Ahmed Boudghene Stambouli
email: stambouli_ahmed@yahoo.fr

Pierre-Yves Bard
email: pierre-yves.bard@univ-grenoble-alpes.fr

Emmanuel Chaljub
email: emmanuel.chaljub@univ-grenoble-alpes.fr

Peter Moczo
email: moczo@fmph.uniba.sk

Jozef Kristek
email: kristek@fmph.uniba.sk

Svetlana Stripajova
email: stripajova@fmph.uniba.sk

Capucine Durand
email: capucine.durand@univ-grenoble-alpes.fr

Djawad Zendagui
email: d_zendagui@mail.univ-tlemcen.dz

Boumédiène Derras
email: b_derras@mail.univ-tlemcen.dz

2D/1D AGGRAVATION FACTORS: FROM A COMPREHENSIVE STUDY TO ESTIMATION WITH A NEURAL NETWORK MODEL

Keywords: Aggravation factors, ground motion, numerical simulation, neural networks, 2D valleys

The aim of this paper is to derive "aggravation factors" (AGFs) quantifying the difference between 2D site response and the corresponding 1D estimate for numerous quantities of engineering interest such as response spectra and ten scalar ground motion intensity parameters (GMIPs). The raw results are a huge collection of AGFs derived by numerical simulation for a total 154692 receivers located at the surface of 894 valleys exhibiting a trapezoidal (131) or triangular (18) shape, with six different velocity contrasts. A simple statistical analysis first allows to identify the respective order of magnitude of these AGFs for each GMIP. In a second step, a neural network approach is used to provide a set of analytical relationships quantifying the coupled effects of a limited number of geometrical and mechanical parameters for all the considered GMIPs, and different positions within the valley. The AGFs are found to be component dependent, with larger values for out-of-plane component (SH waves) compared to the in-plane one (SV waves). They are GMIPs dependent, being the largest for Arias Intensity (I A ) and peak spectral amplification factor (SAF), intermediate for the Cumulative Absolute Velocity (CAV), and the smallest for all the other indicators. The geometry has a significant control on the AGF. For embanked valleys, the highest AGF occur in the center because of constructive interferences, steep edge slopes have significant but very localized effects on the edges, while gentle edge slopes have significant, long distance effects because of their energetic diffraction power. Diffraction away from the lateral slopes also implies a 10-20% variability of the motion on the rocky edges. Finally, results show also that mechanical parameters within the valley do affect the AGF.

INTRODUCTION

Alluvial valleys or basins are characterized by lateral thickness variations which have been shown to generate peculiar wave propagation phenomena (diffraction of surface waves, possible focusing of body waves, vertical and lateral reverberations) leading to increased wave trapping and interferences, and significant differences (increased duration, generally overamplification, sometimes deamplification) with respect to the case of horizontally stratified layers ("1D soil columns"). Such effects have been qualitatively predicted by theory for about 3-4 decades, and have been actually observed in real recordings or damage distribution (for instance in Kobe in 1995). However, they are only very rarely accounted for even in site-specific studies, because of a) the cost of the required geophysical surveys to constrain the geo-mechanical characteristics of the underground structure not only underneath but also around the target site, b) the insufficient number of well-documented observations that prevents any statistical treatment for a purely empirical prediction, and c) the lack of comprehensive enough parameter study that would allow to identify the key controlling parameters and to quantify their effects. This was explicitly the goal of a subtask of the NERA EC infrastructure project (2010-2014, www.nera-eu.org) to take advantage in the recent improvements in computing facilities, software accuracy and storage capacity, to perform a large number of computations for a wide variety of geometrical characteristics, velocity contrasts, and receiver locations within the valley, in order to derive statistically meaningful relationships describing the gross dependence of the overamplification on the main site "meta-parameters". The routine engineering practice to account for effects of subsurface conditions is either to consider the building code provisions based on site classification and the associated pre-defined spectral shapes (most often derived as a function of the "V S30 " parameter), or to perform 1D site response studies taking into account the local soil column. Both approaches have in common to be based only on the local soil structure, i.e. implicitly assuming a 1D site response. Therefore, in order to be consistent with the usual practice and to propose the simplest possible procedures to account for subsurface geometry effects, it was decided to quantify their effects only in terms of "aggravation factors (AGFs)" (Chavez-Garcia and Faccioli, 2000;Chavez-Garcia, 2007) describing the ratio between 2D (or 3D) and 1D amplifications for a variety of representative ground motion parameters. Such an approach has been adopted in several recent studies, such as [START_REF] Makra | Site effects and design provisions: the case of Euroseistest[END_REF][START_REF] Makra | Parametric analysis of the seismic response of a 2D sedimentary valley: implications for code implementations of complex site effects[END_REF] for the Euroseistest site (Mygdonian basin, Greece), [START_REF] Paolucci | Non-dimensional site amplification functions for basin edge effects on seismic ground motion[END_REF] for a family of hypothetical basin edges (walls or wedges), [START_REF] Kumar | Importance of quantification of local site effects based on wave propagation in seismic microzonation[END_REF], [START_REF] Narayan | Effects of strong lateral discontinuity on ground motion characteristics and aggravation factor[END_REF], [START_REF] Hasal | Effect of edge slope on soil amplification at a two dimensional basin model[END_REF], [START_REF] Gelagoti | Nonlinear dimensional analysis of trapezoidal valleys subjected to vertically propagating SV waves[END_REF] and [START_REF] Vessia | Relevant features of the valley seismic response: the case study of Tuscan Northern Apennine sector[END_REF] for hypothetical basins in the non-linear and linear equivalent cases, and [START_REF] Riga | Aggravation factors for seismic response of sedimentary basins: A codeoriented parametric study[END_REF] for a series of partial results obtained within the framework of the NERA project.

The paper successively describes the overall design of the numerical simulations (section 2), an outline of the post-processing from the raw results to the design of neural network models (section 3), and a few examples describing the dependence of the AGFs as a function of the site geomechanical parameters (section 4).

NUMERICAL SIMULATIONS

Overview

The basic idea and goal is to quantify the changes in amplification (increase = overamplification, or decrease = deamplification) compared to the 1D case, and to model their dependence as a function of a limited number of easily available geometrical and mechanical parameters. In that aim, the work was divided in the successive steps listed below: a) Selection of the models to be computed : geometry, mechanical characteristics, incident wavefield b) Selection of a representative number of input accelerograms c) Selection of the (surface) receivers where to compute the resulting motion d) Selection of the ground motion parameters for which the 2D/1D AGFs are computed. e) Selection and checks of the numerical simulation software f) Computations for all the considered cases of the time-domain response for a simple, pulse-like, short-duration signals, in both 2D and 1D cases from each receiver g) Linear convolution of the pulse response at each receiver l or each valley j with the selected input accelerograms i and derivation of the AGF for each considered ground motion parameter GMP k : AGF(R ljki ) = GMP k-2D (R lji ) / GMP k-1D (R lji ) h) Averaging these AGFs over all the considered input accelerograms to derive a mean AGF m (R ljk ) i) Using a neural network approach to correlate AGF m (R ljk ) to valley geometrical (width, thickness, etc.) and mechanical (velocity profile) characteristics, and also receiver location at valley surface. This analysis has been done here only in the case of 2D, linear response.

Design of valley models

Geometric configurations

It was decided to consider a set of trapezoidal and triangular valleys with a broad range of thickness and widths, and various slope angles on each edge, as indicated in Figure 1. More specifically, a) 6 values are considered for valley width W from 500 m to 20 km, b) 6 values are considered for valley maximum thickness H, from 30 m to 1 km, c) 6 sets are considered for slope angles: 4 symmetrical cases with slope angles equal to 10°, 20°, 45° and 65°, and two non-symmetrical cases with one edge angle at 10°, and the other at 45° or 65°. In total, this resulted in 149 geometries, as only 27 (H, W) combinations were considered as indicated in Figure 1. Out of them, 131 have a trapezoidal shape, and 18 a triangular shape with a maximum thickness lower than or equal to the H value. The advantages of such a geometry and parameter set are that it is simple, while it allows investigating the effect of the thickness/width "shape ratio" and of the sloping angles on each edge; in addition, a quick survey of the available cross-sections from real valleys indicated that it is not uncommon to have triangular shapes with some dissymmetry. Of course, real alluvial valleys may have more complex geometries, which may require site-specific computations: the present models are simply intended to provide a first order estimate of the expected level of AGFs.

Mechanical characteristics

Considering the large thickness values, velocity profiles with a velocity gradient were considered more realistic. Their functional form is provided in Figure 1: it is controlled by the velocity at surface V S0 and at a 1 km depth V S1 , and the exponent describing the depth dependence: a value of 0.5 was considered reasonable. The values at surface and depth were then tuned to have V S30 values providing integer velocity contrast with the underlying bedrock, which was considered homogeneous with a constant S-wave velocity equal to 1 km/s. The bedrock / sediment surface velocity contrast thus ranges from 2 to 8 (2, 3, 4, 5, 6 and 8). The total number of considered geomechanical cases was thus 894.

The damping was tuned to the velocity with the "rule-of-thumb" relation Q S = V S /10 (i.e., the damping ς = 0.5/Q S is decreasing with increasing depth). The unit mass was taken as linearly related to the Swave velocity through the relationship unit mass ρ (z) = 1600 + 0.6 (V S (z) -100) in the sediments, and ρ b = 2500 kg/m 3 . The P-wave velocity was taken equal to 1.5 km/s in the sediments (considered as water saturated), and 2 km/s in the bedrock. The quality factor for P-waves was taken according to the following formula : Q P (z) = Min (2*Q S (z), V P (z)/10).

Figure 1 : Geometry of the trapezoidal and triangular valleys considered and gradient type velocity profiles considered for the sedimentary deposit

Input wavefield and accelerograms

The seismic excitation has been basically defined as vertically incident plane S-waves, in order to provide a meaningful comparison with the routine 1D analysis. The motion may be in-plane (incident "SV" waves) or out-of-plane (incident "SH" waves). As the AGFs are looked for on several ground motion parameters (peak values, response spectra, duration, etc., see below) that are not related linearly with their analog on input motion, it is needed to consider several realistic input accelerograms, in order to get robust estimates on the corresponding average AGFs. Our option has been to select a relatively large set of accelerograms (26) in the RESORCE database [START_REF] References Akkar | Reference database for seismic ground-motion in Europe (RESORCE)[END_REF] on the basis of their frequency contents, as it has been repeatedly shown that amplification

Geometry of 2D valleys

Surface receivers

The ground motion has been computed at a minimum of 101 receivers within the valley, with a maximum spacing of 50 m : for valleys having a width smaller than 5km, the receiver spacing thus ranges from 5 m (w = 500 m) to 50 m (w= 5 km), and for valleys wider than 5 km (i.e., 10 or 20 km wide), the number of surface receivers was extended to 200 and 400, respectively. 10 additional receivers have been considered on each side, on the outcropping bedrock, with a spacing equal to w/20 (i.e., from 25 m to 1000 m, over distances from 125 m to 5 km).

Ground motion intensity parameters

The main ground motion intensity parameter (GMIP) considered in all analysis was the acceleration response spectra at a suite of periods/frequencies. Ten additional, scalar, GMIP were also systematically computed as detailed in [START_REF] Kristek | Computation of amplification factor of earthquake ground motion for a local sedimentary structure[END_REF] :

• Peak time domain values (PGA, PGV)

• Short period [Fa, around 0.1 s: geometrical average in the range 0.05 -0.2 s] and intermediate period [Fv, around 1 s: average in the range 0.5 -2 s] amplification factors. • Spectral intensity SI [average in the range 0.10 -2.5 s], Cumulative Absolute Velocity (CAV), Arias Intensity (I A ), Trifunac-Brady duration D TB (5-95 % and 5-75 %), and root mean square acceleration a rms (5-95 % duration only).

COMPUTATION OF THE AGFS AND DERIVATION OF PREDICTIVE MODELS

Example of "raw results"

The GMIPs listed above have been computed for each receiver and each input accelerogram, in both the 1D and 2D cases. The 2D over 1D AGF (i.e., the ratio of the 2D value over the 1D value) has been computed for each input signal, and then averaged over the whole set of input signals. Both the average and the associated signal-to-signal variability (standard deviation) have been saved. For each receiver, this AGF has been computed as a ratio GMIP(2D)/GMIP(1D), and thus averaged geometrically over the whole set of input accelerograms for all GMIP but the Trifunac-Brady duration D TB , for which the 2D-1D changes have been considered through the duration increase D TB (2D) -D TB (1D), which were then arithmetically averaged over the 26 input accelerograms. Examples of such averages for some valleys are displayed in Figure 3, providing some hints on the following (mainly qualitative) results:

• The AGFs are parameter dependent: "energy-related" GMIP (I A , CAV) generally exhibit larger values (up to 3-4), while high-frequency indicators (PGA, 0.1s amplification factor Fa, a rms ), exhibit lower values.

Site location: Definition of zones

We obtained a huge collection of average AGFs (about 150000 receivers, 10 severity index + frequency dependent AGFs at 100 frequencies). These AGFs have been archived in summary files with all the needed metadata. In order to propose acceptable AGF to the engineering community, it is needed to establish a simple correspondence between the gross geo-mechanical characteristics of the valley, the site position (near the edge or near the center) and the value of this AGF. For the work presented here, we have characterized the site position at the surface by defining eight (08) different zones as displayed in wfc and wee, which are the width of the western edge, of the central flat part, and of the eastern edge, respectively.

• wwe = H / tg(α 1 )

• wee = H / tg(α 2 )

• wfc = max [0, w -H/tg(α 1 ) -H/tg(α 2 )]

The "central flat zone" may have a nul width for gentle slope angles and too large H/w shape ratios: in such cases, the valley is triangular instead of trapezoidal, with a local maximum depth Z max and a position xwe with respect to the western edge given by :

• Z max = w / [1./tg(α 1 ) + 1./tg(α 2 )] • xwe = w / [1 + tg(α 1 ) / tg(α 2 )]
The 8 zones are then defined as follows:

• BR corresponds to the outcropping bedrock (10 receivers, 5 on each side) Minimum and average values of the same quantities were also defined and derived in an analog way.

•
In the present paper however, only the maximum values of each zone are investigated.

A maximum of 894 such values were thus derived for each considered valley (H, w, α 1 , α 2 + velocity profile) and a given type of motion (in-plane/SV or out-of-plane/SH). The statistical distribution of these parameters were derived and are presented in [START_REF] Kristek | Identification of key site features for site effect evaluation: extensive numerical sensitivity studies of sedimentary basin structures[END_REF]. We simply illustrate here in Figure 5 the dependence of the maximum values of each considered AGF on zone (amplitude parameters on top and duration increase on bottom), and reproduce the main observations listed in [START_REF] Kristek | Identification of key site features for site effect evaluation: extensive numerical sensitivity studies of sedimentary basin structures[END_REF]:

• The "out-of-plane" AGFs are almost systematically larger than the in-plane one. This has been checked for all the amplitude parameters, and is consistent with the results obtained by Moczo et al. (2018) for a set of real valleys • The largest AGF values correspond to the I A , the SAF (reaching up to about 4), and the CAV (slightly exceeding a value of 2). • As to the locations prone to higher AGFs it concerns mainly the inner valley zones, (flat central part + zones W1 and E1). • A noticeable result from the mean values (not shown here) is the trend to decreased amplitude values on the very edges, especially when the underlying slopes are very steep. • Finally the duration results indicate duration increase that may significantly exceed 10 s (for signals which are 10 to 30 s long), with a maximum occurring almost systematically in valley center (FE), and a trend to duration decrease on the very edges (zones W2 and E2) (see Figure 5). • One may notice also non-negligible effects even on the side rock sites: this corresponds to the waves partly reflected in the bedrock on the sloping interface. The mean values are very close to 1, basically from 1.04 to 1.12, but may exceed 1.2 in exceptional cases: such effects are likely contribute to the aleatory variability of ground motion on rock.

Building neural network models

The goal is to establish simple, approximate relationships providing a satisfactory estimate as the AGFs for different zones in the valley (edge, central part) as a function of the geomechanical characteristics, with a special attention to the short and intermediate period amplification factors Fa and Fv, which are handy to modify the reference spectra. Up to now 7 GMIPs have been considered: I A , CAV, AFMAX, PGA, PGV, Fa and Fv, as illustrated in Figure 6a.

A special kind of neural network has been selected for this work, the radial basis function (RBF) architecture (Figure 6b). In essence, the RBF approach consists in a "guided interpolation" of the training data, on the basis of the Euclidian distances between a given case and a representative set of all the input data used for the training/learning phase. As usually for neural networks, the architecture consists in an input layer, providing the input values, i.e. the explanatory variables for the model, an output layer with the output value to be predicted (here the AGF), and in-between one "hidden layer" implementing a series of combinations of the input values with an a priori functional form and adjustable weights, and thus providing intermediate values then combined to provide the model output Boudghene [START_REF] Stambouli | Deriving amplification factors from simple site parameters using generalized regression neural networks: implications for relevant site proxies[END_REF].. The main elements of the implementation in the present case are provided below:

• The input layer receives the 5-value input vector (X k ) describing the valley geometry (thickness to width ratio hwr = H/w, edge slope angles α 1 and α 2 ) and site "mechanical parameters (V s30 and f 0 ). • The output layer corresponds to the value of the considered AGF.

• The hidden layer, i.e. the "processing unit", consists of a limited number of "central neurons" CN j {j=1, J}, which are 5-value vectors carrying the same information (hwr j , α 1,j , α 2,j , V s30,j and f 0,j ): their values are obtained during the training phase to optimize the sampling of the input vector set. • The processing for each neuron of the hidden layer consists simply in computing the Euclidian distance between the considered input vector and the considered "central neuron", and deriving the weight w j to be associated to this key neuron through a "radial basis function" in the form of a Gaussian with a width parameter b 1 to be tuned by the training phase :

𝑑𝑖𝑠𝑡 ! = 𝐶𝑁 !,! -𝑋 ! ! ! !!! and 𝑤 ! = 𝑒 !(!"#$ ! * !!) !
• The next and last step performed between the hidden and last layer is to combine linearly these Euclidian distance based weights with reference values a j (to be adjusted in the training phase)

𝐴𝐺𝐹 !"# = 𝑎 ! + 𝑤 ! . 𝑎 ! ! !!!
A model is therefore characterized by a total of 6 J + 2 free parameters (a 0 , b 1 , CN j,k and w j ) which are tuned to minimize the root mean square distance between the actual output values of the training set and the model predictions.

Overview of models and their performance

Such models have been derived for the 7 AGFs indicated in Figure 6a, and for six different zones (4 for the trapezoidal case: FC, FW∪FE, W1∪E1, W2∪E2, and 2 for the triangular case: W1∪E1, W2∪E2). 42 [(4+2)×7] models have thus been derived up to now. Their performance may be "measured" by different indicators such as the reduction of variance between initial AGF values, and the model residuals, or the coefficient of determination between actual AGF values and their predictions. The latter is illustrated in Figure 6c for the 6 zones (abscissa), and the 7 considered parameters (different curves): they are found to range from 0.84 to 94, which are fairly good values, and witness the satisfactory performance of these models to correctly predict the actual AGFs.

EXAMPLE MODEL PREDICTIONS

It is not easy to display the functional dependence in case of five input parameters. Given the limited space, only two examples are displayed in Figure 7 and Figure 8. They both display the dependence of the AGF for I A , for the zone W1UE1 of triangular valleys, and for the central zone FC of trapezoidal valleys, respectively. In every instance, the evolution of the AGF with the "main parameter", i.e., the shape ratio H/w, is displayed, for 8 different parameter combinations: there are 4 curves corresponding to four different values of one parameter, the three other parameters being constant and equal to some specific fractile value (10%, F10, median, F50, or 90%, F90) as listed in Table 1. For the "inner edge" zone in triangular valleys (Figure 7), these plots illustrate the major importance of the shape ratio (AGIA increases with H/w), of the sediment softness (AGIA increases with decreasing velocity), the significant -and somewhat unexpected -effect of the "near" slope angle (α 1 ) with larger AGFs for lower sloping angles (cases B, C and H), the conversely very limited impact of the "far" slope angle (α 2 ) (only small differences between cases D and E), and the limited effect of the fundamental frequency (cases F and G). The same conclusions roughly hold also for the central zone of trapezoidal valleys (Figure 8), with a slightly lower impact of the (more distant) sloping angle α 1 , and slightly larger impact of the fundamental frequency. Such conclusions should be considered only as examples of the potentialities of the models obtained with the neural network approach, but they cannot be generalized to AGFs for other parameters. A paper is in preparation that will provide the formulas for each zone and each parameter.

A) D) G) B) E) H) C) F)
Figure 7 : Example predictions for the Arias Intensity, for the W1UE1 zone in triangular valleys. The different cases considered are listed in Table 1 and explained in the text.

CONCLUSIONS

The results from this comprehensive set of computations have been analyzed in a purely statistical way to better understand the key parameters, and modeled using a RBF neural network approach with radial basis function architecture. Both the statistical analysis and the RBF quantitative models lead to the following conclusions:

• The AGFs are component dependent, with larger values for SH, out-of-plane motion compared to SV, in-plane motion. This may be explained by the transfer of energy to the vertical component, as outlined in the other sections of this report • The AGFs are parameter dependent : they are generally the largest (up to 3-4) for Arias Intensity and the peak spectral amplification factor, intermediate (up to around 2-2.5) for the Cumulative Absolute Velocity, and the smallest (up to 1.5 -2) for all the other indicators.

V s30 (fractile10%) V s30 (fractile40%) V s30 (fractile70%) V s30 (fractile90%) 0.2 0.3 0.
V s30 (fractile10%) V s30 (fractile40%) V s30 (fractile70%) V s30 (fractile90%) A) D) G) B) E) H) C) F)
Figure 8 : Example predictions for the I A , for the FC zone in trapezoidal valleys. The different cases considered are listed in Table 1 and explained in the text

• The ground motion within the valley may be significantly prolongated, up to 10 to 15 seconds.

• The diffraction away from the lateral sloping interfaces implies a slight contamination of the motion on the rocky edges, with an increase of outcropping rock motion, which may be up to 20-30%, especially in the case of steep lateral slopes. This probably contributes to the within-event aleatory variability of rock ground motion Large AGFs (exceeding 2) in the spectral domain correspond to • Large velocity contrasts (V S,bedrock /V S30 > 5)

• Embankment ratios Z max / Width larger than 0.08 • Sites located within the central, constant thickness part of the valley (zones FW, FC and FE), or very close to it (inner parts of W1 and E1) • Relatively steep slopes (larger than 20°) when considering sites in the central part of the valley (FW, FC, FE), and any kind of slope angles (including gentle ones) for the "inner-edge" zones W1 and E1. • In addition, the 42 RBF models developed up to now (7 GMIP, 6 zones) provide simple formulae to estimate the aggravation for each zone and ground motion parameter as a function of a relatively small number of parameters involving the valley geometry (shape ratio h/w, edge slope angles α 1 and α 2 , and local mechanical characteristics (V S30 and f 0 ). One important further step would be to identify, in tight connection with the engineering community, some relevant threshold values for the AGFs on each of the various GMIPs considered here, and to identify the geo-mechanical configurations in which one may expect such thresholds to be exceeded. Another useful contribution would be to compare this set of AGFs corresponding to "virtual", simple shaped valleys, with those derived for real valleys [START_REF] Moczo | Key structural parameters affecting earthquake ground motion in 2D and 3D sedimentary structures[END_REF].

ACKNOWLEDGMENTS

The simulation work has been funded by the European Commission FP7 under the Grant Agreement No 262330 (NERA Project, www.nera-eu.org/), and and also by the Slovak Research and

  spectral ordinates are sensitive to the predominant frequency of the input motion: their spectra displayed in Figure2indicate that their dominant frequencies span a wide range from around 1 Hz to beyond 10 Hz.

Figure 2 :

 2 Figure 2 : Normalized response spectra of all the input accelerograms considered (black lines). The solid red curves correspond to the average normalized shape, ± one standard deviation

•Figure 3 :

 3 Figure 3 : Example results for AGFs and their variability along the valley surface and their sensitivity to a) GMIPs, b) sediment softness and c) edge slope angle. In each case, the x-axis represents the normalized position along the valley cross-section (x=0.5 corresponds to valley center, x=0 or 1 to valley edges. For all cases, the sensitivity is shown only for SH incidence) and around a given "base case valley". a): symmetrical trapezoidal valley, h=250m, W = 1 km, softest sediments, steep edge slope angles 45°). b): Intermediate period (1 s) amplification Fv and symmetrical, trapezoidal valley, h=120m, W = 1 km, moderate edge slope angles 20°), c): AGFs for the intermediate period (1 s) amplification Fv, trapezoidal valley (h=120m, W = 1 km, soft sediments, all kinds of slope angles) c)

Figure 4 .NERANERA

 4 The total valley width W is first separated in three sub-widths wwe,Effects on various severity parametersExample : h=250, W = 1000, V S30 =125 bardpi /data/risques/bardpi/d12/artsism/nera/AGF2D1D/results1_AGF_SH f024.sgf Sat Oct 4 17:53:31 2014 risques/bardpi/d12/artsism/nera/AGF2D1D/results1_AGF_SH f018.sgf Sat Oct 4 17:52:54 2014

  W2 and W1 are two equal-width zones located over the western (left) slope of the valley; their width is wwe/2 =Min {0.5 H/tg(α 1 ), 0.5 w/[1+tg(α 1 )/ tg(α 2 )]} • Similarly, E2 and E1 are two equal-width zones located over the eastern (right) slope of the valley; their width is wee/2 =Min {0.5 H/tg(α 2 ), 0.5 w/[1+tg(α 2 )/ tg(α 1 )]}. • FW, FC and FE are three equal-width zones located in the central, constant thickness part of the valley. For trapezoidal valleys, such zones may be reduced to one single point, with a local thickness Z max .

Figure 4 :•Figure 5 :

 45 Figure4: Definition of valley zones. The total valley width W isseparated in seven zones. Four correspond to the edges above the sloping interface: the extreme edges (W2 and E2), the "inner edges" (W1 and E1). In case of trapezoidal shape, the central, flat part is divided in three equal width parts, FW and FE correspond to the parts closer to the edges ("outer central part"), FC correspond to the central part

Figure 6 :

 6 Figure 6 : Neural network analysis: considered AGFs (left), general architecture of a RBF (right) and performance of the 42 different neural networks for the 6 zones and 7 AGFs (bottom)

4NERA(

  Network of European Research Infrastructures for Earthquake Risk Assessment and Mitigation) , for 98 this we have considered only one combination built on 3 geometrical parameters describing as 99 (hwratio,alpha1,alpha2) and two soil parameters describing as (resonance frequency(f0), Vs30) from where a total of 5 100 parameters are considered on inputs parameters, So a total of 7 parameters are considered for the output parameters 101 describing such as (PGA, PGV, Fa, Fv, Afmax, Cav, IA) . The table below describes all parameters to predict in 102 term aggravation factor. 103 Table1 descriptions of output parameters (aggravation factor).

  , the same work are performed for each zones {((W1)U(E1)), ((W2)U(E2)), ((FE)U(FW)) and 107 FC}and for all valleys types trapezoidal or triangular, so a total 42 model RBF is considered for all zone and 108 each outputs parameters (AGF).

7 Figure 2

 72 Figure 2 Illustration of coefficient correlation for all zones and all outputs parameters (top) and variance reduction for 131all zones and all outputs parameters ( bottom). 132Other information is done in table 2 corresponding to number of key neurons in hidden layer of RBF which are 133 determined in training phase .Their number are between 3 and 18 for all model RBF computed, it explains well that 134 one can calculate the aggravation manually, there is one of powerful of this approach. 135 1.4 Overview of obtained results

Table 1 :

 1 Input parameter values considered for cases displayed in Figures7 and 8

	Plot name	H/w	α 1	α 2	V S30 (m/s)	f 0 (Hz)
	A	Continuous	Median (15°, 45°)	Median ` (15°, 45°)	124, 198, 331, 498 Median (0.5, 0.4)
	B	Continuous	F90 (45°)	Median	Same	Median
	C	Continuous	F10 (10°)	Median	Same	Median
	D	Continuous	Median	F90 (45°)	Same	Median
	E	Continuous	Median	F10 (10°)	Same	Median
	F	Continuous	Median	Median	Same	F90 (2.0, 1.4)
	G	Continuous	Median	Median	Same	F10 (0.16)
	H	Continuous 10°, 20°, 45°, 65° Median Median (240, 220)	Median

•

  The geometry has a significant control on the AGF. o For embanked valleys, the highest AGFs occur in the center because of constructive interferences o Steep edge slopes have large effects (with AGFs lower than 1, i.e., deamplification effects), but only very locally just over the valley edges o Gentle edge slopes have significant, long distance effects because of their energetic diffraction effect • The mechanical characteristics within the valley do affect the AGF o The AGFs are generally found the largest for the largest velocity contrasts, in relation with the improved efficiency of lateral wave trapping o Increase in damping induces decrease of the AGF, especially for high-frequency indicators (not shown here, but obtained from a small, parallel sensitivity study, and supported by the SIGMA results)


Habib Ammari
email: habib.ammari@math.ethz.ch

Mihai Putinar
email: mputinar@math.ucsb.edu

AND Andries Steenkamp
email: andriess@student.ethz.ch

Faouzi Triki
email: faouzi.triki@univ-grenoble-alpes.fr

Identification of an Algebraic Domain in Two Dimentions from a Finite Number of its Generalized Polorization Tensors

Keywords: . 1991 Mathematics Subject Classification. Primary: 35R30, 35C20. Key words and phrases. inverse problems, generalized polarization tensors, algebraic domains, shape classification

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Introduction

Given a conductivity contrast, the generalized polarization tensors (GPTs) of a bounded Lipschitz domain are an infinite sequence of tensors. The GPTs form the basic building blocks for the far-field behavior of the electric potential. Recently, many works have shown that the GPTs can be used to efficiently recover geometrical properties of the underlying shape. In fact the knowledge of the full set of GPTs determines uniquely the shape of the domain as proved in [START_REF] Ammari | Properties of the generalized polarization tensors[END_REF]. When the domain is transformed by a rigid motion or a dilation, the corresponding GPTs change according to certain rules. It is possible to construct as combinations of GPTs invariants under these transformations. This property makes GPTs suitable for the dictionary matching problem [START_REF] Ammari | Target identification using dictionary matching of generalized polarization tensors[END_REF][START_REF] Hu | Visual pattern recognition by moment invariants[END_REF]. The GPTs have also been used in various areas of applications such as imaging, cloaking, and plasmonics. We refer the reader to [START_REF] Ammari | Enhancement of near cloaking using generalized polarization tensors vanishing structures: Part I: The conductivity problem[END_REF][START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF][START_REF] Ammari | Expansion methods[END_REF][START_REF] Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Kang | Layer potential approaches to interface problems[END_REF][START_REF] Milton | The Theory of Composites[END_REF][START_REF] Triki | Mathematical modeling of the Photoacoustic effect generated by the heating of metallic nanoparticles[END_REF] and references therein for further information about these applications.

Since the GPTs appear naturally in imaging a small conductivity inclusion from boundary potential measurements, the amount of information about the shape of the inclusion encoded in the first tensors is richer than any other geometrical quantities. Recent numerical studies [START_REF] Ammari | Generalized polarization tensors for shape description[END_REF][START_REF] Ammari | The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion[END_REF] show that by using only the first few terms of GPTs reasonable approximation of the true shape can be recovered. Complete geometric identification of a conductivity inclusion from the knowledge of its first GPT is known to be possible for ellipsoid shapes. In fact if the contrast is given, only the first polarization tensor is needed to retrieve the major and minor axis of an ellipse [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF]. For arbitrary shapes, it is proposed to approach them using ellipseequivalent identification. This consists simply of determining the shape of an ellipse with the same first polarization tensor as that of the targeted inclusion [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF]. The results of this approach are quite surprising since the recovered equivalent ellipse seems to hold much more information on the shape than anticipated. For example, the equivalent ellipse contains more knowledge than the first two or three Dirichlet Laplacian eigenvalues of the inclusion. An interesting question is whether one could recover other shapes of an inclusion from the knowledge of a finite number of its GPTs. In view of [START_REF] Ammari | Shape reconstruction of nanoparticles from their associated plasmonic resonances[END_REF], inclusions with algebraic shapes represent good candidates for such an identification problem. To specify our terminology, an inclusion has algebraic shape if it is a bounded open subset of Euclidean space whose boundary is real algebraic, i.e., contained in the zero set of finitely many polynomials.

In this paper, we are interested in the inverse problem of recovering the shape of an algebraic inclusion given a finite number of its GPTs. We consider shapes unique up to rigid motions, that is orthogonal transformations.

The paper is organized as follows. In Section 2 we introduce the notion of GPTs of an inclusion and their relation to far-field expansion of the fields associated to a piecewise constant conductivity. The inverse problem in question is stated in Section 3, where a review of recent results in the recovery of the shape of an inclusion from the knowledge of all the GPTs is also given. Section 4 is dedicated to an introduction to real algebraic domains. Some basic notions of real algebraic geometry are recalled here. Our main identification result is stated in Theorem 5.1. The detailed proof of the main result as well as a uniqueness result are provided in Section 5. Section 6 is devoted to the generalization of the concept of ellipse-equivalent approach to higher-order GPTs. This generalization takes advantage of the density of algebraic domains in the set of smooth inclusions. We apply the main result in Section 6 by constructing a shape recognition algorithm and demonstrating its optimality by means of a few well chosen examples. The paper is concluded with some discussions in Section 7.

Generalized polarization tensors

Let D be a bounded Lipschitz domain in R 2 , of size of order one. Assume that its boundary ∂D contains the origin. Throughout this paper, we use standard notation concerning Sobolev spaces. For a density φ ∈ H -1/2 (∂D), define the Neumann-Poincaré operator (NPO):

K * ∂D : H -1/2 (∂D) → H -1/2 (∂D), by K * ∂D [φ](x) = 1 2π p.v. ∂D x -y, ν(x) |x -y| 2 φ(y) dσ(y), x ∈ ∂D,
where p.v. denotes the principal value, ν(x) is the outward unit normal to ∂D at x ∈ ∂D, and , denotes the scalar product in R 2 . The spectral properties of the Neumann-Poincaré operator have proven interesting in several contexts [START_REF] Ammari | Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF][START_REF] Bonnetier | Homogenization of the eigenvalues of the Neumann-Poincaré operator[END_REF][START_REF] Bonnetier | Pointwise bounds on the gradient and the spectrum of the Neumann-Poincare operator: The case of 2 discs[END_REF][START_REF] Bonnetier | On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2d[END_REF]. Due to Plemelj-Calderón identity and energy estimates, the spectrum of K * ∂D is real [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF][START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. When D is smooth (with C 1,α boundary), K * ∂D is compact, hence its spectrum consists of a sequence of eigenvalues that accumulates to 0 [START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. When D is Lipschitz, the following proposition characterizes the resolvent set ρ(K * ∂D ) of the NPO [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF][START_REF] Fabes | The spectral radius of the classical layer potentials on convex domains[END_REF]. Proposition 2.1. We have

C \ (-1/2, 1/2] ⊂ ρ(K * ∂D ). Moreover, if |λ| ≥ 1/2 , then (λI -K * ∂D ) is invertible on H -1/2 0 (∂D) := {f ∈ H -1/2 (∂D) : f, 1 -1/2,1/2 = 0}.
Here, , -1/2,1/2 denotes the duality pairing between H -1/2 (∂D) and H 1/2 (∂D).

For |λ| > 1/2 and a multi-index α = (α 1 , α 2 ) ∈ N 2 , where N is the set of all positive integers, define φ α by

φ α (y) := (λI -K * ∂D ) -1 [ν(x) • ∇x α ] (y), y ∈ ∂D.
Here and throughout this paper, we use the conventional notation:

x α = x α1 1 x α2 2 , α = (α 1 , α 2 ) ∈ N 2
, and |α| = α 1 + α 2 . We also use the graded lexicographic order:

α, β ∈ N 2 verifies α ≤ β if |α| < |β|, or, if |α| = |β|, then α 1 ≤ β 1 or α 1 = β 1 and α 2 ≤ β 2 .
The GPTs M αβ for α, β ∈ N 2 (|α|, |β| ≥ 1), associated with the parameter λ and the domain D are defined by

M αβ (λ, D) := ∂D y β φ α (y) dσ(y). (1)
As we said before, the GPTs are tensors that appear naturally in the asymptotic expansion of the electrical potential in the presence of a small inclusion D of conductivity contrast 0 ≤ k = 1 ≤ +∞. The parameter λ is related to the conductivity k via the relation

λ = k + 1 2(k -1)
.

The fact that 0 ≤ k ≤ +∞ implies that |λ| ≥ 1/2, and hence (λI -K * ∂D ) is invertible on H -1/2 0 (∂D). Assume now that the distribution of the conductivity in R 2 is given by

Υ = kχ(D) + χ(R 2 \ D),
where χ denotes the indicator function. For a given harmonic function h in R 2 , we consider the following transmission problem:

∇ • Υ∇u = 0 in R 2 , u(x) -h(x) = O(|x| -1 )
as |x| → ∞.

The electric potential u has the following integral representation (see, for instance, [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF])

u(x) = h(x) + S ∂D (λI -K * ∂D ) -1 [∂ ν h(x)] , x ∈ R 2 ,
where S ∂D : H -1/2 (∂D) → H 1/2 (∂D) is the single layer potential given by

S ∂D [φ](x) = 1 2π ∂D Γ(x -y)φ(y) dσ(y), x ∈ R 2 .
Here, Γ is the fundamental solution of the Laplacian,

Γ(x) := 1 2π ln |x|.
It possesses the following Taylor expansion

Γ(x -y) = ∞ |α|=0 (-1) |α| α! ∂ α Γ(x)y α , y ∈ D, |x| → +∞,
where

∂ α = ∂ α1 x1 ∂ α2 x2 , and α! = α 1 !α 2 ! for α = (α 1 , α 2 ) ∈ N 2 .
Then, the far-field perturbation of the voltage potential created by D is given by [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF]

u(x) -h(x) = ∞ |α|,|β|=1 (-1) |α| α!β! ∂ α Γ(x)M αβ ∂ β h(0) as |x| → +∞. (2)
From the asymptotic expansion (2), we deduce that the knowledge of M αβ for α, β ∈ N 2 (|α|, |β| ≥ 1) is equivalent to knowing the far-field responses of the inclusion for all harmonic excitations.

Shape reconstruction problem

In this section, a brief review of recent results in GPT based inclusion shape recovery is given. We first introduce the harmonic combinations of the GPTs. Positivity and symmetry properties of the GPTs are proved using their harmonic combinations [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF]. A harmonic combination of the GPTs M αβ is

α,β a α b β M αβ ,
where α a α x α and β b β x β are real harmonic polynomials. We further call such a α and b β harmonic coefficients. For example, if a α and b β are any two harmonic coefficients, we have the following symmetry property:

α,β a α b β M αβ = α,β a α b β M βα .
The following uniqueness result has been proved in [START_REF] Ammari | Properties of the generalized polarization tensors[END_REF].

Theorem 3.1. If all harmonic combinations of GPTs of two domains D and D with parameters λ and λ, are identical, that is

α,β a α b β M αβ (λ, D) = α,β a α b β M αβ ( λ, D),
for all pairs a α and b β of harmonic coefficients, then D = D and λ = λ. Theorem 3.1 says that the full knowledge of (harmonic combinations of) GPTs determines uniquely the domain D and the parameter λ.

Recall that the first-order polarization tensor, M αβ with |α| = |β| = 1, of any given inclusion is a 2×2 real valued and symmetric matrix. Remarking that the polarization tensors produced by rotating ellipses with size coincide with the set of 2 × 2 real valued symmetric matrices, it is known that the first-order polarization tensor yields the equivalent ellipse (see for instance [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF] and references therein). The equivalent ellipse of D is the ellipse with the same first-order polarization tensor as D. However, it is not known explicitly what kind of information on D and λ the higher-order GPTs carry. The purpose of this work is to study the possibility of using higher-order GPTs for shape description. Our idea is based on first deriving a set of dense domains that can be identified from finitely many GPTs. We show that good candidates for such a dictionary is the class G of algebraic domains of size one. Then, by approximating the target domain using a sequence of algebraic domains in G we obtain a powerful tool for describing the shape of inclusions if only a finite number of GPTs is available. Next, we introduce the concept of real algebraic domains.

Real algebraic domains

In the present section we consider the class of bounded open subsets of Euclidean space R 2 with real algebraic boundary. We adopt the following definition.

Definition 4.1. An open set G in R 2 is called real algebraic if there exists a finite number of real coefficient polynomials g i (x), i = 1, • • • , m such that ∂G ⊂ V := {x ∈ R 2 : g 1 (x) = • • • = g m (x) = 0}.
The ellipse is a simple example of an algebraic domain, since its general boundary coincides with the zero set of the quadratic polynomial function

g(x) = |α|≤2 g α x α
for given real coefficients (g α ) |α|≤2 and proper signs in the top degree part. We further denote by G the collection of bounded algebraic domains and of size of order one. The differential structure of the boundary ∂G is then well known: it consists of algebraic arcs joining finitely many singular points, see for instance [START_REF]de la Puente Real Plane Alegbraic Curves[END_REF].

It is tempting to also impose connectedness of the respective sets, but this constraint is not accessible by the elementary linear algebra tools we develop in the present note, so we drop it. However, we call "domains" all elements G ∈ G.

Following [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF] we focus on a particular class of algebraic domains which are better adapted to the uniqueness and stability of the shape inverse quest. Let

G * := G ∈ G : G = int G . (3)
An element of G * is called an admissible domain, although it may not be connected.

The assumption that G = int G implies that G contains no slits or ∂G does not have isolated points. If G ∈ G * , the algebraic dimension of ∂G is one, and the ideal associated to it is principal. To be more precise, ∂G is a finite union of irreducible algebraic sets X j , j ∈ J, of dimension one each. The reduced ideal associated to every X j is principal:

I(X j ) = (P j ), j ∈ J
for instance see [START_REF] Bochnak | Real Algebraic Geometry[END_REF]Theorem 4.5.1]. We assume that each P j is indefinite, i.e. it changes sign when crossing X j . Therefore one can consider the polynomial g = j∈J P j , vanishing of the first-order on ∂G, that is |∇g| = 0 on the regular locus of ∂G. According to the real version of Study's lemma (cf. Theorem 12 in [START_REF]de la Puente Real Plane Alegbraic Curves[END_REF]) every polynomial vanishing on ∂G is a multiple of g, that is I(∂G) = (g). We define the degree of ∂G as the degree of the generator g of the ideal I(∂G). For a thorough discussion of the reduced ideal of a real algebraic surface in R d , we refer to [START_REF] Dubois | Algebraic theory of real varieties[END_REF].

In the sequel, we denote by g(x) the single polynomial vanishing on ∂G which is the generator of I(∂G) and satisfying the following normalization condition g α * = 1, where α * = max gα =0 α.

Assume G ∈ G * . If the degree d of ∂G and moments (up to order 3d) of the Lebesgue measure on G are known, then as shown in [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF], the coefficients of g of degree d that vanishes on ∂G are uniquely determined (up to a constant). More precisely, it is shown in [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF] that g is the generator of a one-dimensional kernel of a matrix whose entries are obtained from moments of the Lebesgue supported measure by G. That is, only finitely many moments (up to order 3d) are needed to recover the minimal degree polynomial vanishing on G. It turns out that computing g reduces to a solving a system of linear equations.

We stress that the main result contained in the present article identifies the minimal degree polynomial g, and not the exact boundary of the admissible domain G. To give a simple example, consider the defining equation of the boundary

(4) g(x 1 , x 2 ) = (x 3 1 -x 1 )(x 3 2 -x 2 )
. The following algebraic domains ( 5)

G 1 = (x 1 , x 2 ), x 1 > 0 or x 2 > 0 , G 2 = {(x 1 , x 2 ), x 1 > 0 or x 2 < 0 , G 3 = {(x 1 , x 2 ), x 1 < 0 or x 2 < 0 , G 4 = {(x 1 , y 2 ), x 1 < 0 or x 2 > 0 ,
are all admissible domains, sharing the same minimal degree defining function of the boundary.

Even when restricting the class of domains to those possessing an irreducible boundary, we may encounter pathologies. Without recalling cumbersome details, Example 28 in [START_REF]de la Puente Real Plane Alegbraic Curves[END_REF] produces a series of polynomials whose zero sets may contain curves and isolated points, but some of the isolated points are not irreducible components. The connectedness of G is also tricky, as for instance the intricate nature of the topology of the zero set of a lemniscate reveals: the curve

k j=1 (x 1 -a j ) 2 + (x 2 -b j ) 2 = r, (6)
has k distinct connected components for small values of r > 0, where the poles (a j , b j ) are mutually distinct.

The main objective of our article, comparable to that in [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF], is to isolate a finite pool of domains (we may call them bounded "chambers") from which we can select the shape of G and further on determine the parameter λ, both inferred from the knowledge of finitely many GPTs. It would be extremely interesting to unveil how the additional information encoded in the GPTs allows to select the correct chamber among the many potential candidates.

Main results

Let R[x] be the ring of polynomials in the variables x = (x 1 , x 2 ) and let R N [x] be the vector space of polynomials of degree at most N (whose dimension is r N = (N + 1)(N + 2)/2). For a polynomial function

p(x) ∈ R N [x]
, it has a unique expansion in the canonical basis

x α , |α| ≤ N of R N [x], that is, p(x) = |α|≤N p α x α
for some vector coefficients p = (p α ) ∈ R r N . The following is the main result of the paper.

Theorem 5.1. Let G ∈ G * with ∂G Lipschitz of degree d, and let g(x) = |α|≤d g α x α , be a polynomial function that vanishes of the first-order on ∂G, satisfying I(∂G) = (g), g α * = 1, and g(0) = 0, where α * = max gα =0 α. Then, there exists a discrete set Σ ⊂ C 0 := C \ [-1/2, 1, 2], such that g = (g α ) ∈ R r d is the unique solution to the following normalized linear system:

p = (p α ) ∈ R r d ; |β|≤d M αβ (λ, G)p β = 0 for |α| ≤ 2d; p α * = 1, α * = max pα =0 α for λ ∈ C 0 \ Σ. (7)
Proof. The proof of the theorem has two main steps. In the first step, we show that g satisfies the normalized linear system [START_REF] Ammari | Expansion methods[END_REF]. The second step consists in proving that it is indeed the unique solution to that system.

Step 1. For |λ| > 1 2 , recall from (1) the general expression of the GPTs (8)

M αβ (λ, G) := ∂G (λI -K * ∂G ) -1 [ν(x) • ∇x α ] y β dσ(y).
Since g(x) vanishes on ∂G, we infer ( 9)

∂G (λI -K * ∂G ) -1 [ν(x) • ∇x α ] g(y) dσ(y) = 0, ∀α ∈ N 2 ,
which combined with (8) leads to the desired system:

|β|≤d M αβ (λ, G)g β = 0, ∀α ∈ N 2 . ( 10
)
Step 2. Assume that p ∈ R r d [x] satisfies the system [START_REF] Ammari | Shape reconstruction of nanoparticles from their associated plasmonic resonances[END_REF]. Our objective is to prove that p coincides with g.

Denote by C := C\(-∞, -2]∪[2, +∞), and let µ = λ -1 ∈ C . Define M(µ) to be the rectangular matrix with coefficients: (µM αβ ) |α|≤2d,|β|≤d , that is

M αβ (µ) := ∂G (I -µK * ∂G ) -1 [ν(x) • ∇x α ] y β dσ(y). (11)
Obviously, the following equality holds

M(µ)p = 0. Lemma 5.1. The function µ → M(µ) ∈ L (R r d , R r 2d
) is a holomorphic matrix-valued function on C , and ker(M(0)) = {cg; c ∈ R}.

Proof. Considering the properties of the resolvent set in Proposition 2.1, µK * ∂G is a contraction operator for µ small enough. In fact it can be easily verified that K * ∂G ≤ 1/2 where the norm of K * ∂G is taken in the energy space [START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. Moreover, the Neumann series

(I -µK * ∂G ) -1 = ∞ j=0 µ j (K * ∂G ) j ,
has a holomorphic extension in the resolvent set C . Next, we investigate the kernel of M(0). More precisely, M(0)p = 0 is equivalent to

∂G ν(y) • ∇q(y)p(y) dσ(y) = 0, ∀q ∈ R d [x]. (12)
Since g generates the ideal associated to ∂G and ∂G is Lipschitz, we have ν(x) = ∇g ∇g on the regular part of the curve ∂G. Then (12) becomes ∂G ∇g(y) • ∇q(y)p(y)

1 ∇g dσ = 0, ∀q ∈ R d [x].
By taking q(x) = x α g(x) for α ∈ N 2 satisfying |α| ≤ d, and considering the fact that g(x) vanishes on ∂G, one finds

∂G ∇g y α p(y) dσ = 0, ∀α ∈ N 2 , |α| ≤ d, which in turn implies that ∂G ∇g q(y)p(y) dσ = 0, ∀q ∈ R d [x].
Then, taking q = p in the last inequality gives p(y) = 0 on ∂G. Consequently, p = cg for some real constant c, which is the desired result.

At this point we return to the proof of the main theorem. The analytic family of matrices M (µ) annihilates g for all values of the parameter µ. Moreover, for µ = 0 we saw that M (µ) has maximal rank, that is its kernel is spanned by g. Since maximal rank is constant on a Zariski open subset of the parameter domain, we infer that dim kerM (µ) = 1 for all µ in C , except a discrete subset Σ.

We provide some details of the proof for the convenience of the general readership. Let H = {cg; c ∈ R} ⊥ be the sub-vector space in R r d orthogonal to ker(M(0)). Denote the restriction of M(µ) to H by M(µ) ∈ L(H, R r 2d ). Since the Hilbert space H is µ independent, the matrix-valued function µ → M(µ) inherits the same regularity as the function µ → M(µ), i.e., it is holomorphic on C . Then, as a direct consequence of Lemma 5.1, the restriction of M(0) to H, denoted by M(0) ∈ L(H, R r 2d ), is injective. Recall that a linear bounded operator is injective if and only if it has a left inverse [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]. Then, there exists a left inverse denoted by L 0 ∈ L(R r 2d , H), that only depends on ∂G satisfying

L 0 M(0) = I d ,
where I H is the identity matrix acting on H. Now, let

H -→ H T(µ) = L 0 M(µ).
Then, µ → T(µ) is holomorphic on C , and by construction it verifies T(0) = I H . Thus, we deduce from Steinberg Theorem that T(µ) is invertible everywhere on C except at a discrete set of values Σ [START_REF] Kato | Perturbation theory for linear operators[END_REF]. Hence, L 0 becomes the left inverse of the matrix M(µ) for all µ ∈ C \ Σ. Consequently, M(µ) is injective for all µ ∈ C \ Σ. Since M(µ) is the restriction of M(µ) to H which is the orthogonal space to the vector g in R r d , we obtain that ker(M(µ)) = {cg; c ∈ R} for all µ ∈ C \ Σ. Then, for µ ∈ C \ Σ, (12) implies p = cg for some real constant c. Using the normalization condition, we obtain p = g, and hence p = g for all λ ∈ C \ [-1/2, 1, 2] outside the set Σ := {λ ∈ C; λ -1 ∈ Σ}.

Remark 5.1. We propose here a direct proof of the uniqueness result in Theorem 3.1 in the case where all the GPTs of the algebraic domain G are known. In [START_REF] Ammari | Properties of the generalized polarization tensors[END_REF], the proof of uniqueness for general shapes is based on the relation between the far-field expansion and the Dirichlet-to-Neumann operator.

Let p e be the extension of the vector p ∈ R r d [x] by zero in the canonical basis x α , |α| > d. Assume that p ∈ R r d [x] satisfies the system [START_REF] Ammari | Expansion methods[END_REF],

then β M αβ (λ, G)p e β = 0, ∀α ∈ N 2 ,
and consequently,

∂G (λI -K * ∂G ) -1 [ν(x) • ∇x α ] p(y) dσ(y) = 0, ∀α ∈ N 2 .
Similarly, we have

∂G ν(y) • ∇y α (λI -K ∂G ) -1 [p(x)] dσ(y) = 0, ∀α ∈ N 2 ,
where K ∂G : H 1/2 (∂D) → H 1/2 (∂D) denotes the adjoint of K * ∂G . Since Γ(z -y) has the following expansion

Γ(z -y) = ∞ |α|=0 (-1) |α| α! ∂ α Γ(z)y α , y ∈ G, |z| → +∞, we deduce that ∂G ν(y) • ∇Γ(z -y)(λI -K ∂G ) -1 [p(x)] dσ(y) = 0, ∀z ∈ R 2 \ G,
which implies that

(λI -K ∂G ) -1 [p(x)] = 0 on ∂D.
Since |λ| > 1 2 , λI -K ∂G is invertible and so p vanishes completely on ∂G. Consequently, p = cg for some real constant c. Using the normalization condition, we obtain p = g. Following the discussion in Section 4, we need to add a supplementary criteria to be able to identify uniquely the domain G from its minimal polynomial g. Let Ω r be a bounded domain in R 2 , containing a ball of center zero and radius r > 0 large enough. Let G * 0 be the set of polynomial functions g ∈ G * such that there exists a unique Lipshitz algebraic domain G containing zero and with size one satisfying ∂G ⊂ {g = 0} ∩ Ω.

Corollary 5.1. Let G ∈ G * 0 , G ∈ G * with respectively ∂G and ∂ G of degree d. Let g(x) = |α|≤d g α x α and g(x) = |α|≤d g α x α be respectively polynomial functions that vanish respectively of the first-order on ∂G, and ∂ G satisfying I(∂G) = (g), g α * = 1, g(0) = 0, and I(∂ G) = ( g), g α * = 1, g(0) = 0. Let λ be fixed in C 0 such that λ / ∈ Σ, where the set Σ(∂G) is as defined in Theorem 5.1. Then, the following uniqueness result holds:

(M αβ (G, λ)) |α|≤2d,0<|β|≤d = (M αβ ( G, λ)) |α|≤2d,0<|β|≤d iff G = G. ( 13
)
Proof. The result is a direct consequence of Theorem 5.1. Since the generalized polarization tensors coincide, and λ / ∈ Σ, we can deduce from Theorem 5.1 that g = g. Since g ∈ G * 0 we finally obtain G = G.

Remark 5.2. From applications point of view, the assumption G ∈ G * 0 in Corollary 5.1, is somehow related to the fact that in the inverse problem of identifying small inclusions from boundary voltage measurements, the location and the convex hull are well determined [START_REF] Ammari | Expansion methods[END_REF]. Our method will allow the recovery of the shape up to a certain precision fixed by the highest order of the considered GPTs. The approach can be seen as an extension of the equivalent ellipse approach [START_REF] Ammari | Polarization and moment tensors with applications to inverse problems and effective medium theory[END_REF]. The assumption G ∈ G * 0 can be dropped when the regular locus of ∂G coincides with ∂G. For example, when G is a lemniscate with a large enough level set constant such that ∂G contains all the complex roots of its associated complex polynomial (for r large enough in (6)).

Remark 5.3. In Theorem 5.1, λ is used implicitly. However, if the domain G is sufficiently well approximated by G then one can also attempt to recover λ by solving the following minimization problem:

λ = argmin M αβ (λ, G) -M αβ ( λ, G) ; M αβ = 0.

Approximation by algebraic domains

For possibly non algebraic boundaries ∂G, we describe a simple procedure to compute a polynomial g whose level set {x : g(x) = 0} approximates ∂D. We expect better approximation to be found among higher degree polynomials. Domains enclosed by real algebraic curves (henceforth simply called algebraic domains) are dense, in Hausdorff metric among all planar domains. A very particular case is offered by domains surrounded by a smooth curve. They can be approximated by a sequence of algebraic domains. This observation turns algebraic curves into an efficient tool for describing shapes [START_REF] Fatemi | Sampling and reconstruction of shapes with algebraic boundaries[END_REF][START_REF] Keren | Describing complicated objects by implicit polynomials[END_REF][START_REF] Taubin | Parametrized families of polynomials for bounded algebraic curve and surface fitting[END_REF]. Note that an algebraic domain which in addition is the sub level set of a polynomial of degree d can be determined from its set of two-dimensional moments of order less than or equal to 3d [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF]. On a related topics, an exact reconstruction of quadrature domains for harmonic functions was proposed in [START_REF] Gustafsson | Reconstructing planar domains from their moments[END_REF] with the advantage of providing a potential type function (similar to the ubiquitous barrier method in global optimization), which detects the boundary of any complicated shape without having to make a choice among different potential chambers. More details about the approximation theory concepts related to this framework can be found in [START_REF] Gustafsson | Putinar Hyponormal Quantization of Planar Domains[END_REF].

Theorem 5.1 suggests a strategy to approximately recover information on the boundary ∂D when the latter is not algebraic. We will follow the approach developed in [START_REF] Lasserre | Algebraic-exponential Data Recovery from Moments[END_REF] with interior moments. By considering the GPTs (M αβ ) |α|≤2d,|β|≤d , one may compute the polynomial g ∈ R d [x] with coefficients g ∈ R r d such that g is the most suitable singular vector corresponding to the smallest in absolute value singular value of (M αβ ) |α|≤2d,|β|≤d .

Building on this, we also suggest an algorithm for shape recognition. The algorithm has three steps: (i) recovering g for some degree d polynomial; (ii) checking to see if the recovered polynomial has bounded level set, and (iii) accounting for scaling and rotations via minimization problem.

We stress that not all simple real algebraic sets in R 2 , such as a triangle, are the sublevel set of a single polynomial. The reader should be aware that the algorithms below identify the minimal polynomial g vanishing on the boundary of an admissible planar domain G, but by no means this implies G = {x ∈ R 2 ; g(x) > 0}.

Even worse, our numerical procedure of plotting the potential boundary of G is based on a curve selection process, which may not detect in a single shot all irreducible components of ∂G. The intricate details of amending these weak points of our numerical schemes and experiments will be addressed in a forthcoming article.

We start by showing how orthogonal transformations on the domain relate to linear operation on the underlying coefficients. We use this relation to define a minimization scheme for finding the similarity between reference and target polynomials. Start by writing the algebraic domain as follows:

G := {x ∈ R 2 : g(x) = x t [k] g 2 [k,k] x [k] + 2k-1 j=0 g [j] x j = 2k j=0 g [j] x [j] = 0},
where g is a polynomial of even degree d = 2k (necessary) written as a sum of polynomial forms (homogeneous) and the superscript t denotes the transpose. With the notation explained by Definition 6.1.

Let x t k := [x k 1 , x k-1 1 x 1 2 , ..., x 1 1 x k-1 2 , x k 2 ], g [k,
k] be a (k + 1) -by -(k + 1) matrix, and g [j] be a row vector of j + 1 real coefficients. Note that g [2k] represents the coefficients of the leading form and g [k,k] is its quadratic form [START_REF] Taubin | Parametrized families of polynomials for bounded algebraic curve and surface fitting[END_REF].

The ability to write any polynomial in this form is a consequence of Euler's theorem [START_REF] Taubin | Parametrized families of polynomials for bounded algebraic curve and surface fitting[END_REF]. This formulation is used in [START_REF] Taubin | Parametrized families of polynomials for bounded algebraic curve and surface fitting[END_REF] to prove the following results. Lemma 6.1. If g [k,k] is non-singular, then G is bounded and non-empty. Lemma 6.2. All odd degree forms have unbounded level sets.

In order to recover shapes, we must first define what it means for two shapes to be the same. We avoid the difficulty of defining a similarity measure and simply state that a shape should be invariant under rotations and scaling. This invariance takes the form of a matrix operation on the underlying coefficients. Let O s (2) := {A = sR with R being a rotation matrix and s > 0}. The following result holds. Lemma 6.3. Consider a transformation A = (a i,j ) i,j∈{1,2} ∈ O s (2). Let x = Ax. Then

x [d] = A [d] x [d]
and

A(G) = {x ∈ R 2 : g(x) = 2k j=0 g [j] A -1 [j] x [j] = 0}
with the (d + 1) × (d + 1) matrix A [d] having entries given by

(A [d] ) h,j≤d = j p=0 c p (h)d j-p (h) ; c j (h) = d -h j a d-h-j 1,1 a j 1,2 : d j (h) = h j a h-j 2,1 a j 2,2 .
Here, c j and d j are extended by zeros to be defined up to j ≤ d.

Proof. We write

x t [d] = [(a 1,1 x 1 + a 1,2 x 2 ) d , (a 1,1 x 1 + a 1,2 x 2 ) d-1 (a 2,1 x 1 + a 2,2 x 2 ), ..., (a 2,1 x 1 + a 2,2 x 2 ) d ].
For the h-th entry, we have the following:

(a 1,1 x 1 + a 1,2 x 2 ) d-h (a 2,1 x 1 + a 2,2 x 2 ) h = d-h j=0 d -h j a d-h-j 1,1 a j 1,2 x d-h-j 1 x j 2 h j=0 h j a h-j 2,1 a j 2,2 x h-j 1 x j 2 = x d-h 1 d-h j=0 c j y j x h 1 h j=0 d j y j = d j=0 j p=0 c p (h)d j-p (h) x d-j 1 x j 2 =: A [d],h x [d] ,
where

y j := x -j 1 x j 2 , c j (h) = d-h j a d-h-j 1,1
a j 1,2 and d j (h) = h j a h-j 2,1 a j 2,2 are extended by zeros to be defined up to j ≤ d.

With this result, we can first suggest a simple algorithm for matching GPTs to a predefined real algebraic shape. The goal is to recover the matrix A as it gives the relation between reference and observed shapes. Algorithm 6.1 Shape matching Input: The GPTs (M αβ ) |α|≤2d,|β|≤d and the coefficients g of the reference shape. Procedure:

• Solve g using Theorem 5.1;

• Check if Lemma 6.1 holds for g;

• Find A := argmin A∈Os(2) 2k j=0 ( g [j] -g [j] A [j] ) and match := 2k j=0 ( g [j] -g [j] A [j]
) . Output: A and match . Example 6.1. (Shape search real algebraic domains). The following 6 plots show real algebraic domains with degrees ranging from 6 to 2. The GPTs were calculated from algebraic domains and then the algebraic domains were recovered again via Theorem 5.1. The first shape is a reference shape, see Figure 6.1.

The coefficients are minimized under rotations and scaling. The algorithm correctly matches the second shape.

In the general case, we propose a shape reconstruction algorithm for not necessary algebraic domains. • Let β (1) , β (2) , . . . , β (r d ) be an enumeration of the β's;

• Let α (1) , α (2) , . . . , α (r 2d ) be an enumeration of the α's;

• Construct a matrix [L] 1≤i≤r 2d ,1≤j≤r d := M α (i) β (j) ; • Find the singular vector [ g (1) , . . . , g (r d ) ] t corresponding to the smallest in absolute value singular value of L; • Set g (j) = g β (j) . Output: The coefficients g of the reconstructed real algebraic shape. Example 6.2. (Approximating non-algebraic domains) As stated earlier, algebraic domains can approximate any planar domain. However, some shapes may require an infinite series of polynomials to be described. We approximate the shapes of a triangle, a diamond and a flower with one petal missing. The GPTs were calculated from these diametrically defined shapes which in turn was used to recover equivalentpolynomials, see Figure 6.2. Remark 6.1. Note that a higher degree polynomial may not always yield a better approximation. Example 6.3. (Recovering non-connected real algebraic domains) The proposed reconstruction method is powerful enough to reconstruct non-connected domains. Consider the polynomial given in [START_REF] Ammari | Enhancement of near cloaking using generalized polarization tensors vanishing structures: Part I: The conductivity problem[END_REF]. Below, we plot the real level sets of (5) for values -0.1, 0 and 0.1. On the 0.1-level two components of the domain are selected and their GPTs are computed. From these GPTs, a polynomial is recovered via Algorithm 6.2. The

Conclusion

In this paper, we have introduced a new tool for identifying shapes from finite numbers of their associated GPTs. We have shown in Theorem 5.1 and Corollary 5.1 that the knowledge of the r d × r 2d GPTs of a real algebraic domain having a boundary of degree d is sufficient to identify it. This is a very promising path since it is now possible to recover intricate shapes with the knowledge of only a finite number of GPTs. These results confirm that GPTs are good shape descriptors and can be harnessed to great effect by even simple algorithms. We believe that the number r d × r 2d of required GPTs can be dramatically reduced, and this will be the subject of a future work.

Figure 6 . 1 .

 61 Figure 6.1. Reconstruction of real algebraic shapes. Algorithm 6.2 Approximating general domains with algebraic ones Input: The GPTs (M αβ ) |α|≤2d,|β|≤d obtained from a non-algebraic domain. Procedure:• Let β(1) , β(2) , . . . , β (r d ) be an enumeration of the β's;• Let α(1) , α(2) , . . . , α (r 2d ) be an enumeration of the α's;• Construct a matrix [L] 1≤i≤r 2d ,1≤j≤r d := M α (i) β (j) ; • Find the singular vector [ g(1) , . . . , g (r d ) ] t corresponding to the smallest in absolute value singular value of L; • Set g (j) = g β (j) . Output: The coefficients g of the reconstructed real algebraic shape.

Figure 6 . 2 .

 62 Figure 6.2. Reconstruction of non-algebraic shapes.

Figure 6 . 3 .

 63 Figure 6.3. Reconstruction of non-connected domains.

The work of Faouzi Triki was supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde), and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).