortical sulci recognition[END_REF]). For each retained sulcus and for each subject, sulcus width or opening (the average distance between both banks) was estimated as the ratio of CSF volume and surface area of the sulcus [START_REF] Le Guen | Genetic influence on the sulcal pits: On the origin of the first cortical folds[END_REF], [START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF].

II-B. Genetics

Genotyping data in UKB (UK Biobank Axiom Array) contains 820, 967 SNPs. In such data, for a given SNP, the variant status is obtained without knowing if it lays on the paternal or maternal chromosome for a heterozygote subject. This raises an issue when one wants to use the chain of consecutive SNPs. In the 2018 release of UK Biobank, the so-called "phased data" are available for the 500,000 subjects. With this preprocessing, the succession of SNPs alleles is inferred in contiguous small regions of maternal or paternal chromosomes. Based on the results of a GWAS [START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF] where SNP rs864736 is found to be associated with the opening of several sulci, we chose a genomic region of 55.8 kbp on chromosome 1, which contained all the SNPs in linkage desequilibrium with rs864736 (i.e., SNPs which are supposed to be inherited together from the parents). This region consists in 18 SNPs. Using the "phased data" of our 15,612 subjects in this region, we derived all the haplotypes with a length of 3 to 18 SNPs.

II-C. Haplotype multivariate association analysis

The interplay between neuroimaging and genetic data is uncovered Regularized Generalized Canonical Correlation Analysis (RGCCA), a general framework for multiblock data analysis [START_REF] Tenenhaus | Regularized Generalized Canonical Correlation Analysis[END_REF], [START_REF] Tenenhaus | Regularized generalized canonical correlation analysis: A framework for sequential multiblock component methods[END_REF].

The first block, denoted X 1 , is related to neuroimaging and is defined by p 1 = 96 sulci measured on n = 15, 612 individuals. The second block X 2 is related to genetic information and is defined by p 2 = 604 haplotypes measured on the same set of n individuals. RGCCA aims to find block components y j = X j w j ; j = 1, 2 (where w j is a column vector with p j elements) summarizing the relevant information between and within the blocks. In this context, RGCCA is defined as the following optimization problem :

max w1,w2 cov 2 (X 1 w 1 , X 2 w 2 ) s.t. w j Ŝjj w j = 1, j = 1, 2. (1)
where Ŝjj = τ j I + (1-τj ) n-1 X j X j and τ j is a scalar between 0 and 1. Ŝjj can be considered as a shrinkage estimate of the true variance-covariance matrix Σ jj [START_REF] Ledoit | A well conditioned estimator for largedimensional covariance matrices[END_REF]. [START_REF] Schäfer | A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics[END_REF] gives an analytical formula for the optimal τ j that minimizes the mean square error between Σ jj and its estimate Ŝjj .

II-D. Bootstrap procedure and missing data imputation

A balanced bootstrap procedure [START_REF] Gleason | Algorithms for balanced bootstrap simulations[END_REF] is used to assess the reliability of estimated weights. For that purpose B = 2000 bootstrap samples are considered. Some sulci were not detected in all individuals, therefore a simple regression imputation strategy is used to avoid missing values in each bootstrap sample of sulci opening data. A simple regression model is built to predict each opening value from the covariates Age, Sex, and the 10 first components of UK Biobank-provided MDS. Residuals where reported in a new n × p 1 matrix, where subjects with missing sulci (i.e. where not accounted for in the regression model) are set to 0. This procedure allows both to impute missing values and remove effects of covariates which are confounding factors in our case. Finally, each residual bootstrap sample is standardized within each block in order to make the variables comparable. To make blocks comparable, each block was divided by the square root of its number of variables [START_REF] Tenenhaus | Regularized generalized canonical correlation analysis: A framework for sequential multiblock component methods[END_REF]. The RGCCA package (freely available at CRAN : cran.r-project.org) was then used to yield the weights vectors w b 1 and w b 2 for each bootstrap sample b = 1, . . . , 2000.

II-E. Confidence intervals and variable selection

In order to assess the reliability of estimated weights, αlevel confidence intervals (C j k,α , j = 1, 2) were derived for each element w 1k (corresp. to the k th sulcus) of w 1 and w 2k (corresp. to the k th haplotype) of w 2 . Non-parametric (C j,N P k,α ) and parametric (C j,P k,α ) methods were considered to estimate these confidence intervals.

For C j,N P k,α , the empirical bootstrap distribution of w j,k was used to estimate the confidence interval. In order to be as restrictive as possible, we chose to only look at

C j,N P k,αmin = min(w b j,k ) b∈ 1,B , max(w b j,k ) b∈ 1,B
, which corresponds to a level α min = 2/B = 10 -3 . For C j,P k,α , similarly to [START_REF] Tenenhaus | Kernel Generalized Canonical Correlation Analysis[END_REF], we estimated the mean and variance for each element w 1k and w 2k across the bootstrap samples, from which we derived confidence intervals under the assumption that the weights estimation exhibited asymptotic normality.

Since simultaneous multiple bootstrap confidence intervals estimates were desired with an overall confidence coefficient 1 -α for each block X j , j = 1, 2, we constructed each interval with confidence coefficient 1 -(α/p j ), j = 1, 2. The procedure is similar to a Bonferroni correction. For nonparametric estimation, there is no advantage in exploring levels lower than α min since their confidence intervals correspond to the full distribution.

Nevertheless, non-parametric estimation still offers practical advantages as being closer to the real distribution, thus, we chose to look at three confidence intervals : C j,N P k,αmin , C j,P k,5%,BF and C j,P k,10%,BF . The last two confidence intervals are Bonferroni corrected as mentioned earlier. This correction is likely very conservative w.r.t. the independence of the p j variables since many haplotypes are subsets of others. For these reasons we chose to explore the two levels of confidence α = 0.05 and α = 0.1.

A weight element w jk is considered as relevant if zero is excluded from its confidence interval, since the probability of this weight being zero is lower than the level of confidence associated to this interval.

III. RESULTS

Figure 1 represents the weights w j , j = 1, 2 computed with RGCCA. Only relevant weights according to C j,N P k,αmin are shown. The ones associated to a single star were relevant for C j,P k,10%,BF , the ones with a double star were significant according to both C j,P k,10%,BF and C j,P k,5%,BF .

III-A. Selected variables for imaging block

Figure 1 (top, left) shows a barplot of the weights associated with the 2 selected features of the imaging block. The selected variables correspond to the bilateral Posterior Calloso-Marginal Sulci, of which the opening of the former was reported significantly associated with rs864736. Variables were relevant according to the three confidence intervals considered. There were 2 supplemental imaging features (right Subcallosal and Superior precentral sulci, not shown) relevant according to C j,P k,10%,BF . However, they were excluded since they were not selected using C j,N P k,αmin . the haplotype that starts on position #4 with 5 reference alleles and a single alternative allele at position #9. Selected haplotypes included various combinations of variants (from 3 to 10), however none of the haplotypes included variants between position 11 to 18. [4-000001] and are relevant to C j,P k,5%,BF . If the confidence level is decreased to 10% using C j,P k,10%,BF , 3 additional haplotypes appear : [7-0011], [START_REF] Shen | Variation in longitudinal trajectories of cortical sulci in normal elderly[END_REF][START_REF] Riviere | Brainvisa: an extensible software environment for sharing multimodal neuroimaging data and processing tools[END_REF][START_REF] Le Guen | Genetic influence on the sulcal pits: On the origin of the first cortical folds[END_REF][START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF][START_REF] Fischer | Morphologist 2012: the new morphological pipeline of brainvisa[END_REF][START_REF] Perrot | Cortical sulci recognition and spatial normalization[END_REF][START_REF] Borne | A patch-based segmentation approach with high level representation of the data for cortical sulci recognition[END_REF] and . Only the last one has a positive weight.

III-B. Selected variables for genetic block

III-C. Interpretation findings

We will interpret the sign of the weights using haplotype and Left Posterior Calloso-Marginal Sulcus (FCMpost left) as an example. These both variables have negative weights in the model meaning that they are negatively correlated to their block component y j , j = 1, 2. However, over the B = 2000 bootstrap samples, correlation between y 1 and y 2 was always negative. To summarize, the presence of haplotype is associated to a lower opening for FCMpost left : haplotypes with a negative weight have a protective effect on the sulcus opening w.r.t aging. Opposite conclusions are drawn for haplotypes with a positive weight in the model. However, interpretation of the values of these weights is not straightforward. To assess the magnitude of the association with the phenotypes we rely on explained variance in the following section.

IV. DISCUSSION

Previous studies by our group identified SNP rs864736 (and marginally rs59084003) as significantly associated with sulci opening and grey matter thickness for left Posterior Calloso-Marginal, Intra-Parietal and Central sulci. RGCCA was run two last times, first with only relevant variables according to C j,P k,5%,BF (RGCCA-5%) then with variables based on C j,P k,10%,BF (RGCCA-10%). For comparison purposes, we computed the explained variance using univariate analysis of the most significant variant (rs864736) and the sulci opening most associated with it (FCM posterior left and right). Several haplotypes had better or similar explained variance than rs864736 (see top and middle rows of Table I). Moreover, in RGCCA-10%, variance explained by the genetic block component y 2 = X 2 w 2 (where here, X 2 is only composed of selected variables based on C j,P k,10%,BF ) largely outperformed the single-SNP model. When we evaluated the variance of the imaging block component y 1 , as explained by the full genetic block component y 2 , both RGCCA-5% and RGCCA-10% outperformed the single-SNP model.

Several selected haplotypes are subsets of each other, and differ by one variant. If the SNP they differ from is not frequent, they represent very similar variables. To evaluate the impact of such co-linear variables, haplotype variables were excluded if the percentage of subject that differs for two haplotypes is less than 1%. In this case, only the variable for most frequent haplotype was kept. On this new dataset, we used the same bootstrap procedure to derive a new C j,P k,5%,BF . In a similar fashion to RGCCA-5%, we then derived a third model. For this model, 3 variables in the genetic block were selected ([2-01000001], and [START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 uk biobank participants with neuroimaging data[END_REF][START_REF] Fischer | Morphologist 2012: the new morphological pipeline of brainvisa[END_REF][START_REF] Perrot | Cortical sulci recognition and spatial normalization[END_REF][START_REF] Borne | A patch-based segmentation approach with high level representation of the data for cortical sulci recognition[END_REF]). Variables in the imaging block remained the same and their associated weights were similar to RGCCA-5%. The combination of the 3 genetic variables exhibits similar explanatory power as RGCCA-10%, with an explained variance of 0.64 and 0.7 for left and right Calosso-Marginal sulci, respectively.

Work in progress includes application of sparse versions of RGCCA [START_REF] Tenenhaus | Variable selection for generalized canonical correlation analysis[END_REF], [START_REF] Löfstedt | A general multiblock method for structured variable selection[END_REF] and comparisons with the presented results.

V. CONCLUSION AND FUTURE WORKS

We proposed a multivariate model for haplotype associations with multiple quantitative traits that successfully recovered previously known associations, and gained substantial knowledge regarding the genomic region and associated sulci. We present three new findings : 1) only the genomic region located before rs864736 and rs59084003 seems to be implicated in the association ; 2) haplotype combinations are explanatory variables regarding Calloso-Marginal sulcus in both hemispheres ; and 3) an alternate allele at the third position (rs504473) seems to be associated with an antagonistic effect w.r.t rs864736 and rs59084003. Future works will extend this approach to gene clusters, gene pathways and larger intergenic regions to detect regulating patterns that interact with the observed phenotypes.

This method relies on a critical variable selection procedure based on the bootstrap folds. This procedure has shown to be sensitive to strongly co-linear variables, therefore we intend to propose several developments that could enhance this step. First, using a tree-like representation of haplotypes, we could regularize or combine variables, thus allowing us to keep more observations for the model estimation. Second, using block sparsity and regularization, multivariate procedures such as sparse group-lasso could better account for co-linearity of the variables. In the context of imaging genetics, we argue that insights provided by multivariate approaches are key in uncovering the complex interactions between genes, structure and function.

Fig. 1 .

 1 Fig. 1. (Left) : Weights w1 associated with the selected variables of the imaging block. Selected features were the bilateral Posterior Calloso-Marginal Sulci ; (Top, Right) : Weights w2 associated with the selected variables of the genetic block. All displayed variables were selected using C j,N P k,α min (see text for detail) with ( * * ) : variables selected with parametric confidence level C j,P k,5%,BF and ( * ) : variables selected with parametric confidence level C j,P k,10%,BF . (Bottom, Right) : SNP composition of selected haplotypes : light grey bars show the extent of the sequence and dots indicate the location of alternative alleles (see text for details).

Figure 1 (

 1 Figure 1 (top, right) depicts the weights associated with the 12 selected haplotypes (in decreasing order) according to C j,N P k,αmin . Figure 1 (bottom, right) gives the composition of the haplotypes : selected sequences of variants are represented as a grey box. In each sequence, grey dots indicate the alternate alleles. Haplotypes are named as follows : [index of starting SNP -sequence of variants], e.g. [4 -000001] refers to


Yann Le Guen

Slim Karkar

Antoine Grigis

Cathy Philippe

Jean-François Mangin

Vincent Frouin
email: bvincent.frouin@cea.fr

Heritability of surface area and cortical thickness: a comparison between the Human Connectome Project and the UK Biobank dataset

Keywords: heritability, cortical thickness, surface area

Heritability of the regional thickness/surface in human cortex is established. Yet the estimates vary substantially depending on cohorts or the information they are derived from: pedigree or genotyping. Here we present three heritability studies of the cortex phenotype in two cohorts showing: i) both pedigree and genotyping or ii) genotyping only. We obtained clearly correlated heritability values between studies with a shift appearing between cohorts. Spatial pattern of heritability remains highly consistent across cohorts Index Terms-imaging-genetic

INTRODUCTION

The heritability of the cortical features from structural MRI have been extensively reported in the literature [START_REF] Hofer | Genetic Determinants of Cortical Structure (Thickness, Surface Area and Volumes) among Disease Free Adults in the CHARGE Consortium[END_REF]- [START_REF] Elliott | Genome-wide association studies of brain imaging phenotypes in UK Biobank[END_REF]. In this work, we proposed to benchmark the heritability of two common cortical features, namely the surface area and cortical thickness, in the two largest cohorts currently available: the Human Connectome Project (HCP) and the UK Biobank. These two features have been shown to have low genetic correlation [START_REF] Panizzon | Distinct genetic influences on cortical surface area and cortical thickness[END_REF] and thus provide complementary information on the underlying genetic which shapes the human brain cortex. We noticed that estimates reported by pedigree studies were constantly higher than the ones reported by population of unrelated individuals [START_REF] Hofer | Genetic Determinants of Cortical Structure (Thickness, Surface Area and Volumes) among Disease Free Adults in the CHARGE Consortium[END_REF], [START_REF] Van Der Lee | Gray matter heritability in familybased and population-based studies using voxel-based morphometry[END_REF]. In this study, we replicated this observation and attempted to rule out the hypotheses that these discrepancies are due to: (i) the lack of information in genotyped common variants, (ii) the un-modelled common environment. Additionally, we also discussed the genotype and age interaction which has been shown to affect heritability estimates [START_REF] Chouinard-Decorte | Heritable changes in regional cortical thickness with age[END_REF].

MATERIAL AND METHODS

Subjects, MRI Acquisition, Genetic Information

Human Connectome Project data

In this work, we used data from the HCP, details are available in the HCP reference manual S1200. In our analysis, we included 820 subjects (383/437 M/F), labelled as Caucasian with 69 individuals from the Hispanic ethnicity. The pedigree is composed of 191 twin pairs (127 monozygotic with 123 siblings, and 64 dizygotic with 64 siblings and 1 half sibling), 190 siblings, 1 half sibling and 59 unpaired individuals, aged between 22 and 36 years old (µ ± σ = 29.0 ± 3.6 years). Unpaired individuals did not contribute to the genetic parameter estimation but allowed for a more accurate estimation of mean and variance effects. MR images were acquired by using a Siemens 3T scanner housed at Washington University in St Louis, using MPRAGE and SPACE sequences for respectively T1 and T2 weighted images which both have a 0.7 mm isotropic resolution. Genotyping data of HCP subjects were obtained under dbGaP appl. #17771 (dbGaP Study Accession: phs001364.v1.p1).

UK Biobank project data

The present analyses were conducted under UK Biobank data appl. #25251. The UK Biobank is a health research resource that aims to improve the prevention, diagnosis and treatment of a wide range of illnesses. Between the years 2006 and 2010, about 500,000 people aged between 45 and 73 years old, were recruited in the general population across Great Britain. In this work, we used the data released on January 2018, consisting of 20,060 subjects with a T1-weighted MRI. The UK Biobank genetic data underwent a stringent quality control protocol, which was performed at the Wellcome Trust Centre for Human Genetics [START_REF] Bycroft | Genome-wide genetic data on ~500,000 UK Biobank participants[END_REF]. We restrained our analysis to people identified by UK Biobank as belonging to the main white British ancestry subset. Additionally, we excluded from our analysis subjects with high missingness, high heterozygosity, first degree related individuals or sex mismatches. 14,538 subjects passed the genotyped and imaging quality control protocols, and were identified as belonging to the British ancestry.

Image processing

We used the T1w and T2w volumes, preprocessed by the HCP, from each individual subject's MR data as inputs of the HCP Freesurfer pipeline, which is based on Freesurfer (v.5.3.0) with a number of enhancements specifically designed to capitalize on HCP data. We used the T1w volumes, preprocessed by the UK Biobank, from each individual subject's MR data as inputs of the Freesurfer pipeline (v. 6.0.0). For both cohorts, we extracted the cortical thickness and surface measures for each region of interest (ROI) of the Destrieux atlas (files ?h.aparc.a2009s.annot) [START_REF] Destrieux | Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature[END_REF]. This atlas contains 74 areas per hemisphere and displayed on Fig. 1.

To correct for multiple comparisons, our significance threshold for the p-value associated to our heritability estimate is p < 0.05 / (74×2×2) ≈ 1.7•10 -4 (Bonferroni correction, nb areals×nb hemispheres×nb features).

Heritability computation

We used two different ways to estimate the heritability based on the two types of genetic information. The first one uses the pedigree information in the HCP cohort and the second one uses the genotyped information from common genetic variants in the HCP and UK Biobank. The variance components method, as implemented in the Sequential Oligogenic Linkage Analysis Routines (SOLAR) [START_REF] Almasy | Multipoint Quantitative-Trait Linkage Analysis in General Pedigrees[END_REF], was used for the heritability estimations of the features namely the average cortical thickness. The covariance matrix Ω for a pedigree of individuals is given by: Ω = 2•Φ•σg² + I• σe², where σg² is the genetic variance due to the additive genetic factors, Φ is the kinship matrix representing the pairwise kinship coefficients among all individuals, σe² is the variance due to individual-specific environmental effects, and I is the identity matrix, assuming all environmental effects are uncorrelated among family members. Narrow sense heritability is defined as the fraction of the phenotype variance σp² attributable to additive genetic factors: h² = σg²/ σp². Significance of the heritability is tested by comparing the likelihood of the model in which σg² is constrained to zero with the one of a model in which σg² is estimated. Before testing for the significance of heritability, phenotypes values for each individual within the HCP cohort were adjusted for the following covariates: sex, age, age², age×sex interaction, age²×sex interaction. Inverse Gaussian transformation was also applied to ensure normality of the measurements. An additional variance parameter can be added to model the effect of common environment, which is associated with any non-genetic factors shared between the individuals like living in the same household. In this case, the covariance matrix can be written as: Ω = 2•Φ•σg² + H• σc² + I• σe², where H is the structuring matrix for σc², the variance due to common environment effects. H contains zeros and ones depending upon whether a pair of individual shares the same household. The household information is not directly available in HCP data. Thus, we assumed that two individuals share the same household if they have the same family id. We force the common environment to remain in the estimated model using -keephouse SOLAR option. To estimate the heritability from the single nucleotide polymorphisms (SNPs) genotyped information, we used the Genome Complex Trait Analysis (GCTA) method [START_REF] Yang | GCTA: A Tool for Genome-wide Complex Trait Analysis[END_REF].

In the UK Biobank, we considered the genotyped SNPs variants common to the UKBiobank and UKBileve arrays (details at http://www.ukbiobank.ac.uk). In order to compute the kinship matrix of the population, specific SNPs were selected with PLINK v1.9 [START_REF] Purcell | PLINK: A Tool Set for Whole-Genome Association and Population-Based Linkage Analyses[END_REF] with: missing genotype = 5% (70,783 variants excluded), Hardy-Weinberg equilibrium test (hwe) = 10 -6 [START_REF] Yang | GCTA: A Tool for Genome-wide Complex Trait Analysis[END_REF]318), and minor allele frequency (maf) = 1% (102,559). We kept the SNPs in moderate linkage disequilibrium with variation inflation factor 10 within a window of 50 SNPs (92,081 variants excluded). Then, we computed the genetic relationship matrix with GCTA using the 507,515 SNPs left. The amount of phenotypic variance captured by this matrix is estimated using a mixed-effects linear model. As covariates, we included the gender, the genotyping array type, the age at the MRI session and the 10 genetic principal components provided by UK Biobank to account for population stratification. Similarly for the HCP cohort, we considered the genotyped SNPs variants from the Infinium Multi-Ethnic Genotyping array (2,052,643 variants). Using the same filtering procedure with PLINK 705,738 SNPs remained to build the relationship matrix. As covariates, we included the gender, age, a dummy covariate for ethnicity and 5 genetic principal components, that we computed using multidimensional scaling with PLINK. Higher order components accounted for family structures and were thus not included. Note that GCTA usually recommends excluding related individuals in the analysis, however in our case there were not enough individuals remaining. The GCTA estimates in HCP should therefore be taken with caution.

In short, heritability was estimated three times in HCP from pedigree with SOLAR accounting or not for common environment and from the genotyping data with GCTA. As regards the UK Biobank, the heritability is computed only from the genotyping data with GCTA.

RESULTS AND DISCUSSION

We estimated the heritability of cortical features obtained from the standard outputs of Freesurfer. First, we used popular methods in the imaging genetics community to compute these estimates [START_REF] Hofer | Genetic Determinants of Cortical Structure (Thickness, Surface Area and Volumes) among Disease Free Adults in the CHARGE Consortium[END_REF]. However, we noted a discrepancy between the heritability estimates obtained using the HCP/pedigree and UK Biobank/genotyping for both surface area and cortical thickness (Fig. 2). We observe constantly higher estimates in HCP compare to the UK Biobank lead us to believe that there is either a bias in one of the analysis or environmental conditions differ drastically. Note that heritability is not an absolute measure and is specific to a particular population and environment.

Second, we investigated if using the GCTA method on HCP/genotyping, which estimates the relationship matrix based on common variants information, would result in smaller estimates due to the missing information about rare variants difference among individual. Yet, the heritability estimates obtained with HCP/genotyping almost equal to the ones computed with HCP/pedigree. The estimates obtained with GCTA were slightly higher than the ones with SOLAR (Fig 3., 2 nd column). We hypothesized that it might be because the genetic relationship coefficient is more accurate when estimated with genotyping information. For example, siblings shares on average 50% and thus have ½ value in the relationship matrix used by SOLAR. The GCTA kinship estimation allows to estimate if this coefficient is higher (eg: 52%) or lower (eg: 48%) than this a priori coefficient.

Third, when we modelled the shared environment in HCP the heritability estimates remained highly similar to the ones without taking into account the common household (Fig 3

., 3 rd column).

For the surface area: in the HCP the heritability values range from 0.15 to 0.80, and an average of 0.46; in UK Biobank from 0.08 to 0.44, and an average of 0.26 (Fig. 2 a,b). For the average cortical thickness: in the HCP the heritability values range from 0.12 to 0.80, and an average of 0.48; and in UK Biobank from 0.05 to 0.42, and an average of 0.22 (Fig. 2 c,d). Thus, the heritability estimates in HCP tends to be more than 20% higher than the ones in the UK Biobank on average. However, there is strong a correlation between the estimates obtained in the two cohorts (Fig 3,1 st column), which lead us to be confident in the overall regional pattern of heritability.

We know that these brain features are highly dependent on age with a widening of the cortical sulci and thinning of grey matter thickness over time [START_REF] Le Guen | eQTL of KCNK2 regionally influences the brain sulcal widening: evidence from 15,597 UK Biobank participants with neuroimaging data[END_REF]. Thus, there might be an effect that reduces the heritability. We accounted for the age as covariate but its effect on the brain is likely to result from complex interaction with genome [START_REF] Chouinard-Decorte | Heritable changes in regional cortical thickness with age[END_REF]. Additionally, we observed that the phenotypic variance for the cortical thickness increases with age. Thus, because the effect of the genes×age are likely non additive and that the phenotypic variance increases, the heritability values computed in a senior cohort (µ = 62.6 years old) are lower than in a young adult cohort (µ = 29.0 years old).

CONCLUSION

This work emphasizes that heritability estimates computed based on a priori genetic relationship or genotyping material are consistent. The discrepancy between heritability computed from HCP and UK Biobank cohorts might be due to genes×age interactions not taken into account by the heritability, which only measures the additive genetic effects. Age-related gray matter shrinkage in the UK Biobank, not present in the mid-20 HCP cohort, may further reduce the heritability estimates. Finally, it is important to note that the pattern of heritability remains consistent across cohorts.

Figure 1 .

 1 Figure 1. Destrieux brain atlas parcellation.

Figure 2 (

 2 Figure 2(a) Heritability of the surface area estimated with SOLAR in the HCP cohort. (b) Heritability of the surface area estimated with GCTA in the UK Biobank cohort. (c) and (d) equivalent of (a) and (b) respectively for the heritability estimates of the average cortical thickness. All displayed estimates are Bonferroni corrected (p < 1.7•10 -4 ).

Figure 3 .

 3 Figure 3. 1 st line: Surface area Comparison of heritability estimates obtained with SOLAR on HCP and the ones obtained: (i) wtih GCTA on UK Biobank (p < 0.05 corrected), (ii) with GCTA on HCP (p < 0.05 corrected), (iii) with SOLAR on HCP trying to model the shared environment between subjects (p < 0.05 uncorrected). 2 nd line: Cortical thickness average same comparisons as for the 1 st line.

  Data were provided by the HCP, funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. This project has received funding from the European Union's Horizon 2020 Research and Innovation Program under Grant Agreements No. 720270 (HBPSGA1) and No. 785907 (HBPSGA2). It was supported by the CEA DRF Impulsion program (SRAVS project 2017, P.I. Vincent Frouin).


Thierry Daudé
email: thierry.daude@u-cergy.fr

Niky Kamran
email: nkamran@math.mcgill.ca

Francois Nicoleau
email: francois.nicoleau@univ-nantes.fr

On non-uniqueness for the anisotropic Calderón problem with partial data

Keywords: Inverse problems, Anisotropic Calderón problem, Unique continuation principle. 2010 Mathematics Subject Classification. Primaries 81U40, 35P25; Secondary 58J50. Contents

We show that there is non-uniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order ρ < 1 at the end where the measurements are made. More precisely, we construct a toroidal ring (M, g) which is not a warped product manifold, and we show that there exist in the conformal class of g an infinite number of Riemannian metrics g = c 4 g such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric g and do not satisfy the unique continuation principle.

The anisotropic Calderón problem on compact connected Riemannian manifolds with boundary is one of the most famous examples of inverse problems for an elliptic equation. The original problem that Calderón considered was whether one can recover the physical properties of a medium (like its electrical conductivity) by making only voltage and currents measurements at its boundary. This latter problem can be naturally formulated as a problem of geometric analysis in terms of the Dirichlet-to-Neumann map, (DN map in short), for the Laplace-Beltrami operator on Riemannian manifolds with boundary. We refer to the surveys [START_REF] Guillarmou | The Calderón inverse problem in two dimensions[END_REF][START_REF] Kenig | Recent progress in the Calderón problem with partial data[END_REF][START_REF] Salo | The Calderón problem on Riemannian manifolds, Inverse problems and applications: inside out[END_REF][START_REF] Uhlmann | Electrical impedance tomography and Calderón's problem[END_REF] for a description of the current state of the art on the general anisotropic Calderón problem and also to [START_REF] Astala | Calderón's inverse problem for anisotropic conductivities in the plane[END_REF][START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF][START_REF] Dos | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Dos | The Calderón problem in transversally anisotropic geometries[END_REF][START_REF] Guillarmou | Inverse problems for Einstein manifolds[END_REF][START_REF] Guillarmou | Calderón inverse problem with partial data on Riemann surfaces[END_REF][START_REF] Isakov | On uniqueness in the inverse conductivity problem with local data[END_REF][START_REF] Imanuvilov | Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets[END_REF][START_REF] Katchalov | Inverse boundary spectral problems[END_REF][START_REF] Katchalov | Equivalence of time-domain inverse problems and boundary spectral problem[END_REF][START_REF] Kenig | The Calderón problem with partial data on manifolds and applications[END_REF][START_REF] Kenig | The Calderón problem with partial data[END_REF][START_REF] Lassas | The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary[END_REF][START_REF] Lassas | On determining a Riemannian manifold from the Dirichlet-to-Neumann map[END_REF][START_REF] Lee | Determining anisotropic real-analytic conductivities by boundary measuremements[END_REF] for important contributions to the question of uniqueness. On one hand, the uniqueness issue in the Calderón problem is still far from being completely understood in the case of smooth Riemannian manifolds of dimension greater or equal than 3, and remains a major open problem. On the other hand, some counterexamples to uniqueness in the case in which the Dirichlet and Neumann data are measured on disjoint subsets of the boundary were found in our recent papers [START_REF] Daudé | Non uniqueness results in the anisotropic Calderón problem with Dirichlet and Neumann data measured on disjoint sets[END_REF][START_REF] Daudé | On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets[END_REF][START_REF] Daudé | A survey of non-uniqueness results for the anisotropic Calderón problem with disjoint data[END_REF].

The main goal of this paper is to give some non-uniqueness results for the anisotropic Calderón problem with partial data, (i.e in the case where the Dirichlet and Neumann measurements are made on the same open subset Γ of the boundary), for a class of metrics whose coefficients are smooth in the interior of the manifold and Hölder continuous on the subset of the boundary where the measurements are made.

In order to state our main result, let us first recall the definition of the DN map (see for instance [START_REF] Lee | Determining anisotropic real-analytic conductivities by boundary measuremements[END_REF] for the geometric formulation of the DN map for smooth Riemannian manifolds that we use here and [START_REF] Alessandrini | Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities[END_REF] for the formulation of the DN map corresponding to the original Calderón problem in terms of anisotropic conductivities with coefficients which are only L ∞ ). Let M be an n-dimensional smooth compact connected manifold with smooth boundary ∂M . We assume that this manifold M is equipped with a Riemannian metric g = (g ij ) with measurable bounded coefficients satisfying (in local coordinates) the uniform ellipticity condition i,j

g ij (x)ξ i ξ j ≥ c|ξ| 2 for a.e x ∈ M and ξ ∈ R n , (1.1)
where the constant c is strictly positive and where g ij is the inverse of (g ij ).

On the Riemannian manifold (M, g) we consider the Laplace-Beltrami operator ∆ LB , given in local coordinates by

∆ LB = -∆ g = - 1 |g| ∂ i |g|g ij ∂ j .
where |g| = det (g ij ) is the determinant of the metric tensor (g ij ), and where we use the Einstein summation convention. It is well-known that the Laplace-Beltrami operator -∆ g with Dirichlet boundary conditions on ∂M is self-adjoint on L 2 (M, dV ol g ) and that 0 is not an eigenvalue of -∆ g .

Let us consider the Dirichlet problem

-∆ g u = 0, on M, u = ψ, on ∂M. (1.2)
A classical result (see for instance [START_REF] Alessandrini | Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Salo | The Calderón problem on Riemannian manifolds, Inverse problems and applications: inside out[END_REF][START_REF] Taylor | Partial Differential Equations, I. Basic theory[END_REF]) ensures that for any ψ ∈ H 1/2 (∂M ), there exists a unique weak solution u ∈ H 1 (M ) of the Dirichlet problem (1.2). We recall that u is a weak solution of

(1.2) if M du, dw g dV ol g = 0 for all w ∈ H 1 0 (M ), (1.3)
where du, dw g is the pointwise scalar product of the one-forms du, dv on M induced by g and given in local coordinates by du, dw g = g ij ∂ i u ∂ j w, and if the trace of u on the boundary is equal to ψ. So, we can define the DN map as an operator Λ g from H 1/2 (∂M ) to H -1/2 (∂M ) by

Λ g ψ|φ = M du, dv g dV ol g , for all ψ, φ ∈ H 1/2 (∂M ), (1.4)
where u is the unique weak solution of the Dirichlet problem (1.2), v is any element of H 1 (M ) such that v |∂M = φ, and •|• is the standard L 2 duality pairing between H 1/2 (∂M ) and its dual. In the case where the metric g and the function ψ are smooth, this definition coincides with the usual one

Λ g (ψ) = (∂ ν u) |∂M , (1.5)
where (∂ ν u) |∂M is the normal derivative of u with respect to the unit outer normal ν on ∂M .

As mentioned earlier, we are interested in the case in which the Dirichlet and Neumann data are measured on the same non-empty open subset Γ of the boundary ∂M . Let us introduce the subspace of H 1/2 (∂M ) defined by:

H 1/2 co (Γ) = {f ∈ H 1/2 (∂M ) | suppf ⊂ Γ}. (1.6)
The partial DN map is defined in a weak formulation as the operator Λ g,Γ such that

Λ g,Γ (ψ)|φ = M du, dv g dV ol g , for all ψ, φ ∈ H 1/2 co (Γ), (1.7)
where u is the unique weak solution of the Dirichlet problem (1.2), and where v is any element of H 1 (M ) such that v |∂M = φ. As previously, for smooth metrics g and smooth boundary data ψ, the partial DN map is simply given by:

Λ g,Γ (ψ) = (∂ ν u) |Γ . (1.8)
In its simplest form, the anisotropic Calderón problem with partial data can be stated as follows:

If a pair of partial DN maps Λ g1,Γ and Λ g2,Γ coincide, is it true that the metrics g 1 and g 2 are the same? Because of several natural and geometric gauge invariances, the answer to the question stated above turns out to be negative. These lead to refined formulations of the Calderón problem that we shall present shortly, and that constitute the actual statement of this inverse problem. Indeed, it results from the definition (1.2) -(1.7) that the partial DN map Λ g,Γ is invariant when the metric g is pulled back by any diffeomorphism of M whose restriction to Γ is the identity, i.e. ∀φ ∈ Diff(M ) such that φ |Γ = Id, Λ φ * g,Γ = Λ g,Γ .

(1.9)

In dimension two, there is another gauge invariance of the DN map due to the conformal invariance of the Laplace-Beltrami operator. More precisely, recall that in dimension 2

∆ cg = 1 c ∆ g ,
for any smooth function c > 0. Therefore, we have in dimension

2 ∀c ∈ C ∞ (M ), c > 0 and c |Γ = 1, Λ cg,Γ = Λ g,Γ , (1.10)
since the unit outer normal vectors ν cg and ν g are identical on Γ.

It follows that the appropriate question (called the anisotropic Calderón conjecture with partial data) to address is the following:

Let M be a n-dimensional smooth compact connected manifold with smooth boundary ∂M . Let g, g denote Riemannian metrics on M with measurable bounded coefficients and let Γ be an open subset of ∂M . Suppose that Λ g,Γ = Λ g,Γ .

Does it follow that g = g up to the gauge invariance (1.9) if n ≥ 3, and up to the gauge invariances (1.9)-(1.10) in dimension n = 2.

One may also consider a simpler inverse problem by assuming that the Riemannian manifolds (M, g) and (M, g) belong to the same conformal class, that is g = cg for some positive smooth function c on M . In that case, the anisotropic Calderón problem amounts to the following statement Let M be a n-dimensional smooth compact connected manifold with smooth boundary ∂M . Let g denotes a Riemannian metric on M with measurable bounded coefficients and let Γ be an open subset of ∂M . Let c be a smooth positive function on M . If

Λ cg,Γ = Λ g,Γ ,
then is it true that c = 1, on M ? (1.11)

In fact, according to (1.9), the assumption Λ cg,Γ = Λ g,Γ should entail the question: does there exist a diffeomorphism φ : M -→ M with φ | Γ = Id such that φ * g = cg?

(1.12)

But, as was proved by Lionheart [START_REF] Lionheart | Conformal uniqueness results in anisotropic electrical impedance imaging[END_REF] for smooth metrics, any diffeomorphism φ : M -→ M which satisfies φ * g = cg and φ |Γ = Id for a non-empty open subset Γ of ∂M is the identity on the whole manifold M . Thus, the condition (1.12) may therefore be replaced by the condition (1.11).

Finally, there exists a last version of the anisotropic Calderón problem with partial data on Γ involving an external potential. Consider the Dirichlet problem for the Schrödinger equation on

(M, g) with potential V ∈ L ∞ (M ) (-∆ g + V )u = 0, on M, u = ψ, on ∂M, (1.13)
where ψ ∈ H 1/2 co (Γ). We assume that 0 does not belong to the Dirichlet spectrum of -∆ g + V . Then, there exists a unique weak solution u ∈ H 1 (M ) of (1.13) (see for instance [START_REF] Dos | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Salo | The Calderón problem on Riemannian manifolds, Inverse problems and applications: inside out[END_REF]). As previously, this allows us to define in the same way (i.e. in a weak sense) the partial Dirichlet-to-Neumann map Λ g,V,Γ (ψ) for all ψ ∈ H 1/2 co (Γ). For smooth Riemannian metrics g, it is well-known that there is a close connection between the anisotropic Calderón problem for Schrödinger operators and the anisotropic Calderón problem within the conformal class of a fixed metric g. It is based on the transformation law for the Laplace-Beltrami operator on a n-dimensional Riemannian manifold (M, g) under conformal changes of the metric, that is

-∆ c 4 g u = c -(n+2) (-∆ g + q g,c ) c n-2 u , (1.14)
where the potential q g,c is given by

q g,c = c -n+2 ∆ g c n-2 . (1.15)
As a by product, we get for instance the following result for smooth metrics (see [START_REF] Daudé | On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets[END_REF], Proposition 1.1):

Proposition 1.1. Let Γ be any fixed open set of ∂M . Assume that c is a smooth positive function on M such that c = 1 on Γ and ∂ ν c = 0 on Γ. Then Λ c 4 g,Γ = Λ g,qg,c,Γ . (1.16)
In particular, if the potential q g,c = 0, i.e. if the conformal factor c satisfies additionally

∆ g c n-2 = 0, on M, (1.17)
we get immediately

Λ c 4 g,Γ = Λ g,Γ .
Note that in dimension n = 2, (1.17) is automatically satisfied and (1.16) corresponds simply to the gauge invariance (1.12). In dimension n ≥ 3 and for smooth metrics g, the unique continuation principle implies that c = 1 on M (remember that not only (1.17) is satisfied, but c must be identically 1 on Γ and ∂ ν c must be 0 on Γ). Thus, if we want to use the result in Proposition 1.1 to obtain a counterexample to uniqueness for the anisotropic Calderón problem with partial data, we need to construct a metric g in such a way the Laplace-Beltrami operator -∆ g does not satisfy the unique continuation principle.

We recall that, in dimension n ≥ 3, the unique continuation principle holds for a uniformly elliptic operator on a domain Ω if the coefficients of the principal part of this operator are locally Lipschitz continuous, whereas in dimension n = 2, the unique continuation principle holds if the coefficients of the principal part are L ∞ (see for instance [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, III, Grundlehren der matematischen Wissenschaften[END_REF][START_REF] Hörmander | Uniqueness theorems for second order elliptic differential equations[END_REF][START_REF] Tataru | Unique continuation for pde's[END_REF]). Nevertheless, in dimension n = 3, if the coefficients of the principal part are only Hölder continuous, there exist examples of nonunique continuation. The first such example was given in 1963 by Pliś [START_REF] Pliś | On non-uniqueness in Cauchy problem for an elliptic second order differential equation[END_REF], and later in 1972, a sharper counterexample was found by Miller [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF] for an elliptic equation in divergence form. This divergence form is very well adapted with our Riemannian setting. So, the main and basic idea of our paper is to construct a metric g on a suitable manifold M such that the Laplacian ∆ g is nothing but Miller's elliptic operator and the conformal factor c is very close to Miller's solution.

But, before giving this construction, our first task is to slightly extend Proposition 1.1 for metrics g having coefficients in L ∞ (M ) since, in this case, the potential q g,c only has a distributional sense. In other words, we have to write Proposition 1.1 (with q g,c = 0) in a weak sense. To do this, we remark that, for a smooth metric g and for a smooth conformal factor c, the conditions

∂ ν c = 0 on Γ and ∆ g c n-2 = 0 on M are equivalent to M d(c n-2 ), dw g dV ol g = 0 , ∀w ∈ H 1 (M ) such that supp w |∂M ⊂ Γ, (1.18)
thanks to the Green's formula:

M ∆ g (c n-2 )w dV ol g + M d(c n-2 ), dw g dV ol g = ∂M ∂ ν c n-2 w dσ g , (1.19)
where dV ol g denotes the Riemannian volume element and dσ g denotes the volume element induced by g on ∂M .

Now, we can state the following extension of Proposition 1.1 for metrics with bounded measurable coefficients which is one of the main arguments of our counterexamples for the anisotropic Calderón problem with partial data. Proposition 1.2. Let Γ be any fixed open set of ∂M . Assume that c is a smooth positive function on M such that c = 1 on Γ and such that

M d(c n-2 ), dw g dV ol g = 0 , ∀w ∈ H 1 (M ) such that supp w |∂M ⊂ Γ.
(1.20)

Then, Λ c 4 g,Γ = Λ g,Γ .
(1.21)

Proof. For any ψ, φ ∈ C ∞ 0 (Γ), the partial DN map Λ c 4 g,Γ is given by the relation:

Λ c 4 g,Γ ψ | φ >= M du, dw c 4 g dV ol c 4 g , (1.22)
where u ∈ H 1 (M ) is the unique weak solution of (1.2) associated to the metric c 4 g with u |∂M = ψ and w ∈ C ∞ (M ) is any extension of φ. Note that the existence of such an extension is given for instance in the monograph [START_REF] Lee | Introduction to Smooth Manifolds[END_REF], Corollary 6.27, together with the Tietze's extension theorem. As a consequence, the function c n-2 uw ∈ H 1 (M ) and its trace on the boundary has its support in Γ. Now, a straightforward algebraic calculation gives:

M du, dw c 4 g dV ol c 4 g = M d(c n-2 u), d(c n-2 w) g dV ol g - M d(c n-2 ), d(c n-2 uw) g dV ol g . (1.23)
Thus thanks to the hypothesis (1.18), we get:

Λ c 4 g,Γ ψ | φ = M d(c n-2 u), d(c n-2 w) g dV ol g . (1.24)
Let us now prove that v = c n-2 u is a weak solution of (1.2). Indeed, since u is the unique weak solution of (1.2) associated to the metric c 4 g, we have for any ϕ ∈ C ∞ 0 (M ), It follows that v = c n-2 u is a weak solution of (1.2) for the metric g which satisfies v |∂M = ψ since c = 1 on Γ.

Then, using the definition of the partial DN map Λ g,Γ and (1.24) again, we get immediately

Λ c 4 g,Γ ψ | φ = Λ g,Γ ψ | φ for all ψ, φ ∈ C ∞ 0 (Γ). (1.27)
We conclude the proof by a standard density argument.

Statement of the main result

Let us introduce first some notations. We consider the n-dimensional manifold

M = [0, 1] × T n-1 ,
where T n-1 denotes the (n -1)-dimensional torus, (n ≥ 3). This manifold has the topology of a cylinder.

Using the standard toroidal coordinates, we can also interpret M as a toroidal ring (see [START_REF] Giannotti | A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term[END_REF], Remark 2.5). Note that the boundary of M is disconnected and consists in the disjoint union of two copies of T n-1 , (which we call ends in this paper), more precisely

∂M = Γ 0 ∪ Γ 1 , Γ 0 = {0} × T n-1 , Γ 1 = {1} × T n-1 .
Our main result is the following:

Theorem 1.1. There exists a Riemannian metric g on M whose coefficients are smooth in [0, 1) × T n-1 and Hölder continuous of order ρ < 1 on the end Γ 1 , and there exist an infinite number of smooth positive conformal factors c which are not identical to 1 on M , such that the following partial DN maps on Γ 1 are identical: Λ c 4 g,Γ1 = Λ g,Γ1 .

(1.28)

As we have said earlier, the proof of this theorem is rather simple and relies on Miller's famous counterexample to unique continuation for a uniformly elliptic equation in divergence form in dimension 3 (see [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF]). Assume for a moment that the dimension of our manifold M is 3 and let us summarize the strategy of the proof. We consider first a metric g on M such that ∆ g is precisely the uniformly elliptic operator from Miller's construction. Note in passing that the elliptic operator from [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF] naturally lives on the cylinder M = [0, 1] × T 2 as was noticed by Gianotti in [START_REF] Giannotti | A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term[END_REF]. Then, using Miller's solution of this elliptic PDE, we shall construct in section 2 an infinite family of smooth conformal factors c satisfying the assumptions of Proposition 1.2, i.e c = 1 on Γ 1 , and ∆ g c n-2 = 0, ∂ ν c = 0 on Γ 1 in the following weak sense:

M d(c n-2 ), dw g dV ol g = 0 , ∀w ∈ H 1 (M ) such that supp w |∂M ⊂ Γ 1 .
(1.29)

This leads automatically to counterexamples to uniqueness for the Calderón problem with partial data in dimension 3 since the metrics g and c 4 g are not isometric (see the proof of Theorem 1.1). The proof in the case of higher dimensions is similar.

Remark 1.1. It is important to stress that, even though two of the coefficients (namely A 1 (t) and A 3 (t)) of the elliptic operator ∆ g are only Hölder continuous functions, ∆ g c n-2 is classically well-defined since these two functions are not differentiated with respect to t in the expression of ∆ g , (see [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF], Theorem 1, or Proposition 2.1 in this paper). In other words, the equation ∆ g c n-2 = 0 on M can be also understood in the classical sense.

Remark 1.2. All the derivatives of the conformal factors c at the end Γ 1 are equal to zero as one would expect from boundary determination results (see [START_REF] Kang | Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operators[END_REF]).

Remark 1.3. We emphasize that this theorem is of a local nature. We cannot obtain with the same approach a counterexample for the anisotropic Calderón problem with global data, i.e when Γ = ∂M . Indeed, if the smooth conformal factor satisfies c = 1 on ∂M and

M < d(c n-2 ), dw > g dV ol g = 0 , ∀w ∈ H 1 (M ), (1.30)
then, choosing w = c n-2 in (1.30), we obtain immediately that c is identical to 1 on M . An alternative interpretation is to say that 0 is not a Dirichlet eigenvalue of the Laplace-Beltrami operator ∆ g .

A brief history of known results on the anisotropic Calderón problem

In this last part of the Introduction, we give a brief and non-exhaustive survey of some of the most important known results on the anisotropic Calderón conjecture. In dimension 2, the anisotropic Calderón conjecture for global and partial data has been been settled positively for compact connected Riemannian surfaces in [START_REF] Lassas | On determining a Riemannian manifold from the Dirichlet-to-Neumann map[END_REF][START_REF] Lee | Determining anisotropic real-analytic conductivities by boundary measuremements[END_REF]. We also refer to [START_REF] Astala | Calderón's inverse problem for anisotropic conductivities in the plane[END_REF][START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF] for similar results for global and partial data in the case of anisotropic conductivities which are only L ∞ on bounded domains of R 2 . In dimension n ≥ 3, if the Riemannian manifold is real analytic, compact and connected, with real analytic boundary, a positive answer for global (i.e when Γ = ∂M ), and partial data has been given in [START_REF] Lee | Determining anisotropic real-analytic conductivities by boundary measuremements[END_REF][START_REF] Lassas | On determining a Riemannian manifold from the Dirichlet-to-Neumann map[END_REF][START_REF] Lassas | The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary[END_REF]. Similarly, the global anisotropic Calderón problem has been answered positively for compact connected Einstein manifolds with boundary in [START_REF] Guillarmou | Inverse problems for Einstein manifolds[END_REF].

If the background metrics are not assumed to be analytic, the general anisotropic Calderón problem in dimension n ≥ 3 is still a difficult open problem, whether one is dealing with the case of full or partial data. Nevertheless, some important results have recently appeared in [START_REF] Dos | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Dos | The Calderón problem in transversally anisotropic geometries[END_REF][START_REF] Kenig | The Calderón problem with partial data on manifolds and applications[END_REF], for special classes of smooth compact connected admissible Riemannian manifolds with boundary. By definition, admissible manifolds (M, g) are conformally transversally anisotropic,

M ⊂⊂ R × M 0 , g = c(e ⊕ g 0 ),
where (M 0 , g 0 ) is an n-1 dimensional smooth compact connected Riemannian manifold with boundary, e is the Euclidean metric on the real line and c is a smooth strictly positive function in the cylinder R × M 0 . It has been shown in [START_REF] Dos | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Dos | The Calderón problem in transversally anisotropic geometries[END_REF] that for admissible manifolds and under some geometric assumptions on the transverse manifolds M 0 (see for instance [START_REF] Dos | The Calderón problem in transversally anisotropic geometries[END_REF] for a precise statement), the conformal factor c is uniquely determined from the knowledge of the DN map. These results have been further extended to the case of partial data in [START_REF] Kenig | The Calderón problem with partial data on manifolds and applications[END_REF]. We also refer to [START_REF] Guillarmou | Calderón inverse problem with partial data on Riemann surfaces[END_REF][START_REF] Isakov | On uniqueness in the inverse conductivity problem with local data[END_REF][START_REF] Imanuvilov | The Calderón problem with partial data in two dimensions[END_REF] for additional results in the case of local data and to the surveys [START_REF] Guillarmou | The Calderón inverse problem in two dimensions[END_REF][START_REF] Kenig | Recent progress in the Calderón problem with partial data[END_REF] for further references.

Let us also mention several papers dealing with the Calderón problem for more singular metrics or conductivities in dimension n ≥ 3. Haberman and Tataru [START_REF] Haberman | Uniqueness in Calderón's problem with Lipschitz conductivities[END_REF] showed uniqueness in the global Calderón problem for uniformly elliptic isotropic conductivities that are Lipschitz and close to the identity. The latter condition was relaxed by Caro and Rogers in [START_REF] Caro | Global uniqueness for the Calderón problem with Lipschitz conductivities[END_REF]. In dimensions 3 and 4, these results were slightly improved by Haberman in [START_REF] Haberman | Uniqueness in Calderón's problem for conductivities with unbounded gradient[END_REF] to the case of conductivities that belong to W 1,n . Related to the partial Calderón problem, Krupchyk and Uhlmann in [START_REF] Krupchyk | The Calderón problem with partial data for conductivities with 3/2 derivatives[END_REF] proved that an isotropic conductivity with -roughly speaking - 3 2 derivatives in the L 2 sense is uniquely determined by a DN map measured on possibly very small subset of the boundary.

Finally, we conclude this introduction mentioning the series of papers on invisibility by Greenleaf, Kurylev, Lassas and Uhlmann (see [START_REF] Greenleaf | On non-uniqueness for Calderón's inverse problem[END_REF] for the original paper, [START_REF] Astala | The borderlines of invisibility and visibility in Calderón's inverse problem[END_REF] for an extension of this work in dimension 2 and [START_REF] Greenleaf | Invisibility and inverse problems[END_REF][START_REF] Uhlmann | Inverse problems: seeing the unseen[END_REF] for thorough surveys in this field). In these works, some counterexamples to uniqueness to the global Calderón problem were described. These counterexamples are obtained for a class of metrics that are highly singular at a given closed hypersurface lying within the manifold in the sense that the metric degenerates or blows up at this hypersurface. This is a situation in sharp contrast to the one obtained in the present work in which the metrics remain positive definite and Hölder continuous everywhere in the manifold, (and even smooth in the interior). Coming back to invisibility, we also refer to our recent paper [START_REF] Daudé | The anisotropic Calderón problem for singular metric of warped product type : the borderline between uniqueness and invisibility[END_REF] where similar non-uniqueness results were obtained for singular warped product metrics on the same class of manifolds M as the ones used in this paper.

Counterexamples to uniqueness

Miller's counterexample to unique continuation principle

In this section, we recall the remarkable counterexamples obtained by Miller [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF] to the unique continuation principle, counterexamples that were slightly improved later by Mandache [START_REF] Mandache | On a counterexample concerning unique continuation for elliptic equations in divergence form[END_REF]. We say that a partial differential equation P (x, D)u = 0 on a domain Ω possesses the unique continuation property if the equality u = 0 in some ball within Ω implies the equality u = 0 on Ω.

In dimension n ≥ 3, the unique continuation property holds for a uniformly elliptic operator on a domain Ω if the coefficients of the principal part of this operator are locally Lipschitz continuous (see for instance [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators, III, Grundlehren der matematischen Wissenschaften[END_REF][START_REF] Hörmander | Uniqueness theorems for second order elliptic differential equations[END_REF][START_REF] Tataru | Unique continuation for pde's[END_REF]). If the coefficients are only Hölder continuous, then there exist examples of nonunique continuation. The first one was given in 1963 by Pliś [START_REF] Pliś | On non-uniqueness in Cauchy problem for an elliptic second order differential equation[END_REF]. He considered a uniformly elliptic equation on a domain of R 3 having the form:

3 i,j=1 a ij (x) ∂ 2 ij u + 3 i=1 b i (x) ∂ i u + c(x)u = 0, (2.31)
where the coefficients of this equation are Hölder continuous with order less than 1. The coefficient of zero-order term c(x) has no constant sign and might explain the existence of this counterexample. In 1972, a sharper counterexample (without zero-order term) was found by Miller [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF]. He constructed a smooth solution u(t, x, y) of a uniformly elliptic equation in divergence form:

div (A ∇u) = 0, (2.32)
where the (3 × 3) symmetric matrix A is given by

A =   1 0 0 0 1 + a 1 (t, x, y) + A 1 (t) a 2 (t, x, y) 0 a 2 (t, x, y) 1 + a 3 (t, x, y) + A 3 (t)   . (2.33)
This matrix A has its eigenvalues in [α, α -1 ] with elliptic constant α ∈]0, 1[. More precisely, Miller proved the following result:

Theorem 2.1 (Miller [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF]). There exists an example of nonunique continuation on the half-space E = [0, +∞[×R 2 for a uniformly elliptic equation 4. All functions u, a 1 , a 2 , a 3 are periodic in x and y with period 2π.

∂ 2 t u + ∂ x ((1 + a 1 + A 1 )∂ x u) + ∂ x (a 2 ∂ y u) + ∂ y (a 2 ∂ x u) + ∂ y ((1 + a 3 + A 3 )∂ y u) = 0 in E. ( 2
5. Although the coefficient matrix A is only Hölder continuous at t = T , u is a classical (as well as weak) solution of (2.34) on E.

We emphasize that this theorem can be improved as follows: modifying slightly Miller's initial proof, the coefficients A 1 (t), A 3 (t) can be actually constructed in such a way that they are Hölder continuous functions of fixed order ρ < 1 , (see [START_REF] Miller | Non-unique continuation for uniformy parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients[END_REF], Remarks p. 115), as it occurs in Mandache's paper [START_REF] Mandache | On a counterexample concerning unique continuation for elliptic equations in divergence form[END_REF]. But, we prefer to use Miller's work rather Mandache's paper [START_REF] Mandache | On a counterexample concerning unique continuation for elliptic equations in divergence form[END_REF] since it is not clear for us that the coefficients of the elliptic operator constructed by Mandache are smooth on [0, T [. It is also important to say again that this function u is a classical solution of the elliptic equation (2.34) since we do not differentiate the Hölder functions A 1 (t) and A 3 (t) with respect to t in the elliptic equation (2.34).

Moreover, since the solution u(t, x, y) is periodic in (x, y) with period 2π, as was observed by C. Giannotti in [START_REF] Giannotti | A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term[END_REF], Miller's solution can be considered as a solution to an elliptic equation on the toroidal ring M = [0, T ] × T 2 where T 2 is the usual 2-dimensional torus.

Note that, in the following section, and in order to simplify the notation, we assume (without loss of generality) that T = 1.

Construction of the counterexamples on a toroidal ring of R 3

In this section, we consider a Riemannian manifold (M, g), which has the topology of a cylinder M = [0, 1] × T 2 . We denote (t, x, y) a global coordinate system on M . The manifold M can be interpreted as a toroidal ring (see [START_REF] Giannotti | A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term[END_REF], Remark 2.5). We equip this manifold with the following Riemannian metric:

g = Ddt 2 + (1 + a 3 + A 3 )dx 2 -2a 2 dxdy + (1 + a 1 + A 1 )dy 2 , (2.35)
where the coefficients a 1 , a 2 , a 3 , A 1 , A 3 are given by Theorem 2.1, and

D = det A = (1 + A 1 (t) + a 1 (t, x, y))(1 + A 3 (t) + a 3 (t, x, y)) -a 2 2 (t, x, y). (2.36)
Clearly, (M, g) is not a warped-product manifold and we have |g| (g -1 ) = A. Note that this metric is well-defined on M since all the coefficients are periodic in the variables x, y with period 2π. We recall that the boundary ∂M of M is disconnected and consists in the disjoint union of two copies of T 2 , precisely

∂M = Γ 0 ∪ Γ 1 , Γ 0 = {0} × T 2 , Γ 1 = {1} × T 2 .
We emphasize that our metric g is smooth everywhere inside the manifold (precisely on M \ Γ 1 ) and Hölder continuous of order ρ < 1 on the end Γ 1 . Thanks to Theorem 2.1 and as it was observed before, Miller's solution is a classical harmonic function for the Laplace Beltrami operator ∆ g , i.e it satisfies the Laplace equation in the classical sense ∆ g u = 0 in M.

(2.37)

Moreover, since the solution u(t, x, y) is smooth on E and is vanishing for t ≥ 1, all the derivatives of u are also equal to zero at t = 1. In particular, one has:

u |Γ1 = 0 , ∂ ν u |Γ1 = 0, (2.38)
where ∂ ν is the normal derivative on Γ 1 with respect to the unit outer normal vector ν on ∂M . Now, let us define our infinite family of conformal factors c(t, x, y). We set

c ǫ (t, x, y) = 1 + ǫ u(t, x, y), (2.39)
where ǫ > 0 is chosen sufficiently small to ensure that c ǫ (t, x, y) ≥ 1 2 on the compact manifold M . Proof of Theorem 1.1 in dimension 3. The conformal factors c ǫ are smooth on M . Moreover, thanks to Theorem 2.1, they are not identically 1 on M . Using (2.37) and (2.38), these conformal factors c satisfy obviously and in the classical sense:

∆ g c ǫ = 0 in M , c ǫ|Γ1 = 1 , ∂ ν c ǫ|Γ1 = 0.
(2.40)

Then, thanks to Green's formula, these conformal factors c satisfy the assumptions of Proposition 1.2, and one obtains thus immediately: Λ c 4 ǫ g,Γ1 = Λ g,Γ1 .

(2.41)

Now assume that, for all ǫ > 0, there exists a diffeomorphism (depending on ǫ) φ : M -→ M such that φ |Γ1 = Id and φ * g = c 4 ǫ g. In particular, since φ is a diffeomorphism, we see that

V ol g (M ) = V ol φ * g (M ) = V ol c 4 ǫ g (M ). Hence we must have M [(1 + ǫu) 6 -1] |g| dx = 0 for all ǫ > 0. (2.42)
The term of order 2 of this polynomial in the variable ǫ must be equal to 0, i.e

M u 2 |g| dx = 0, (2.43)
which is not possible since u is not identically equal to zero. Thus, there exists an infinite number of conformal factors c ǫ such that the metrics g and c 4 ǫ g are not isometric and we see that we have found counterexamples to uniqueness for the partial anisotropic Calderón problem in dimension 3.

Generalization in the n-dimensional case

The above construction can be generalized in a straightforward way to the n-dimensional case with n ≥ 3. Indeed, let us consider the manifold M = [0, 1] × T n-1 where T n-1 stands for the (n -1) dimensional torus. We denote (t, x 1 , x 2 , ..., x n-1 ) a global coordinate system on M . As in the previous section, we introduce the coefficient

D(t, x 1 , x 2 ) = (1 + A 1 (t) + a 1 (t, x 1 , x 2 ))(1 + A 3 (t) + a 3 (t, x 1 , x 2 )) -a 2 2 (t, x 1 , x 2 ), (2.44)
where the coefficients a 1 , a 2 , a 3 , A 1 , A 3 are still given by Theorem 2.1 with the identifications x 1 = x and x 2 = y. Now, we equip this manifold M with the Riemannian metric

g = D 1 n-2 dt 2 + D -1 (1 + A 3 + a 3 )dx 2 1 -2a 2 dx 1 dx 2 + (1 + A 1 + a 1 )dx 2 2 + n-1 i=3 dx 2 i .
(2.45)

As before, the metric g is smooth inside the manifold M and only Hölder continuous at the end Γ 1 = {1} × T n-1 . Then, we define the conformal factors (which will not depend on the variables x i for i ≥ 3) by c ǫ (t, x 1 , ..., x n-1 ) = (1 + ǫ u(t, x 1 , x 2 ))

1 n-2 , (2.46)
where u(t, x 1 , x 2 ) is Miller's solution given in Theorem 2.1 and ǫ > 0 is small enough to ensure that c ǫ > 0 on M . Using the same arguments as in the previous section, one has:

Λ c 4 ǫ g,Γ1 = Λ g,Γ1 , (2.47)
which implies that there is non-uniqueness for the partial anisotropic Calderón problem in dimension n ≥ 3.

Remark 2.1. In the previous non-uniqueness results for the anisotropic Calderón problem with partial data, we considered smooth compact connected cylindrical manifolds equipped with a metric whose coefficients are only Hölder continuous, and having two ends. If we remove the assumption of smoothness for the manifold, then we can allow a connected boundary for M and obtain probably counterexamples to uniqueness in the partial Calderón problem. More precisely, let us consider the product manifold M = [0, 1] × Ω where Ω is any connected bounded open set of R n-1 with smooth boundary. Note that the boundary of M is now connected and given by: ∂M = {0} × Ω ∪ {1} × Ω ∪ ((0, 1) × ∂Ω) .

Clearly, we lose the smoothness of the manifold since M has corners. Nevertheless, if the Dirichlet and the Neumann data are measured on Γ = {1} × Ω, one can probably use the previous constructions to get counterexamples in this new setting.

1 Introduction 1 . 1 1 Introduction 1 . 1

 111111 The anisotropic Calderón problem with partial data . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A brief history of known results on the anisotropic Calderón problem . . . . . . . . . . . . The anisotropic Calderón problem with partial data

M du, dϕ c 4 g

 4 dV ol c 4 g = 0. (1.25) Using again the relation (1.23) (with w replaced by ϕ) and (1.18), we get for any ϕ ∈ C ∞ 0 (M ), M d(c n-2 u), d(c n-2 ϕ) g dV ol g = 0 (1.26)

  .34) 1. The classical solution u(t, x, y) is C ∞ on E, identically zero for t ≥ T > 0, but not identically zero in any open subset of [0, T [×R 2 . 2. The coefficients a 1 (t, x, y), a 2 (t, x, y), a 3 (t, x, y) are C ∞ on E and are identically zero for t ≥ T . 3. The coefficients A 1 (t), A 3 (t) are Hölder continuous of order 1 6 on [0, ∞[, C ∞ on [0, T [, and identically zero for t ≥ T .

* Research supported by the JCJC French National Research Projects Horizons, No. ANR-ANR-16-CE40-0012-01 † Research supported by NSERC grant RGPIN 105490-2011 ‡ Research supported by the French National Research Project GDR Dynqua 1


Giovanni Azzolina

Eric Collet

Céline Mariette

Marco Cammarata

Elzbieta Trzop

Mathias Sander

Matteo Levantino

Kosuke N Nakagawa

Hiroko Tokoro

Shin-Ichi Ohkoshi

] Elzbietatrzop

Roman Bertoni

Single laser shot photoinduced phase transition of rubidium manganese hexacyanoferrate investigated by X-ray diffraction

Keywords: Charge transfer, Magnetic properties, X-ray diffraction, Photoinduced phase transition

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Introduction

The possibility to tune the physical and chemical properties of materials with light irradiation opens the doors to many different technological applications like optical sensors or memory devices as examples. Optical control of materials was a key scientific challenge for many years especially down to ultrafast timescales. [START_REF] Basov | [END_REF] Molecular-based materials show various types of functionalities that may switch under photoexcitation, [2] like ferroelectric or insulator-metal charge-transfer systems, [3] photochromic materials [4] or spin-crossover complexes. [5] Prussian Blue Analogues (PBA) consisting in cyanide-bridged bimetallic assemblies [6] are also very promising materials undergoing photoinduced phase transitions (PIPT). Understanding the underlying physical mechanisms driving PIPT is mandatory in order to harvest the peculiar properties of these materials. Various ultrafast studies were performed in cyanidebridged bimetallic assemblies, using time-resolved optical, [7] Raman, [8] X-ray diffraction [9] and X-ray spectroscopy. [10] Figure 1. Thermal transition probed through the M.T product (M = molar magnetic susceptibility and T = temperature, top) and the HT phase fraction (bottom) obtained by using Rietveld refinement of full X-ray diffraction pattern.

Here we investigate PIPT in rubidium manganese hexacyanoferrate systems (RbMnFe) with general composition RbxMn[Fe(CN)6](x+2)/3•zH2O), by using X-ray diffraction. In these systems, the charge transfer process is accompanied by a structural Jahn-Teller distortion between a high temperature (HT) cubic ( 𝐹 4 ̅3𝑚 ) phase Fe III (S=1/2)-CN-Mn II (S=5/2) and a low temperature (LT) tetragonal (𝐼 4 ̅𝑚2) phase Fe II (S=0)-CN-Mn III (S=2). This transition is associated with a wide thermal hysteresis loop that can be tuned by changing slightly the chemical composition. [6f, 11] Continuous visible light excitation can drive reversible and magnetic PIPT between these phases at low temperature, [12] and a single laser pulse excitation can also be used for controlling the magnetism. [13] Single-shot laser switching was also observed in the hysteresis domain of bistability: a ns laser pulse (532 nm) can convert the LT phase to the HT phase above a threshold excitation density, and full conversion can be achieved, as observed by infrared spectroscopy. [14] The infrared probe is very sensitive to the electronic and structural reorganizations accompanying the change of electronic state, through the frequency shift and splitting of the CN stretching mode. However, this local probe cannot discriminate the nature of the macroscopic conversion mechanism being either a homogeneous process or a phase separation process.

Here we present an X-ray diffraction study of the photoinduced phase transition of the Rb0.94Mn[Fe(CN)6]0.98•2.5H2O system, induced by a single laser shot inside the thermal hysteresis. We take advantage of the tight relationship between electronic and structural properties for studying the PIPT process and for showing that a long-range ordered phase forms. Microcrystals of the Rb0.94Mn[Fe(CN)6]0.98•2.5H2O system, are particularly interesting for applications, because the bistability region extends above room temperature, as shown by the M.T product vs T plot in the upper part of Fig. 1 (M being the molar magnetic susceptibility and T the temperature). [15] The M.T value ( 4.75 cm 3 .K.mol -1 at 300 K) of the HT phase Fe III (S=1/2)Mn II (S=5/2) decreases around T1/2↓ 185 K on cooling, as the LT phase Fe II (S=0)Mn III (S=2) forms. Conversely, on warming the sample from the LT phase, the M.T value increases around T1/2↑ 300 K, and reaches the one of the HT phase. The width of the thermal hysteresis loop is  115 K. Fig. 2 shows a schematic of the experimental setup at the ID09 beamline of the ESRF synchrotron (experimental section). X-ray pulses of 14.8 keV X-rays were used to monitor the diffraction signal of a thin film of RbMnFe microcrystals as function of temperature. We also measured the diffracted signal before and after single laser shots (560 nm, pulse duration 5 ns) at several fluencies. At this given wavelength, photo-excitation of the material induces LT to HT phase switching. In order to maximize the volume of film probed, a grazing incidence geometry is used. The X-ray data are collected on a 2D detector. The diffraction patterns, showing the diffracted X-ray intensity as a function of the scattering wave vector q (Å -1 ) are obtained by azimuthal integration of the signal diffracted on the detector. Because of the diffraction geometry, the peaks are relatively broad compared to previous studies, [15] due to the  30 µm height of the X-ray beam.

Results and Discussion

For checking the quality of the film and the sensitivity of the experimental set-up for tracking PIPT, we studied first the temperature dependence of the X-ray diffraction signal for monitoring the thermal phase transition. The diffraction patterns measured at 300 K and 140 K are shown respectively in the top and bottom part of Fig. 3. At 300 K the sample is fully in the HT phase, whereas at 140 K it is fully in the LT phase. The HT and LT diffraction signals obtained by Rietveld refinement are shown in red and blue along with the peak indices. The X-ray diffraction data also reveal a contribution to the signal from a known impurity [15] in the solution used (Rb2Mn II [Fe(CN)6]•3.5H2O, see experimental section), plotted in green. On cooling, during the transition from HT to LT phases, the system undergoes a ferroelastic structural phase transition from a cubic 𝐹 4 ̅3𝑚 to a tetragonal 𝐼 4 ̅𝑚2 space group. [15] With this symmetry breaking, the following relationship exists between the lattice parameters of the HT and LT phases: aHT=aLTbLT; bHT=aLT+bLT; cHT=cLT. Consequently, the HT Bragg peaks split during the transition towards Bragg peaks of different Miller indices in the LT phase. This can be observed clearly in the middle part of Fig. 3, showing how the diffracted intensity changes as function of temperature. For example, the (200) Bragg peak of the HT phase splits into the ( 110) and ( 002) peaks of the LT phase. Around 185 K, one can also observe a coexistence region, where both the Bragg peaks from the HT and LT phases are present. The intensity of the HT peaks decreases as the intensity of the LT ones increases during cooling. This indicates the coexistence of long-range ordered HT and LT domains. The shift of the peaks to higher q at low temperature is associated with the global lattice contraction accompanying the phase transition. It is also worth noticing that along the full temperature range there is no noticeable shift of the diffracted intensity along q in the LT or in the HT phases. For the Rietveld refinements we used aHT=10.5606 Å (at 300K), aHT=10.5592Å (at 210K), aLT= 7.0850 Å and cHT=10.5110 Å in agreement with previous results. [15] Indeed the Rb0.94Mn[Fe(CN)6]0.98•2.5H2O system is known to show an intrinsic zero thermal expansion of the HT phase and a weak negative thermal expansion of the LT phase [15] . We extracted the evolution of the HT phase fraction with temperature from the Rietveld refinement, as shown in the bottom part of Fig. 1. On cooling, the diffraction signal of the HT phase decreases around 200 K, where the LT phase appears. This coexistence of HT and LT phases in the hysteresis branch around 185 K is also observed on warming up from the LT phase: the LT phase signal decreases around 280 K where the HT phase appears. Our data closely replicate the wide hysteresis loop shown in the magnetic susceptibility measurements (Fig. 1 top) and the values of T1/2↓185 K and T1/2↑300 K, which were also observed by previous X-ray diffraction experiments with higher resolution. [15] This confirms that our experiment, despite the unconventional diffraction setup, allows for probing the long-range structural reorganization associated with the change of electronic state from LT to HT phases. Hereafter the similar setup is used for tracking PIPT, which can be permanently induced by a single laser shot inside the thermal hysteresis, as evidenced by infrared spectroscopy. [14] In order to study the effect of a single ns laser pulse on the LT phase inside the hysteresis loop, the sample is cooled down to 120 K to ensure the full LT conversion and warmed up again to 240 K. The initial state of the sample checked by X-ray diffraction corresponds to the bottom branch of the thermal hysteresis (LT phase). X-ray diffraction signal is then collected before and after exciting the sample with a 560 nm pulse of light few ns long. This wavelength is used in order to excite both the Mn III d-d and the intervalence transfer bands of the sample that affect the Jahn-Teller distortion and the intervalence metal-to-metal charge transfer from Fe II to Mn III respectively. [16] Figure 4. Comparison between diffraction patterns before light irradiation (blue) and after at  60% (orange) and 100% (red) conversion. The inset shows a zoom of the low q region with the indexed peaks of the HT and LT phases. Note that there is no shift in position, only change in relative intensity.

X-ray diffraction data shown in Fig. 4 confirm that a nanosecond light pulse transforms the system from the LT to the HT phase. Depending on the energy deposited on the sample, different degrees of LT to HT conversion can be reached. Fig. 4 shows the X-ray diffraction patterns in the LT phase before single nanosecond laser shot. After a single laser shot of medium fluence (0.4 mJ.mm -2 ), approximately 60% conversion is reached. With higher fluence of the laser shot (1.02 mJ.mm -2 ), full conversion toward the HT phase occurs. The region of the diffraction pattern below q = 1.5 Å -1 is particularly sensitive to the change in HT/LT fraction. The inset of Fig. 4 shows how the peaks of the HT phase appear while the ones of the LT disappear after ns laser shot, without noticeable shift in the peak position. This PIPT process corresponds therefore to the development of longrange ordered HT domains, and this phase nucleation process is very similar to the one observed during the thermal phase transition. The opposite scenario, namely a more homogeneous and local conversion, would result in a gradual shift of the Bragg peaks and/or peaks widening depending on the strain relaxation around the initial photo-induced "trapped" distortions. Such a scenario is ruled-out, as we clearly observe the coexistence of HT and LT phases Bragg peaks when conversion is partial, as shown in Fig. 4 for  60% conversion. [14] Data in b) correspond to the present X-ray study performed at 240 K and the fractions were obtained by using pseudo-Voigt fitting of the intensities of (111)HT and (101)LT peaks (blue) and by Rietveld refinement of the diffraction pattern (red).

As a consequence, the relative intensities of the (101)LT and the (111)HT Bragg peaks can be trusted to be a good marker of the relative proportion between the HT and LT phases. In Fig. 5, we compare the dependence with the laser pulse fluence of the conversion rate from LT to HT as function of the power of the shot. Fig. 5a shows the data obtained by IR spectroscopy (orange) and reported in a previous work. [14] Fig. 5b shows the present X-ray results, obtained by pseudo-Voigt fit of the (101)LT and (111)HT peaks (blue) and by Rietveld refinement (red) of the diffraction pattern. The IR data, obtained at 295 K, clearly evidenced a threshold fluence below which no permanent switching is observed in the thermal hysteresis, and a non-linear conversion with the fluence of the ns laser pulse used. The present X-ray data obtained at 240 K reproduce well the trend and a full conversion of the LT phase towards the cubic HT phase is reached with a similar laser fluence  1.02 mJ/mm 2 . The complete macroscopic switching mechanism after a laser pulse excitation is still unknown. For spin-crossover materials it was shown that their response to pulsed laser excitation is a multi-step process, [17] which sequentially involves ultrafast trapping of electronic excitation within 100s fs, cooperative elastic switching due to volume expansion (ns) and thermal population of high temperature state by laser heating (µs). In the present case, the fact that the photoresponses are found to be similar in X-ray and IR experiments performed at two different temperature (240 K and 295 K) may indicate that the driving force of the process is not (only) a laser heating process. However, for understanding the relative contribution of electronic and lattice dynamics, and possible elastic and thermal effects, which may be both responsible for the formation of the HT phase, time-resolved studies should be performed to disentangle their respective contributions in the time domain.

Conclusions

We performed X-ray diffraction experiments on a thin film of the charge-transfer RbMnFe magnetic material, in order to study the PIPT process induced by a single shot of nanosecond laser. The experimental observations show that the quantity of photoinduced HT phase varies non-linearly with the laser fluence and that a full conversion from the LT phase to the HT phase is reached at high fluence. The data also show that long-range ordered HT phase domains form after ns laser excitation, giving rise to a phase separation process similar to the one observed at thermal equilibrium. Photo-induced effects and heating process are known to often compete during out-of-equilibrium PIPT induced by a laser flash. However, our experimental data suggest that the process is not only due to laser heating, as the photo-response at 240 K is similar to the one previously reported at 295 K. The present grazing-incident X-ray diffraction technique will be applied for time-resolved studies and combined to other emerging ultrafast techniques, [18] for monitoring in real time the PIPT dynamics.

Experimental Section

Sample preparation:

Plate-shaped microcrystals of Rb0.94Mn[Fe(CN)6]0.98•2.5H2O were synthesized by mixing a poyethilene glycol monolaurate (PEGM) matrix containing MnCl2 (aq) and RbCl (aq) with another PEGM matrix containing K3[Fe(CN)6] (aq) and RbCl (aq). The obtained precipitate was centrifuged, washed in methanol and dried in air as described previously. [15] The formula was confirmed using elemental analysis and infrared spectroscopy. The thin films used for the experiment were obtained by depositing a droplet of a dispersion of microcrystals in ethanol on a thin glass slide (3 x 18 mm) and waiting for the solvant to evaporate. An impurity Rb2Mn II [Fe(CN)6]•3.5H2O is also present, as observed by X-ray diffraction and chemical analysis (Calculated Rb, 21.73; Mn, 13.92; Fe, 13.88% and found Rb, 21.98; Mn, 13.85; Fe, 13.79%). Even if the synthesis conditions are set up in a similar way, the compositions always slightly differ, ca. ± 0.06 difference appears in x (RbxMn[Fe(CN)6]((x+2/)/3)•zH2O). It is very likely due to slight differences in temperature, chemical concentration, and stirring speed. Because of such small variations, we perform for each synthesis elemental analysis, magnetic measurement, X-ray structural analysis, etc. to determine the exact composition and properties of the synthesized samples. Above-mentioned situations are common for analog compounds that show similar phase transitions. The direct relationship between composition and phase transition phenomenon is an important topic that was the scope of previous studies. [11] Experimental setup: The experiment was performed at the ID09 beamline of the ESRF synchrotron. The X-ray energy was selected by regulating the gap of the undulator and a Rumultilayer mirror was used to shape the pulse energy spectrum in order to reach a peak energy of 14.8 keV. The beam was focused using a toroidal mirror and the beam size at the focal point was 150 x 150 µm. The footprint on the sample, due to the grazing angle experimental geometry was 0.150x3 mm 2 . For exciting the sample, we used a EKSPLA NT342 laser with a central wavelength of 560 nm and a pulse duration of  5 ns. Electronically controlled shutter and waveplate were used to select a single laser pulse and set its energy. Two cylindrical lenses were used to shape the laser pulse profile to a size of 0.3x5.6 mm 2 , in order to excite a part of the sample larger than the one probed by X-ray diffraction. We performed the measurements by keeping the sample surface at a grazing angle respect to the X-rays incident direction. This was done to match the penetration depth of the X-rays and the laser. The laser direction was perpendicular to the direction of the X-rays as shown in Fig. 2. The samples were fixed to a goniometer head mounted on a Huber 1-Circle Goniometer (model 410a) with the rotation axis perpendicular both to the X-rays and laser direction. Temperature control during the experiment was achieved using an Oxford Cryosystems Cryostream 800 liquid nitrogen cooler positioned along the rotation axis of the goniometer. The temperature scans were performed using a cooling/heating rate of 6 K/min. We collected the diffraction images using a Rayonix MX170-HS CCD plate detector, with an active area of 170 x 170 mm. The detector was positioned at a distance of around 230 mm from the sample. Each image was obtained by accumulating the diffracted intensity from several X-ray pulses, 100 for the single shot measurements and 1000 for the temperature scans. Measurement: The temperature dependent data were collected while continuously changing the temperature of the sample with the cryostream. The single shot data were collected at 240K. However in order to measure inside the hysteresis loop the sample was first cooled down to 120K and then warmed up to 240K. We collected 20 images before the laser shot and 20 images after. Each measurement was performed on a fresh spot of the sample, initially in the full LT state. Data treatment: The diffraction patterns were obtained by performing an azimuthal integration of the diffraction images using the pyFAI and trx python packages. Before the data treatment the temperature dependent patterns measured within 1K were averaged together, also the 20 patterns collected before and after each single shot were averaged as well. The Rietveld refinement was performed using the software TOPAS-Academic version 6. We refined the peak shapes using a fundamental approach, taking into account the large footprint of the beam on the sample and the effects of preferential orientation on the intensities of the peaks. We also performed a pseudo-Voigt fitting on the intensities of (111)HT and (101)LT peaks of the single shot data in order to estimate the converted fraction after each laser shot.

Figure 2 .

 2 Figure 2. Schematic representation of the setup used for the experiment. The X-ray beam from the synchrotron is diffracted by the film of microcrystals deposited on the substrate. A nanosecond pump laser is used for switching the sample inside the thermal hysteresis.

Figure 3 .

 3 Figure 3. Comparison between the powder diffraction patterns obtained at 300 K and 140 K. The grey lines represent the raw data, the red line represents the HT phase, the green line represents the impurity and the blue line represents the LT phases patterns, obtained from full pattern Rietveld refinement. The peaks are indexed in the HT and LT cells. The central part shows how the diffracted intensity (colour code) changes with temperature.

Figure 5 .

 5 Figure 5. Fraction converted from LT to HT after a single shot as function of the power density. Data in a) are shown in orange and were obtained by IR spectroscopy after ns laser excitation at 295 K.[14] Data in b) correspond to the present X-ray study performed at 240 K and the fractions were obtained by using pseudo-Voigt fitting of the intensities of (111)HT and (101)LT peaks (blue) and by Rietveld refinement of the diffraction pattern (red).

Acknowledgments

Parts of this research were carried out in the frame of the IM-LED LIA (CNRS) and a JSPS Grant-in-Aid for Specially promoted Research 15H05697 and JSPS KAKENHI 16H06521 Coordination Asymmetry. The authors gratefully acknowledge Agence Nationale de la Recherche for financial support under grant ANR-16-CE30-0018 and University Rennes 1 for PhD funding of G.A..