Lionel Salmon
email: frsalmon@lcc-toulouse.fr

Gabor Molnar

Saioa Cobo Santamaria

Pascal Oulié

Michel Etienne

Tarik Mahfoud

Philippe Demont

Akira Eguchi

Haruo Watanabe

Koichiro Tanaka

Ga´bor Molna´r

Saioa Cobo

Pascal Oulie´

Hiroshi Watanabe

Azzedine Bousseksou

Re-investigation of the spin crossover phenomenon in the

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Coordination complexes containing various substituted poly(1-pyrazolyl)borate ligands have been intensively investigated since the pioneering work of Trofimenko. [START_REF] Trofimenko | Scorpionates: The Coordination Chemistry of Polypyrazolylborate Ligands[END_REF] Iron and cobalt derivatives, among several other possible complexes, have received particular attention because their spin state equilibrium is highly sensitive to ligand substitution. [START_REF] Long | [END_REF] The ferrous complex formed with the hydrotris(1-pyrazolyl)borate ligand, [Fe(HB(pz) 3 ) 2 ] (1), where pz = pyrazolyl, is one of the most studied complexes of this family (Fig. 1). In this thermochromic complex, the iron(II) ions undergo a thermallyinduced spin crossover between their 1 A 1 low-spin (LS) and 5 T 2 high-spin (HS) electronic configurations in the 300-450 K temperature range. This high temperature spin crossover (SCO) behavior was first revealed by optical spectroscopy 3 and confirmed later by magnetic susceptibility, Mo¨ssbauer and infrared spectroscopic studies. 4,5 Interestingly, the thermal spin transition curve recorded during the first heating differs drastically from the successive thermal cycles. Grandjean et al. 5 explained this rather unusual observation by shattering the initially microcrystalline sample into a fine powder, which they observed on a melting point stage during the first heating. Indeed, it is well-established that the spin crossover phenomenon is very sensitive to grinding of the sample. 6,7 These mechanical treatments induce various defects in the crystal lattice, leading to more and more gradual and incomplete transition curves. Moreover, it was reported in many instances that single-crystals of spin crossover complexes break apart when going through the spin transition. This phenomenon occurs due to the important volume change which accompanies these transitions. 8 However, crystal fracture leads in general only to slight changes in physical properties. In this context, we have found only one well-documented literature report on self-grinding phenomena in a spin crossover system. Miyazaki et al. 9 revealed that single-crystals of [Fe(DAPP)(abpt)](ClO 4 ) 2 (DAPP = bis(3-aminopropyl)(2-pyridylmethyl)amine and abpt = 4-amino-3,5-bis(pyridin-2-yl)-1,2,4triazole) gradually disintegrate upon thermal cycling, and this morphological change is accompanied by a continuous variation of the physical properties. The occurrence of such a violent self-grinding effect in 1 therefore appears rather surprising. For this reason, we repeated and extended previous studies on this compound by investigating its physical properties and crystal structure in a wide temperature range. These new data confirm most of the observations made by Grandjean et al., 5 but refute their interpretation of the observed magnetic behavior. In the course of this study, interesting and unexpected electrical properties of 1 were also been uncovered, of possible relevance for the technological application of this compound.

Experimental section Synthesis

Hydrotris(1-pyrazolyl)borate ligand 10 and the corresponding [Fe(HB(pz) 3 ) 2 ] iron(II) complex 5 were synthesized according to experimental procedures reported in the literature. The [Fe(HB(pz) 3 ) 2 ] complex was sublimed at 463 K in a vacuum (ca. 0.1 Torr) prior to use, and either a microcrystalline clear violet powder or small single-crystals (depending on the sublimation conditions) were obtained. Elemental analysis calculated for [Fe(HB(pz)

Physical properties

Magnetic susceptibility measurements were carried out at heating and cooling rates of 2 K min À1 in a 2 T magnetic field by means of a Quantum Design MPMS2 SQUID magnetometer. The experimental data were corrected for the diamagnetic contribution. AC conductivity data were recorded in the two-probe geometry using a Novocontrol BDS 4000 broadband dielectric spectrometer coupled to a Quatro Cryosystem. Frequency sweeps (10 À2 -10 6 Hz) were carried out isothermally in the 220-445 K temperature range. Diffuse reflectance measurements were performed using a halogen lamp, a heating and cooling stage (Linkam), and a Princeton Instruments monochromator and CCD camera system (InSight100A).

Thermal analysis

Differential thermal analysis and thermogravimetric (DTA-TG) data were acquired simultaneously using a SETARAM 92-16.18 thermal analyzer. Differential scanning calorimetry (DSC) analysis was carried out on a Netzsch DSC 204 instrument under helium purging gas (20 cm 3 min À1 ) at a heating/cooling rate of 10 K min À1 . Temperature and enthalpy (DH) were calibrated using the melting transition of standard materials (Hg, In, Sn).

Microscope observations

The morphology of samples was observed under an Olympus BX51 optical microscope equipped with a DP12 digital camera. Images of the crystals were recorded during the heating and cooling cycles through the window of a Linkam THMS600 variable temperature stage, and also at room temperature before and after a thermal cycle using a Â5 or Â20 objective lens, respectively.

X-Ray crystallography

A Panalytical MPD XPert Pro powder diffractometer and an Antoon Paar TTK 450 Chamber were used to acquire variable temperature powder diffractograms between room temperature and 453 K. The extraction of peak positions for indexing and cell refinement were performed with the fitting program DICVOL, available in the PC software package Highscore+ supplied by Panalytical. Single-crystal X-ray diffraction (XRD) data were collected at room temperature and 420 K with monochromatic Mo-K a radiation (l = 0.71073) using an Xcalibur (Oxford Diffraction) diffractometer. A total of 12 676 reflections were collected at room temperature, of which 3218 unique reflections were used for the structure determination.

The second collection at 420 K was formed from 13 117 measured reflections, 3411 of which independent reflections were used for the hypothesis and refinement. DIRDIF92 [START_REF] Beurskens | The DIRDIF Program System[END_REF] was used for the structure solutions, SHELXL-97 12 for the refinements and PLATON [START_REF] Spek | PLATON: A Multipurpose Crystallographic Tool[END_REF] for structure analysis. The final R factor (with I 4 2s(I)) had values of 0.0431 and 0.0501 for the 298 and 420 K measurements, respectively. Crystallographic data are given in Table 1. The temperature dependence of the unit cell parameters between 298 and 423 K was determined using a KAPPA-CCD-Enraf-Nonius diffractometer.

Results and discussion

Physical properties

Fig. 2 shows the temperature dependence of the product of the molar magnetic susceptibility (w M ) and the temperature (T) for a powder sample of 1. The magnetic properties of Fe[HB(pz) 3 ] 2 were reported earlier in refs. 4 and 5. The results we obtained are quite similar to these reports. In our case, the w M T product of 1 (''as-prepared'') exhibited a gradual increase from 0.07 cm 3 mol À1 K at 300 K to 1.21 cm 3 mol À1 K at 405 K, then an abrupt increase to 2.40 cm 3 mol À1 K at 410 K and finally a gradual increase to 2.80 cm 3 mol À1 K at 450 K. On the contrary, upon cooling, the w M T product gradually decreased to 0.197 cm 3 mol À1 K at 300 K. In agreement with the initial report of Hutchinson et al., 4 the magnetic susceptibility displayed an apparent hysteresis only for the first thermal cycle. For all subsequent cooling and heating treatments, the magnetic properties retraced the initial cooling curve and not the initial heating curve. The temperature at which the sharp and irreversible transition was observed in our sample is different somewhat from previous reports (410 K vs. 390 K 5 ). One should recognize, however, that the initial state (i.e. the freshly sublimated sample) of this transition is thermodynamically unstable, and obviously no characteristic transition temperature exists for such a system, since the transformation from the metastable to the stable phase depends not only on the temperature, but also on the time. The spin crossover in 1 is also accompanied by spectacular color changes (see ESIw). At room temperature, the ''as-prepared'' sample has a violet color, which fades when it is heated to 470 K. Remarkably, when the sample is cooled back to room temperature, its visual aspect appears very different from the initial one. At the end of the first cycle, the initially pale-violet fine microcrystalline powder becomes coarse, leading to a deeper color and a more brilliant aspect. We have also carried out a microscopic observation of these morphological changes, as shown in Fig. 3. The optical microscope clearly shows that the sample is composed of small, needle-like crystallites of a few mm in size before the first thermal cycle (Fig. 3a), while it reveals the presence of well-defined single-crystals having various sizes up to a few hundred mm after the first cycle (Fig. 3b). Indeed, the in situ observation during the first heating process (see ESIw) demonstrates clearly that at 430 K large single-crystals can already grow within a few minutes from the initially fine powder. This recrystallisation process is certainly related to the fact that around 430 K, the sublimation of the sample slowly begins. These observations are not in agreement with those of Grandjean et al., 5 who actually reported the opposite process, i.e., the disintegration of the sample at the end of the first thermal cycle. In order to obtain more quantitative details of the optical changes, we have acquired variable temperature diffuse reflectance spectra of 1 in the 400-900 nm range. Fig. 4(a) displays the reflectivity spectra at different temperatures during the first two thermal cycles. The spectra are plotted as Àlog(R sig /R ref ), where R sig /R ref is the normalized reflectance signal, since this quantity is almost proportional to the absorbance of the sample, provided that the scattering coefficient is wavelength-independent. The reflectance spectrum of the sample at room temperature is characterized by an intense absorption band centered around 530 nm-responsible for the violet color of the compound-which was assigned to the 1 A 1g -1 T 1g ligand-field absorption of the LS molecules. 14 When the temperature increases the intensity of this band decreases due to the transformation of the singlet LS ground state to the quintet HS state, leading to the bleaching of the samples. During the first heating cycle one can also observe a rather abrupt change of the reflectance signal around 420 K (Fig. 4(b)), which can be straightforwardly correlated with the discontinuous change observed in the magnetic measurements. Moreover, the first heating cycle differs significantly from the subsequent cooling and heating cycles, which nearly follow the same trace. These findings are in good agreement with the report of Grandjean et al. 5 However, Fig. 4(a) and 4(b) very clearly reveal that the reflectance of the ''as-prepared'' sample at 300 K (i.e., in the pure LS state) is not the same as that of the thermally cycled sample. The observed enhancement of the singlet absorption band is, of course, not related to any change of absorption coefficient, but occurs due to the change of the scattering coefficient, which can be traced back to the altered sample granularity (Fig. 3).

Besides the magnetic and optical properties, the SCO phenomenon is known to be accompanied by changes of the electrical properties as well. Actually, most SCO complexes are highly insulating, and pertinent information can be obtained by determining their (quasi-static) dielectric constant. 15 We therefore measured the complex impedance of a powder sample of 1 (placed between two gold-plated brass electrodes) over a large frequency and temperature range. Quite surprisingly we noticed that the conductivity of the sample was relatively high, B6 Â 10 À8 S m À1 , at room temperature. This observation was confirmed on two independently-synthesized batches, proving that it is not due to adventitious contamination but is an inherent property of this complex. Fig. 5(a) displays the frequency dependence of the real part, s 0 (f), of the complex conductivity s*(f), recorded at representative temperatures during the first heating cycle. At each temperature, one can distinguish low and high frequency regions. In both regions, the conductivity exhibits only a slight frequency dependence, but the high frequency part of the spectrum is characterized by a higher conductivity. The critical frequency, f c , which separates the low and high frequency regions, increases continuously with increasing temperature from ca. 1 Hz to 10 kHz between 243 and 363 K, then decreases to B1 Hz at 433 K. Fig. 5(b) shows the thermal variation of the real part of the complex conductivity, s 0 , extracted from the frequencyindependent region at 10 mHz. At this frequency, s 0 can be assimilated to the dc conductivity. The dc conductivity is strongly activated up to ca. 360 K, next starts to decrease slowly up to 415 K and then exhibits an abrupt drop. All other heating and cooling cycles follow a common trace that differs significantly from the first heating curve. Notably, the electrical conductivity (at a given temperature) is smaller by three orders of magnitude when compared to the first heating curve and displays no discontinuities. The conductivity data can be clearly correlated with the magnetic behavior. When the temperature increases, the conductivity of the LS form also increases, but above ca. 360 K, the LS -HS crossover counterbalances the effect of the thermal activation, which means that the HS form of 1 is more insulating than the LS form. Around 415 K, an irreversible transition occurs towards the thermodynamically stable form, in agreement with the magnetic and optical observations. This stable phase is significantly more insulating than the ''as-prepared'' sample, but the LS form also exhibits a higher conductivity than the HS form (at a given temperature). As far as the charge transport mechanism is concerned, in such a low mobility solid, the conductivity should be associated with a chargehopping process. The fact that f c exhibits a similar temperature dependence as the ac conductivity strongly suggests that this frequency corresponds to the hopping frequency. 16 These processes can be straightforwardly analyzed within the Jonscher or electric modulus formalisms, [START_REF] Jonscher | Dielectric Relaxation in Solids[END_REF] but such a detailed study is out of the scope of the present paper and the work will be published elsewhere.

The various physical property measurements (magnetic, optical and electric) corroborate the occurrence of an irreversible transition in 1 during the first heating cycle. However, our visual observations reveal that this transition is followed by a recrystallization of the sample at higher temperatures. This finding suggests that the change of the thermal spin crossover curve from a rather abrupt to a gradual one is not driven by a self-grinding process. Indeed, such an irreversible transition may also be explained by thermal degradation or by polymorphism. In the latter scenario, it is assumed that the as-prepared metastable polymorph is converted to the thermodynamically stable polymorph during the first heating process. In order to assess the validity of these hypotheses, we have carried out different thermal and crystallographic analyses on 1.

Thermal analysis

The DSC curves of a powder sample between 300 and 450 K are shown in Fig. 6. Two heating-cooling sequences were carried out; the first heating mode reveals a very broad endothermic peak corresponding to the gradual spin crossover phenomenon between ca. 340 and 410 K, and a subsequent well-defined endothermic peak around 439 K. The onset of the transition occurring at 430 K corresponds to the abrupt and irreversible change observed in the physical properties. Conversely, in the cooling mode, the DSC curve reveals a single very broad exothermic peak corresponding to the spin conversion from the HS to the LS state. The same broad peak is observed during the second heating-cooling cycle, but without the anomaly observed during the first heating. A simultaneous DTA-TG analysis (Fig. 7) clearly confirms that no mass change is associated with this endotherm. On the other hand, above 480 K, the DTA-TG analysis shows a reduction of sample mass, which corresponds to the beginning of the sublimation of the sample.

Crystal structures

Oliver et al. 18a and other groups 18b,c have described the room temperature (LS) structure of 1, but to our knowledge, no other polymorphs have been published until now. The crystal structures of a freshly obtained [Fe(HB(pz) 3 ) 2 ] single-crystal were resolved by X-ray crystallography at 298 K (LS state) and 420 K (LS + HS mixture) (Fig. 8 and Table 1). The X-ray analysis did not reveal any change in space group over the temperature range 298-420 K. Both structures belong to the monoclinic space group P2 1 /n (Z = 4). The symmetry of the molecule in the solid state is virtually D 3d , and the [FeN 6 ] octahedral coordination sphere is formed of six nitrogen atoms belonging to the pyrazolyl ligands. Selected bond lengths and angles are given in Table 2 andTable 3, respectively (see ESI for further XRD dataw). Since a change from the t 2g 6 LS state to the t 2g 4 e g 2 HS state is predicted to be accompanied by an expansion of the metal-ligand bond lengths from ca. 2.0 to 2.2 A ˚, accurate X-ray crystal analyses provide useful indications of the spin state of the iron(II) ion. In compound 1, each Fe-N bond is lengthened by approximately 5.5-6% upon increasing the temperature, the average Fe-N distances at 298 K and 420 K being 1.982(8) and 2.098(10) A ˚, respectively. The high temperature (420 K) values correspond to a ca. 1 : 1 mixture of HS and LS states, in reasonably good agreement with the magnetic data. Considering the unit cell parameters, as might be expected, the cell volume increases with temperature due to the combined effect of thermal dilatation and spin state change. This increase is accompanied by a rise in the value of the b and c parameters, whereas the a value decreases slightly (Table 4). Depending on the temperature, the bond angles around the nitrogen atoms bonded to the boron atoms do not vary in the same way; some of them decrease whereas others increase. It is worth noting that while the N-N-C angles do not change, the N-N-B angles increase slightly and the C-N-B angles decrease by about 1.5% between 298 to 420 K. This change is consistent with the variation in the Fe-B bond lengths; Fe-B2: 3.085 A ˚at 298 K and 3.154 A ˚at 420 K. Finally, an important point is that above ca. 420 K, the crystals systematically broke apart, in agreement with the observations of Grandjean et al. 5 This can be explained by the loss of cohesion upon increasing the temperature due to the rise in the free volume of the unit cell.

These XRD results indicate a polymorph form of the crystal, in comparison with the results of Oliver et al., 18a who determined at 289 K a different unit cell, characterized by a volume of 2240.3 A ˚3, which is significantly higher than the volume of 2181.0(3) A ˚3 obtained by us at 298 K. At this point, it is worth noting that polymorphism in connection with SCO phenomena has already been observed in certain pyrazolylborate complexes. 19 In order to elucidate the possible polymorphism in 1, we recorded powder X-ray diffractograms of an ''as-prepared'' powder sample between room temperature and 453 K (Fig. 9). At room temperature, the lattice parameters belong to the tetragonal system with a = b = 17.11(2) and c = 7.493(7) A ˚. Above 413 K, a phase transition clearly occurs, but up to 453 K, cell refinement was not possible due to the co-existence of at least two different phases. The variation of the cell parameters and the corresponding volume are reported in the ESIw; a gradual volume increase of ca. 4% between room temperature and 413 K was measured. The sample was then cooled back down to room temperature, but the observed diffraction pattern was clearly different from the initial one recorded at the same temperature. Indeed, the room temperature diffraction pattern obtained after a thermal cycle could be refined by similar lattice parameters as the single-crystals (monoclinic system; a = 9.89(1) A ˚, b = 17.00(2) A ˚, c = 12.87(1) A ˚, V = 2149.6(6) A ˚3 and b = 96.61(1)1), and the powder diffraction pattern generated from the single-crystal data appeared to be very similar to experimentally obtained powder patterns (see ESIw). Following this heating-cooling sequence, a few single-crystals could be isolated and investigated by X-ray crystallography. The structure and space group (P2 1 /n (Z = 4), a = 9.900( 2 These observations lead us to propose the following scenario for the thermal behavior of 1. A freshly sublimed sampledepending on the details of the sublimation process-may consist of either a fine powder or a coarse crystalline material, which is comprised of tetragonal or monoclinic polymorphs of 1, respectively. During the first heating sequence, the initially metastable LS tetragonal polymorph abruptly transforms to the thermodynamically stable HS monoclinic form at around 400-410 K (depending on the heating rate). If the sample is Table 3 Selected bond angles (1) for 1

T = 180 K T = 298 K T = 420 K
allowed to stand at high temperature (ca. 420-430 K) for several minutes, rather large single-crystals of this monoclinic polymorph can be grown. These crystals display the same structure as single-crystals obtained directly after sublimation. Apparently, this crystal growth occurs at the onset of sublimation of the sample. During successive cooling and heating cycles, the compound preserves its monoclinic structure, and LS to HS conversion gradually takes place in this lattice between room temperature and 450 K. However, around 420 K, the crystals break apart, presumably due to large lattice strains; this phenomenon does not significantly influence the material's physical properties. As a perspective, it would certainly be very interesting to solve the structure of the tetragonal polymorph from high quality powder XRD data in order to search for correlations between the crystal structure and the magnetic behavior of the two polymorphs, alongside the ideas proposed by Reger et al. 20

Conclusions

In this work, we have re-investigated the remarkable spin crossover properties of the complex [Fe(HB(pz) 3 ) 2 ]. For the most part, we have confirmed the observations previously published in the literature. However, our results do not support the proposition of Grandjean et al., 5 who explained the irreversibility of the abrupt spin transition observed during the first heating sequence as a self-grinding process. Our novel structural data reveal a rich polymorphism in this compound, and, on the basis of these findings, we propose a new interpretation of its magnetic behavior. In particular, we have associated the irreversible transition with a structural change between a metastable tetragonal and a stable monoclinic form. Additionally, we have revealed that solid [Fe(HB(pz) 3 ) 2 ] exhibits an electrical conductivity that is moderate on an absolute scale but relatively high within the family of spin crossover complexes. Even more importantly, the conductivity changes by 3-4 orders of magnitude when the material passes through the first (irreversible) transition. This property, together with the knowledge that this thermochromic compound is stable to light, air and water, as well as the fact that it can be deposited on surfaces by sublimation, suggests that it might be useful as an active element in ROM-type devices for various applications, alongside the initial ideas of Kahn and Martinez. [START_REF] Kahn | Table 4 Variation of the unit cell parameters of a single-crystal of 1 during the first heating process T/K 298 323[END_REF]

Fig. 1

 1 Fig.1The schematic structure of [Fe(HB(pz) 3 ) 2 ] (1).

Fig. 2

 2 Fig. 2 The temperature dependence of the w M T product of 1 upon two successive thermal cycles (first cycle: closed circles, second cycle: open triangles).

Fig. 3

 3 Fig. 3 Microscope images recorded at 295 K displaying the typical aspect of the same sample a: before and b: after the first thermal cycle.

Fig. 4

 4 Fig. 4 (a) Diffuse reflectance spectra of 1 recorded at various temperatures during the first two thermal cycles. (b) Temperature dependence of the reflectivity of 1 (recorded at 530 nm) upon two successive thermal cycles.

Fig. 5

 5 Fig. 5 (a) The frequency dependence of the ac conductivity of 1 recorded at selected temperatures between 243 and 433 K during the first heating cycle. (b) The temperature dependence of the ac conductivity of 1 (recorded at 10 mHz) after two successive thermal cycles (first cycle: open symbols, second cycle: closed symbols).

Fig. 6

 6 Fig. 6 DSC curves of 1 recorded in heating and cooling modes (two successive cycles; first cycle: full lines, second cycle: dashed lines).

Fig. 7

 7 Fig. 7 Simultaneous DTA-TG curves of 1 in the first heating sequence.

Fig. 8 a

 8 Fig. 8 a: An ORTEP view of the Fe[HB(pz) 3 ] 2 molecule (ellipsoids are drawn at the 50% probability level). b: The crystal packing at 298 K.

  .020(3) A ˚, c = 12.890(3) A ˚, V = 2157.8(8) A ˚3 and b = 96.53(3)1; T = 180 K) of these crystals was found to be comparable with those of crystals obtained by sublimation (see above).

  3 ) 2 ] (481.91 g mol À1 ): calc. C, 44.86; N, 34.88; H, 4.18; found C, 44.61; N 34.53; H, 4.09%.

Table 1

 1 Crystallographic data for 1w

		T = 180 K	T = 298 K	T = 420 K
	Chemical formula		C 18 H 20 N 12 B 2 Fe
	M w a/A ˚9.900(2) b/A ˚17.020(3) c/A ˚12.890(3)	481.9 9.9390(7) 17.0720(16) 12.9410(14)	9.8400(6) 17.6210(19) 13.3150(14)
	b (1), V/A ˚3	96.53(3) 2157.8(8)	96.658(7) 2181.0(3)	97.068(7) 2291.2(4)
	Z	4	4	4
	Space group r calc /g cm À3 m/cm À1	P2 1 /n 1.483 0.734	P 2 1 /n 1.468 0.726	P 2 1 /n 1.397 0.691
	R factor	0.027	0.0409	0.0501
	Weighted R factor	0.0673	0.0804	0.0642

Table 2

 2 Selected bond lengths (A ˚) for 1

		T = 180 K	T = 298 K	T = 420 K
	Fe(1)-N(12)	1.9774(17)	1.980(3)	2.095(4)
	Fe(1)-N(22)	1.9871(17)	1.987(3)	2.097(3)
	Fe(1)-N(32)	1.9789(16)	1.980(4)	2.099(5)
	Fe(1)-N(41)	1.9800(16)	1.974(4)	2.091(5)
	Fe(1)-N(51)	1.9833(16)	1.992(3)	2.113(3)
	Fe(1)-N(61)	1.9770(16)	1.976(3)	2.094(4)

Acknowledgements

The authors are grateful to L. Vendrier, L. Rechignat, J.-F. Meunier (LCC) and J. Jaud (CEMES) for technical assistance. This work was financially supported by the ANR NANOMOL project.


Kaïs Ammari
email: kais.ammari@fsm.rnu.tn

Souleymane Kadri Harouna
email: souleymane.kadri_harouna@univ-lr.fr

Souleymane Kadri Harouna

Boundary value problems initial condition identification by a wavelet based Galerkin method

Keywords: Control, Identification, Wavelet, Numerical Discretization 2010 MSC: 35R30, 35K05

In this study, we introduce two numerical methods to reconstruct, from posterior measurements of the solutions, the initial conditions in the one-dimensional heat conduction problem and the wave propagation. The methods use a wavelet based Galerkin method for the spatial discretization and a spectral decomposition of the basis stiffness matrix to get the numerical solution without time discretization as in classical approches. In fact, according to the proposed methods, setting an error bound and properly selecting the eigenvalues of the discretization system, we are able to reconstruct the initial conditions without exceeding the required error. The applicability and computational efficiency of the methods are investigated by solving some numerical examples with toy solutions and the experimental results confirm their accuracy.

Introduction

Initial condition reconstruction from the solution posterior measurements is part of inverse problems associated to system governed by partial differential equations [START_REF] Payne | Improperly Posed Problems in Partial Differential Equations[END_REF]. In the recent decades, plenty of works have been devoted to this problem due to its importance in many applications in engineering, see [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF][START_REF] Komornik | Fourier series in control theory[END_REF]. Most often, this question arises in practice in the non-destructive testing of materials, the development of the transient elastography technique, river pollution, population dynamics [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF][START_REF] Imperiale | Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains[END_REF], etc. Particularly, in the case of parabolic or hyperbolic model equations, the problem is ill posed and very difficult to tackle [START_REF] Lattès | Méthode de quasi-réversibilité et applications[END_REF][START_REF] Payne | Improperly Posed Problems in Partial Differential Equations[END_REF]. In this work, we aim to study the initial condition reconstruction 10 problem from a numerical point of view.

Although the problem is very old, among many others, Seidman's work [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF] is one of the pioneers in the numerical recovery of the initial temperature from measurements made at later times. In [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF], for the backwards heat equation stabilized by an a priori initial bound, which is a frequency filtering, an estimator is determined for the time intermediate values of the solution. This estimator is optimal with respect to the filtering bound and the observation accuracy. However, to construct this estimator observations in the whole spatial domain are needed and its limit when the time tends to zeros may not always exist.

Recently, using impulse control techniques, again by frequency filtering, Phung et al. [START_REF] Phung | Impulse output rapid stabilization for heat equations[END_REF] have provided a new optimal estimator that compensates the zeros limits problem. Moreover, in [START_REF] Phung | Impulse output rapid stabilization for heat equations[END_REF] observations are only made in a subdomain where the impulses act.

Another approche presented in [START_REF] Devore | Recovery of an Initial Temperature from Discrete Sampling[END_REF] uses measurements at a fixed position. Judiciously positioning the thermometer, DeVore and Zuazua [START_REF] Devore | Recovery of an Initial Temperature from Discrete Sampling[END_REF] show that the initial temperature profile can be completely determined by later time measurements.They also show that with sufficient number of measurements, the initial profile can be reconstructed up to a prescribed accuracy and they provide an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class.

In the context of wave equations and the initial profile recovery, a well documented study can be funded in [START_REF] Komornik | Fourier series in control theory[END_REF], especially in the continuous case and in the Hilbertian framework. For the numerical approaches, we refer to [START_REF] Imperiale | Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains[END_REF] and references therein. The particularity of the method presented in [START_REF] Imperiale | Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains[END_REF] is the use of an iterative back-and-forth nudging algorithm, which is a data assimilation technique [START_REF] Auroux | A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm[END_REF]. Without using gradient descent to minimize a cost function, the method allows to reconstruct the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. Globally, the algorithm converges with perfect measurements and remains robust to noise. But, the initial condition reconstruction on a bounded domain is possible only using artificial transparent boundary conditions for the exterior domain. Such an issue is necessary for many other methods [START_REF] Imperiale | Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains[END_REF].

The approaches of [START_REF] Devore | Recovery of an Initial Temperature from Discrete Sampling[END_REF][START_REF] Komornik | Fourier series in control theory[END_REF][START_REF] Phung | Impulse output rapid stabilization for heat equations[END_REF][START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF] use Fourier basis or the laplacian operator eigenfunction basis with physical boundary conditions. Then, for the numerical computation, it is very difficult to design an efficient numerical method that handle easily the non periodic boundary conditions. Finite element basis was suggested in [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF], but it is known that the condition number of the discretization matrice on this basis increases with the number of the points of the mesh.

Due to their localization both in space and frequency domains, wavelet basis is a good choice for the spatial discretization of partial differential equations.

Moreover, the two scale relation satisfied by the wavelet generator allows to compute the coefficients of the mass and the stiffness matrices of the basis without using a quadrature formula. Numerically, this is done by solving an eigenvalue problem [START_REF] Beylkin | On the representation of operator in bases of compactly supported wavelets[END_REF] and the spectrum of these matrices are bounded according to the resolution j. Thus, an explicit optimal conditioners for the stiffness matrix at each scale is known [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF]. Furthermore, wavelet based method has the advantage of improving the discretization accuracy by increasing the number of wavelet vanishing moments [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF]. To deal with physical boundary conditions, edge wavelets are used and this leads also to fast algorithms as in the periodic, see [START_REF] Harouna | Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related applications[END_REF][START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF].

Then, using a wavelet based Galerkin discretization, the model equations are transformed in ordinary differential equations systems and an exponential integrator method is used for their resolution [START_REF] Hochbruck | Exponential integrators[END_REF]. To compute the matrices exponential, a singular value decomposition (SDV) is applied [START_REF] Strang | Introduction to linear algebra[END_REF]. Therefore, the analytic expression of the solution is known and this avoids the use of an iterative procedure with is often constrained by a CFL condition [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]. From the solution formula, to reconstruct the initial data, it is only necessary to observed the wavelet coefficients of the solution at a certain time. This amounts to know the solution values at the spatial discretization grid points which is supposed to be provided in advance. Then, classical algorithms are used to project the observed solution onto the wavelet basis in order to get these coefficients [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF].

In the present work, two observation strategies are considered. The first one consists to observe a noiseless solution at all the grid points, similar to [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF] and the second one supposed that noisily wavelet coefficients of the solution are accessible. For the heat equation, one can uses the morse pseudo inversion method to get the backward exponential of the arising matrices, however, a lot of attention must be paid to the accuracy to recover the spatial discretization precision. In the present work, according to the fast decay of the exponential of negative numbers, to have an error in 0 < ϵ << 1, it is only necessary to keep the N greatest first eigenvalues of the system, with N ≈ -log(ϵ)/(νt) where ν is the diffusion coefficient and t is the observation time. Otherwise, to reconstruct the wave initial profile and speed, observation at two different times t 1 ̸ = t 2 are required and to overcome the problem of indetermination when t 1 and t 2 are very close, again the morse pseudo inversion method is used to solve the two systems. Although the approach is well known for the heat equation, to best of our knowledge, there is no an efficient numerical method that allows the reconstruction of the wave initial profile using Galerkin discretization and that is not based on the use of Fourier basis. Therefore, one of the contributions of this work is to provide an efficient and stable numerical method for initial conditions reconstruction, easy to implement taking into account non-periodic boundary conditions. The magnitude of the numerical error of the method is the same as the Galerkin discretization error on the exact solution. This error has polynomial decay that depends of the data smoothness and the wavelet basis number of vanishing moments: it is in O(N -s ) with N the number of grid points and s > 0 is at most the used wavelet number of vanishing moments.

The reminder of the paper is as follows. In Section 2, we introduce the wavelet basis construction and then we specify its properties useful to prove some error estimations. Section 3 is dedicated to the wavelet based Galerkin discretization. Following [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF] and using the Jackson and Bernstein inequalities, we give classical a priori error estimates on the wavelet based method for the heat and wave equations. The bounds are the same as for the finite element method, but easier to prove in this context. In Section 4 we present our numerical algorithm for the initial conditions reconstruction. Section 5 illustrates numerical experiments that support the theoretical results of the previous sections.

Wavelet basis

One simpler way to define the wavelet basis is to use the multiresolution analysis framework [START_REF] Daubechies | Ten lectures on Wavelets[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF]. Precisely, by definition a multiresolution analysis of L 2 (R) is a sequence of closed nested subspaces {V j } j∈Z that satisfy:

(i) V j ⊂ V j+1 , ∩ j∈Z V j = {0} and ∪ j∈Z V j = L 2 (R). (ii) ∀f ∈ L 2 (R) : f (x) ∈ V j ⇔ f (2x) ∈ V j+1 and f (x) ∈ V 0 ⇔ f (x -k) ∈ V 0 . (iii)There exists a function φ such that {φ(. -k)} k∈Z is a Riesz basis of V 0 .
The function φ is called scaling function and the refinement relation (ii) leads to:

φ(x) = k∈Z h k √ 2φ(2x -k), h k ∈ R. (1)
The mask {h k } k∈Z is called scaling function filter, its support coincides with the support of φ [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF] and the parameter j defines the resolution.

Relation (i) allows to write:

V j+1 = V j ⊕ W j , (2)
where W j denotes the complement of V j in V j+1 in a topological sense, thus it is not unique. Then, wavelet basis is defined as a basis of W j and using (ii), the wavelet generator ψ ∈ W 0 can be defined by its two-scale relation in V 1 :

ψ(x) = k∈Z g k √ 2φ(2x -k), g k ∈ R. (3)
Similarly, the mask {g k } k∈Z is called the wavelet filter and setting φ j,k = 2 j/2 φ(2 j x -k) and ψ j,k = 2 j/2 ψ(2 j x -k), we have:

V j = span{φ j,k : k ∈ Z} and W j = span{ψ j,k : k ∈ Z}.
To have an orthogonal wavelet basis, it suffices to set W j as the orthogonal complement of V j :

V j+1 = V j ⊕ W j , W j = V j+1 ∩ (V j ) ⊥ . (4)
In general, the space W j is defined as:

V j+1 = V j ⊕ W j and W j = V j+1 ∩ ( Ṽj ) ⊥ , (5)
where { Ṽj } j∈Z is another multiresolution analysis of L 2 (R) with scaling function φ and wavelet ψ. In this case, (V j , Ṽj ) j∈Z is referred as a biorthogonal multiresolution analysis of L 2 (R) [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF][START_REF] Daubechies | Ten lectures on Wavelets[END_REF].

The multiscale projection of a function f ∈ L 2 (R) onto V j and W j are defined respectively by:

P j (f ) = k∈Z ⟨f, φj,k ⟩φ j,k and Q j (f ) = k∈Z ⟨f, ψj,k ⟩ψ j,k . (6)
From ( 4) and ( 5), we have Q j (f ) = P j+1 (f ) -P j (f ) and the multiscale decomposition of f ∈ L 2 (R) reads:

f = P j (f ) + ℓ≥j Q ℓ (f ).
To compute numerically these projections, one uses the fast wavelet transform algorithm and only the masks {h k } k∈Z and {g k } k∈Z are needed [START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF]. Moreover, if the scaling function generator is smooth enough, for f ∈ H s (R) we have the following Jackson and Bernstein inequalities [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF][START_REF] Daubechies | Ten lectures on Wavelets[END_REF]:

∥P j (f ) -f ∥ L 2 (0,1) ≤ C2 -js ∥f ∥ H s (0,1) and ∥P j (f )∥ H s (0,1) ≤ C2 js ∥P j (f )∥ L 2 (0,1) , s > 0. ( 7
)
This wavelet basis construction extends easily to higher dimension using tensor product of the one-dimensional wavelet basis [START_REF] Daubechies | Ten lectures on Wavelets[END_REF]. In this work, we use the construction on the interval [0, 1] satisfying physical boundary conditions [START_REF] Harouna | Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related applications[END_REF][START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF]. Then, compactly support generators are used and due to the dilatation factor, a minimum resolution j min > 0 is fixed to ensure that the supports of all analyzing functions are contained in [0, 1], see [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF]. We can also impose interpolation property in the basis construction [START_REF] Daubechies | Ten lectures on Wavelets[END_REF][START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF][START_REF] Dubuc | Interpolation through an iterative scheme[END_REF], this will facilitate the initialization step of the fast wavelet transform algorithm and if not, quadrature formulas are used [START_REF] Harouna | Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related applications[END_REF][START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF].

Wavelet based Galerkin method for the heat and wave equations

In this section we briefly present the wavelet based Galerkin method for the spatial discretization of the wave and heat equations. The approach is similar to the finite element method [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF] and our objective is to show that we have at worst the same order of accuracy. For the sake of easy understanding, onedimensional equations are considered. The generalization to higher dimensions is straightforward.

Wavelet based Galerkin method for the wave equation

The initial-boundary value problem for the one-dimensional waves propagation that we considered is expressed as:

∂ 2 t u(t, x) = ν∂ 2 x u(t, x), u(0, x) = u 0 (x) and ∂ t u(0, x) = u 1 (x), (8)
where x ∈ [0, 1] and t ∈]0, T ] with T > 0. The wave is supposed to be fixed at boundaries and this implies homogeneous Dirichlet boundary condition: u(t, 0) = u(t, 1) = 0, see [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF].

Fourier's basis allows to compute an explicit numerical solution of ( 8), but such a basis is not compactly supported and the fast Fourier transform algorithm exists only for periodic function. To overcome this, instead of using the finite element method [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF], we will use a wavelet based Galerkin method and we won't need a time discretization. Moreover, the first equation of ( 8) is a linear hyperbolic equation and in the works of the literature, there is a lot of studies on the finite element based method applied to hyperbolic equations [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]. We aim here to recall some well known results on error estimations in the particular case of wavelet based method. Then, we assume that the wavelet basis and the exact solution have the required smoothness in space.

According to the boundary conditions, we also suppose that the approximation spaces (V j ) j≥jmin is a multi-resolution analysis of H 1 0 (0, 1) [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF]. Then, the wavelet based Galerkin method consists in looking for a solution u j ∈ V j of (8) in the following discrete form:

u j (t, x) = Nj k=1 ⟨u, ψj,k ⟩ψ j,k (x) = Nj k=1 d j,k (t)ψ j,k (x). (9)
Inserting 9 into (8) and using integration by part and the boundary condition, we get:

Nj k=1 d ′′ j,k (t) 1 0 ψ j,k (x)ψ j,m (x)dx + νd j,k (t) 1 0 ψ ′ j,k (x)ψ ′ j,m (x)dx = 0, ( 10
)
for m = 1, . . . , N j . Thus, the coefficients [d j,k ] solve this linear system:

A j d ′′ j,k (t) + R j [d j,k (t)] = 0, (11)
where A j and R j denote respectively the mass matrix and the stiffness matrix of the used wavelet basis:

[A j ] k,m = 1 0 ψ j,k (x)ψ j,m (x)dx and [R j ] k,m = ν 1 0 ψ ′ j,k (x)ψ ′ j,m (x)dx. ( 12
)
These matrices are symmetric and positive definite. In practice, one can compute their elements without numerical quadrature by solving an eigenvalue problem [START_REF] Beylkin | On the representation of operator in bases of compactly supported wavelets[END_REF] and optimal diagonal preconditioners exist for these matrices [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF].

One main property of the Galerkin method is that a priori error estimates exist and this allows to prove its convergence. Precisely, in the case of wavelet based method, we have the following proposition: Proposition 1. Let u j and u be solutions of (10) and (8), respectively. If the initial conditions u 0 (x) and u 1 (x) and the wavelet basis are regular enough, then we have:

∥u j -u∥ L 2 (0,1) ≤ C2 -js , (13)
for all j ≥ j min and s > 0.

Proof. Let e j (t, x) = u(t, x) -u j (t, x) be the spatial discretization error. By definition, we have:

∥e j ∥ L 2 (0,1) = ∥u -u j ∥ L 2 (0,1) ≤ ∥P j (u) -u j ∥ L 2 (0,1) + ∥P j (u) -u∥ L 2 (0,1) . ( 14
)
If the solution u ∈ H s (0, 1), using the Jackson inequality [START_REF] Daubechies | Ten lectures on Wavelets[END_REF], the second term of ( 14) is bounded as:

∥P j (u) -u∥ L 2 (0,1) ≤ C2 -js ∥u∥ H s (0,1) .
For the first term, we will use classical a posteriori error estimates for hyperbolic equations [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]. Indeed, let θ j = u j -P j (u) and taking ∂ t [u j -P j (u)] = ∂ t θ j as test function in ( 8) and [START_REF] Dubuc | Interpolation through an iterative scheme[END_REF], we obtain:

⟨∂ 2 t u -ν∂ 2 x u, ∂ t θ j ⟩ = ⟨∂ 2 t u j -ν∂ 2
x u j , ∂ t θ j ⟩, and this rewrites in:

⟨∂ 2 t [u -P j (u)] -ν∂ 2 x [u -P j (u)], ∂ t θ j ⟩ = ⟨∂ 2 t θ j -ν∂ 2 x θ j , ∂ t θ j ⟩. Integration by part gives: 1 2 d dt ∥∂ t θ j ∥ 2 L 2 (0,1) + ν∥∂ x θ j ∥ 2 L 2 (0,1) = ⟨∂ 2 t [u-P j (u)], ∂ t θ j ⟩+ν⟨∂ x [u-P j (u)], ∂ x ∂ t θ j ⟩ ≤ ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) ∥∂ t θ j ∥ L 2 (0,1) + ν∥∂ x [u -P j (u)]∥ L 2 (0,1) ∥∂ x ∂ t θ j ∥ L 2 (0,1) . (15
) After integration in t, we get:

∥∂ t θ j (t)∥ 2 L 2 (0,1) + ν∥∂ x θ j (t)∥ 2 L 2 (0,1) ≤ ∥∂ t θ j (0)∥ 2 L 2 (0,1) + ν∥∂ x θ j (0)∥ 2 L 2 (0,1) (16)
+ 2 t 0 ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) ∥∂ t θ j ∥ L 2 (0,1) + 2 t 0 ν∥∂ x [u -P j (u)]∥ L 2 (0,1) ∥∂ x ∂ t θ j ∥ L 2 (0,1) . Since 2 t 0 ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) ∥∂ t θ j ∥ L 2 (0,1) ≤ 2 t 0 ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) 2 + 1 2 max h∈[0,t] ∥∂ t θ j (h)∥ L 2 (0,1) 2 and 2 t 0 ν∥∂ x [u -P j (u)]∥ L 2 (0,1) ∥∂ x ∂ t θ j ∥ L 2 (0,1) ≤ 2 t 0 ν∥∂ x [u -P j (u)]∥ L 2 (0,1) 2 + ν 2 max h∈[0,t] ∥∂ x ∂ t θ j (h)∥ L 2 (0,1) 2 ,
we deduce that:

1 2 max h∈[0,t] ∥∂ t θ j (h)∥ 2 L 2 (0,1) + ν max h∈[0,t] ∥∂ x ∂ t θ j (h)∥ 2 L 2 (0,1) ≤ ∥∂ t θ j (0)∥ 2 L 2 (0,1) + ν∥∂ x θ j (0)∥ 2 L 2 (0,1) + 2 t 0 ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) 2 + 2 t 0 ν∥∂ x [u -P j (u)]∥ L 2 (0, 1) 2 .
From ( 16) we get:

∥∂ t θ j (t)∥ 2 L 2 (0,1) + ν∥∂ x θ j (t)∥ 2 L 2 (0,1) ≤ 2∥∂ t θ j (0)∥ 2 L 2 (0,1) + 2ν∥∂ x θ j (0)∥ 2 L 2 (0,1) + 4 t 0 ∥∂ 2 t [u -P j (u)]∥ L 2 (0,1) 2 + 4 t 0 ν∥∂ x [u -P j (u)]∥ L 2 (0,1) 2 .
Again using the Jackson estimation [START_REF] Daubechies | Ten lectures on Wavelets[END_REF], we get:

∥∂ t θ j (t)∥ 2 L 2 (0,1) + ν∥∂ x θ j (t)∥ 2 L 2 (0,1) ≤ 2∥∂ t θ j (0)∥ 2 L 2 (0,1) + 2ν∥∂ x θ j (0)∥ 2 L 2 (0,1) + C2 -2js t 0 ∥∂ 2 t u∥ H s (0,1) 2 + t 0 ∥u∥ H s+1 2 .
Poincaré's inequality and the previous estimates give:

∥θ j (t)∥ 2 L 2 (0,1) ≤ C∥∂ x θ j (t)∥ 2 L 2 (0,1) ≤ 2∥∂ t θ j (0)∥ 2 L 2 (0,1) +2ν∥∂ x θ j (0)∥ 2 L 2 (0,1) +C2 -2js . ( 17
) Finally, we have:

∥e j (t)∥ 2 L 2 (0,1) ≤ C ∥∂ t θ j (0)∥ 2 L 2 (0,1) + ν∥∂ x θ j (0)∥ 2 L 2 (0,1) + 2 -2js . ( 18
)
and

∥∂ t e j (t)∥ 2 L 2 (0,1) ≤ C ∥∂ t θ j (0)∥ 2 L 2 (0,1) + ν∥∂ x θ j (0)∥ 2 L 2 (0,1) + 2 -2js . (19)
Then, to have the desired order on the errors e j and ∂ t e j , which is in O(2 -js ), it suffices to take u j (0, x) = P j (u(0, x)) and ∂ t u j (0, x) = P j (∂ t u(0, x)). This will be assumed in the sequel.

Proposition 1 shows that the wavelet based method has the same error as the finite element method [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF] if the mesh size is about h ≈ 2 -j . Then, the objective in the next sections is to build a numerical method that preserves this order of accuracy in the reconstruction and identification of the initial condition u(0, x) and speed ∂ t u(0, x) associated to [START_REF] Dubuc | Interpolation through an iterative scheme[END_REF]. Before going to this, we will also summarize our spatial discretization of the heat equation.

Wavelet based Galerkin method for the heat equation

As done for the wave equation in previous section, the Cauchy's problem for the heat conduction that we deal with is:

∂ t u(t, x) = ν∂ 2 x u(t, x), u(0, x) = u 0 (x), (20)
where x ∈ [0, 1] and t ∈ [0, T ]. Again, homogeneous Dirichlet boundary conditions are assumed: u(t, 0) = u(t, 1) = 0. Other boundary conditions can be used without impacting the method as we will see.

Similarly, the wavelet based Galerkin method for the heat equation consists in searching a solution of (20) denotes u j ∈ V j in the following form:

u j (t, x) = Nj k=1 d j,k (t)ψ j,k (x). (21)
Then, it is easy to see that u j is a solution for the following variational problem:

Nj k=1 d ′ j,k (t) 1 0 ψ j,k (x)ψ j,m (x)dx + νd j,k (t) 1 0 ψ ′ j,k (x)ψ ′ j,m (x)dx = 0, ( 22
)
for m = 1, . . . , N j .

In this case, the computation of u j is thus reduced to the computation of the wavelet coefficients (d j,k ). According to [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF], these coefficients are solution of the following one order differential system:

A j d ′ j,k (t) + R j [d j,k (t)] = 0, (23)
where A j and R j are respectively the mass matrix and the stiffness matrix of the used wavelet basis. Moreover, the spatial discretization error satisfies:

Proposition 2. Let u j and u be solutions of (20) and [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF], respectively. If the initial conditions u 0 (x) and the wavelet basis are regular enough, then we have:

∥u j -u∥ L 2 (0,1) ≤ C2 -js , (24)
for all j ≥ j min and s > 0.

Proof. Let e j (t, x) = u(t, x) -u j (t, x) be the discretization error. Then, we 195 have:

∥e j ∥ L 2 (0,1) = ∥u -u j ∥ L 2 (0,1) ≤ ∥P j (u) -u j ∥ L 2 (0,1) + ∥P j (u) -u∥ L 2 (0,1) . ( 25
)
Using the Jackson inequality [START_REF] Daubechies | Ten lectures on Wavelets[END_REF], we see that the second term in the right hand of ( 25) is bounded by C2 -js for a solution u ∈ H s (0, 1). On the other hand, since:

⟨∂ t u -ν∂ 2 x u, ψ j,m ⟩ = ⟨∂ t u j -ν∂ 2 x u j , ψ j,m ⟩, ∀ ψ j,m ∈ V j , (26)
we have:

⟨∂ t [u -P j (u)] -ν∂ 2 x [u -P j (u)], ψ j,m ⟩ = ⟨∂ t [u j -P j (u)] -ν∂ 2 x [u j -P j (u)], ψ j,m ⟩, ∀ ψ j,m ∈ V j . Denoting θ j (t, x) = u j (t, x) -P j (u(t, x)
) and replacing ψ j,m by θ j in 26, we get:

1 2 d dt ∥θ j (t)∥ 2 L 2 (0,1) + ν 1 0 |∂ x θ j | 2 = ⟨∂ t [u -P j (u)] -ν∂ 2 x [u -P j (u)], θ j ⟩ ≤ ∥∂ t [u -P j (u)]∥ L 2 (0,1) ∥θ j ∥ L 2 (0,1) + ν∥∂ x [u -P j (u)]∥ L 2 (0,1) ∥∂ x θ j ∥ L 2 (0,1) ≤ C∥∂ t [u-P j (u)]∥ L 2 (0,1) ∥∂ x θ j ∥ L 2 (0,1) +ν∥∂ x [u-P j (u)]∥ 2 L 2 (0,1) + ν 4 ∥∂ x θ j ∥ 2 L 2 (0,1) ≤ C 2 ν ∥∂ t [u -P j (u)]∥ 2 L 2 (0,1) + ν∥∂ x [u -P j (u)]∥ 2 L 2 (0,1) + ν 2 ∥∂ x θ j ∥ 2 L 2 (0,1)
, by using the Cauchy-Schwarz inequality and Poincaré's inequality. Therefore:

1 2 d dt ∥θ j (t)∥ 2 L 2 (0,1) + ν 2 ∥∂ x θ j ∥ 2 L 2 (0,1) ≤ C 2 ν ∥∂ t [u -P j (u)]∥ 2 L 2 (0,1) + ν∥∂ x u -∂ x P j (u)∥ 2 L 2 (0,1) ≤ C2 -2js ∥∂ t u∥ 2 H s (0,1) + C2 -2js ∥u∥ 2 H s+1 .
Integration over time gives:

∥θ j (t)∥ 2 L 2 (0,1) + ν t 0 ∥∂ x θ j ∥ 2 L 2 (0,1) ≤ ∥θ j (0)∥ 2 L 2 (0,1) + C2 -2js t 0 ∥∂ t u∥ 2 H s (0,1) + ∥u∥ 2 H s+1 . (27)
Finally, we have:

∥e j (t)∥ L 2 (0,1) ≤ C ∥θ j (0)∥ L 2 (0,1) + 2 -js . ( 28
)
Thus to have e j in O(2 -js ), it suffices to take u j (0, x) = P j (u(0, x)) and this will be assumed in the sequel.

Likewise, the error bound given by Proposition 2 is the same as the finite element method [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]. We will now focus on the reconstruction and identification of initial conditions, with the main objective of having methods whose errors do not exceed those given by Proposition 1 and Proposition 2.

Initial conditions reconstruction

In this section, we present our approach for identifying the initial conditions and then study its numerical accuracy. We start with the wave equation problem.

Wave equation initial conditions identification

Most often, the problem encountered in the wave propagation control is to know the initial conditions. Thus we propose in this section a method that allows to identify the wavelet coefficients of the initial conditions u j (0, x) = P j (u(0, x)) and ∂ t u j (0, x) = P j (∂ t u(0, x)), where u j is the numerical solution of the wavelet based Galerkin method system defined in [START_REF] Dubuc | Interpolation through an iterative scheme[END_REF]. To do this, we assume that we have as data, measurements of the solution on the sptatial grid points and at two different moments 0 < t 1 < t 2 .

In the common approaches, the numerical resolution of (11) uses a time discretization schemes [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]. This is necessary only in the nonlinear case or if the system cannot be diagonalized easily. Then, as we have a linear equations, we will use here a singular value decomposition (SVD) [START_REF] Strang | Introduction to linear algebra[END_REF] in order to transform the system in a diagonal one and thus to compute its solution as in Fourier domain. In the general case, where biorthogonal wavelet basis is used, the A j is not the identity matrix. Thus, the first step of the method consists in transforming [START_REF] Dubuc | Interpolation through an iterative scheme[END_REF]

in d ′′ j,k (t) + A -1 j R j [d j,k (t)] = 0. ( 29
)
The matrix M j = A -1 j R j is symmetric and positive (product of two symmetric and positive matrices). Using the SVD, we obtain D j = C -1 j M j C j , where D j is a diagonal matrix of eigen values of M j and C j is an orthogonal change of basis matrix. Then, setting :

dj,k = C -1 j [d j,k ], (30)
it easy to transform (29) in:

d′′ j,k (t) + D j dj,k (t) = 0. ( 31
)
System (31) is diagonal, then for each k, we have:

d′′ j,k (t) + D j,k dj,k (t) = 0, with D j,k = [D j ] k,k > 0. ( 32
)
According to the differential equations theory, the solution of ( 31) is given by:

dj,k (t) = cos(t D j,k ) dj,k (0) + 1 D j,k sin(t D j,k ) d′ j,k (0). ( 33
)
Relation (33) gives the wavelet coefficients of the Galerkin method solution u j without using a time discretization. Moreover, if we know the value of dj,k (t) at two different times, from (33) we can identify the initial conditions coefficients. Precisely, let us suppose that we have two solutions of (33

) denoted [ dj,k (t 1 )] ̸ = [ dj,k (t 2 )]: dj,k (t 1 ) = cos(t 1 D j,k ) dj,k (0) + 1 D j,k sin(t 1 D j,k ) d′ j,k (0) and dj,k (t 2 ) = cos(t 2 D j,k ) dj,k (0) + 1 D j,k sin(t 2 D j,k ) d′ j,k (0).
Then, if

|∆| = 1 D j,k cos(t 1 D j,k ) sin(t 2 D j,k ) -cos(t 2 D j,k ) sin(t 1 D j,k ) = 1 D j,k sin (t 2 -t 1 ) D j,k ̸ = 0,
a simple calculation gives:

dj,k (0) = 1 |∆| D j,k sin(t 2 D j,k ) dj,k (t 1 ) -sin(t 1 D j,k ) dj,k (t 2 ) (34) and d′ j,k (0) = 1 |∆| -cos(t 2 D j,k ) dj,k (t 1 ) + cos(t 1 D j,k ) dj,k (t 2 ) . ( 35
)
Now we define as reconstructed initial conditions, the conditions whose wavelet coefficients are defined by ( 34) and ( 35) where [ dj,k (t 1 )] ̸ = [ dj,k (t 2 )] are the wavelet coefficients of the provided data. Therefore, in terms of numerical error, 220 the reconstructed initial conditions ūj (0) and ū′ j (0) from dj,k (0) and d′ j,k (0) satisfy: Proposition 3. Let ūj (0) and ū′ j (0) be the reconstructed initial conditions from the wavelet coefficients defined by (34) and (35). Then, we have:

∥ū j (0) -u 0 ∥ L 2 (0,1) ≤ C2 -js and ∥ū ′ j (0) -u 1 ∥ L 2 (0,1) ≤ C2 -js . ( 36
)
Proof. The computation of coefficients dj,k (0) and d′ j,k (0) using ( 34) and ( 35) is a linear system resolution:

dj,k (0) d′ j,k (0) = 1 |∆| 1 √ D j,k sin(t 2 D j,k ) -1 √ D j,k sin(t 1 D j,k ) -cos(t 2 D j,k ) cos(t 1 D j,k ) dj,k (t 1 ) dj,k (t 2 ) .
(37) which can be written as:

dj,k (0) d′ j,k (0) = F (t 1 , t 2 ) u j (t 1 ) u j (t 2 ) , ( 38
)
where F (t 1 , t 2 ) is the matrix of the operator that maps

(u j (t 1 ), u j (t 2 )) to (d j,k (0), d ′ j,k (0))
as defined by (37). Denoting by

dj,k (0) d′ j,k (0) = F (t 1 , t 2 ) P j (u(t 1 )) P j (u(t 2 ))) , (39)
we have:

dj,k (0) -dj,k (0) d′ j,k (0) -d′ j,k (0) ℓ2 ≤ ∥F (t 1 , t 2 )∥ P j (u(t 1 )) -u j (t 1 ) P j (u(t 2 ))) -u j (t 2 ) L 2 (0,1) . (40)
Since from Proposition 1, we have:

∥u j (t) -P j (u(t))∥ L 2 (0,1) ≤ C2 -js , we get: ∥ dj,k (0) -dj,k (0)∥ ℓ2 ≤ C∥F (t 1 , t 2 )∥2 -js and ∥ d′ j,k (0) -d′ j,k (0)∥ ℓ2 ≤ C∥F (t 1 , t 2 )∥2 -js . ( 41
)
Choosing appropriately the times t 1 and t 2 , one can get ∥F (t 1 , t 2 )∥ bounded independently of j, thus the proposition is proved.

From Proposition 3, we see that our approach allows to reconstruct the initial conditions of the wave propagation problem with a numerical error of the same rate as the spatial discretization error. We will now study the reconstruction error for the heat equation.

Heat conduction initial condition identification

Heat conduction is a diffusion process and reversing a diffusion process is very difficult, most often impossible. For the linear problem [START_REF] Payne | Improperly Posed Problems in Partial Differential Equations[END_REF], we aim in this section to provide a numerical method to reconstruct the values of the coefficients d j,k (0) from measurements of the solution u(t, x) at grid points, up to a prescribed accuracy. For easy understanding, we will use the same notations as in the previous section.

Similarly, to compute the solution u j wavelet coefficients d j,k (t), we consider the Galerkin method ordinary differential equation system [START_REF] Shukla | Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem[END_REF] associated to [START_REF] Payne | Improperly Posed Problems in Partial Differential Equations[END_REF] and started from d j,k (0):

     A j d ′ j,k (t) + R j [d j,k (t)] = 0, [d j,k (0)] = [P j (u(0))] . (42)
Then, the solution of ( 42) is given by:

d j,k (t) = e -tMj d j,k (0), with M j = A -1 j R j . (43)
Therefore, using (43) and for all t > 0, it is easy to infer that:

d j,k (0) = e tMj d j,k (t). ( 44
)
Making the change of variable

dj,k = C -1 j [d j,k ], (45) relation (44) becomes dj,k (0)
= e tD j,k dj,k (t) with

D j = C -1 j M j C j . (46)
Since D j,k are positive and tend to infinity with j, the relation (46) is not useful in practice for large j. Then, to control the instability due to the inversion of the large eigenvalues, we propose here to replace relation (44) by:

[ dj,k (0)] = D ϵ (t)[ dj,k (t)], (47)
where, for ϵ > 0, the matrix D ϵ (t) is defined by:

D ϵ (t) = D N (t) 1+ϵ ϵ I Nj -N , (48)
with

D N (t) =    e tDj,1 . . . e tD j, N    and 1 + ϵ ϵ I Nj -N =    1+ϵ ϵ . . . 1+ϵ ϵ    .
Thus, taking ϵ = e -tD j, N , we deduce that:

[ dj,k (0)] -[ dj,k (0)] 2 ℓ2 = [ dj,k (0)] -D ϵ (t)[ dj,k (t)] 2 ℓ2 = Nj k= N +1 1 + ϵ ϵ e -tD j,k -1 2 |d j,k (0)| 2 ,
and since e -tD j,k ≤ ϵ for k ≥ N , we have:

1 + ϵ ϵ e -tD j,k -1 ≤ 1 + ϵ ϵ ϵ -1 = ϵ.
This leads to:

240 ∥[ dj,k (0)] -[ dj,k (0)]∥ 2 ℓ2 ≤ Nj k= N +1 ϵ 2 |d j,k (0)| 2 ≤ ϵ 2 ∥[ dj,k (0)]∥ 2 ℓ2 . (49)
According to (49), we see that relation ( 47) is an approximation of the coefficients dj,k (0) with an error order close to ϵ.

In practice, the coefficients d j,k (t) are computed from some measurements of the temperature u(t, x). If the Deslauriers-Dubuc interpolating wavelet basis [START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF][START_REF] Dubuc | Interpolation through an iterative scheme[END_REF] is used in the discretization of the solution u, we have:

u(t, x) = P j (u(t, x)) + ℓ≥j Q ℓ (u(t, x)), (50)
with

P j (u(t, x)) = Nj k=1 c j,k (t)φ j,k (x). (51)
Then, for 1 ≤ m ≤ N j , we have:

u(t, m/2 j ) = P j (u(t n , m/2 j )) = Nj k=1 c j,k (t)φ j,k (m/2 j ) = c j,m (t n ), (52)
From (52), setting d j,k (t) = c j,k (t), one can compute d j,k (0) according to (44) and (47). In all the cases, if the wavelet basis generator is regular, the error on the reconstructed initial condition ūj (0) is the same as the Galerkin method error:

Proposition 4. Let ūj (0) be the initial condition reconstructed from the wavelet coefficients given by (47). Then, we have:

∥ū j (0) -u 0 ∥ L 2 (0,1) ≤ C2 -js . (53)
Proof. Let us denote by dj,k (0) the wavelet coefficients of the initial condition computed from the data P j (u(t, x)) using (47):

[ dj,k (0)] = D ϵ (t)[P j (u(t, x))].
Then, we have:

∥[ dj,k (0)] -[ dj,k (0)]∥ ℓ2 = ∥[ dj,k (0)] -D ϵ (t)[ dj,k (t)] + D ϵ (t)[ dj,k (t)] -[ dj,k (0)]∥ ℓ2 ≤ ∥[ dj,k (0)] -D ϵ (t)[ dj,k (t)]∥ ℓ2 + ∥D ϵ (t)[ dj,k (t)] -[ dj,k (0)]∥ ℓ2 ,
where [ dj,k (t)] are the coefficients associated to u j (t). Using (49), the term ∥[ dj,k (0)] -D ϵ (t)[ dj,k (t)]∥ ℓ2 is bounded as follows:

∥[ dj,k (0)] -D ϵ (t)[ dj,k (t)]∥ ℓ2 ≤ ϵ∥[ dj,k (0)]∥ ℓ2 .
On the other hand, using the estimation of Proposition 2 we have:

∥D ϵ (t)[ dj,k (t)] -[ dj,k (0)]∥ ℓ2 = ∥D ϵ (t)[u j (t)] -D ϵ (t)[P j (u(t, x))]∥ ℓ2 ≤ ∥D ϵ (t)∥∥u j (t) -P j (u(t, x))∥ L 2 (0,1)
≤ C∥D ϵ (t)∥2 -js .

As 0 < ϵ < 1 and using the definition of D ϵ (t), it is easy to see that:

∥D ϵ (t)∥ ≤ 1 + ϵ ϵ .
Thus, for a suitable constant C 1 > 0, we obtain:

∥[ dj,k (0)] -[ dj,k (0)]∥ ℓ2 ≤ C 1 (ϵ + 1 + ϵ ϵ 2 -js ). (54)
Therefore, if j ≥ -

log( ϵ 2 C 1 (1+ϵ) ) s log(2)
, from (54) we get:

∥[ dj,k (0)] -[ dj,k (0)]∥ ℓ2 ≤ 2C 1 ϵ. (55)
Due to (55), to get the statement of the proposition, it suffices to take:

j ≥ max    - log(ϵ) s log(2) , - log( ϵ 2 C1(1+ϵ) ) s log(2)    . ( 56
)
We will now check, from numerical experiments, if these error bounds are achieved in practice.

Numerical results

This section is devoted to the presentation of numerical results to evaluate the performance and efficiency of the proposed identification methods. All the simulations are executed with MATLAB and the source codes are available on request. The used wavelet basis generator is the Daubechies orthogonal generator with 4 vanishing moments [START_REF] Daubechies | Ten lectures on Wavelets[END_REF]. It is known that the autocorrelation of this generator will give the Deslauriers-Dubuc generator that reproduces polynomials up to degree 4 [START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF][START_REF] Dubuc | Interpolation through an iterative scheme[END_REF]. Boundary conditions are incorporated in the wavelet basis following [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF].

For the wave propagation problem [START_REF] Devore | Recovery of an Initial Temperature from Discrete Sampling[END_REF], as analytical solution we took the function u defined by: u(t, x) = cos(2πt + 2) sin(2πx).

(57)

The initial conditions of this solution are:

u 0 (x) = cos(2) sin(2πx), u 1 (x) = -2π sin(2) sin(2πx)
and the diffusion coefficient is set to ν = 1. Likewise, the exact solution of (20) that we used is defined by:

u(t, x) = e -t sin(2πx), (58)
with the initial condition u 0 (x) = sin(2πx) and diffusion coefficient set to ν = 1/4π 2 . These solutions have the advantage to be considered with homogeneous Dirichlet boundary condition or with periodic boundary condition.

The simulation final time is T = π for the two experiments.

We first investigate the decaying rate of the numerical error of the wavelet based Galerkin method that provides schemes (31) and (43). Since the theoretical order are given by the Proposition 1 and Proposition 2, our objective is to verify if these orders are the same for the numerical solution. Figure 1 shows the plot of the norm of the relative errors on the solutions (57) and (58), according to the number of grid points dim(V j ) = N j ≈ 2 j , where j is the maximal space resolution. The solutions (57) and (58) are C ∞ , then the slopes of the curves obtained correspond to the maximum regularity of the wavelet basis, here we have s = r = 4. This is in good agreement with the theoretical results of Section 3.

Next, we we study the initial conditions reconstruction errors in the case of noiseless data. As given data, measurements of the solution at grid points, we used solution (57) taken at the times t 1 = T /2 and t 2 = T for the wave propagation problem and solution (58) taken at the simulation final time for the heat equation. In both cases, we take the solutions provided by the Galerkin method. This makes that only the error of our reconstruction algorithm is dominant. The parameter ϵ defined in (48) has been set to ϵ = 10 -5 and since s = 4, this requires j ≥ 4.1524 to satisfy the bound given by (56). Thus, the smaller epsilon is, the larger j min must be: we used j min = 5 in all the simulations. For comparison purpose, we plot the same relative errors norm as done to evaluate the Galerkin discretization performance. Figure 2 and Figure 3 show these errors in the case of wave equation and Figure 4 and Figure 2 show these errors in the case of heat equation. Again, the expected decay rates are achieved, which confirms the results of Proposition 3 and Proposition 4. In addition, the choice of ϵ allows to prevent the limited floating point precision in the numerical computation of the matrices exponential for large j. Figure 3 and Figure 5 show the residual errors on the reconstructed initial conditions at grid points k/2 j for k = 0 : 2 j . Due to the non-symmetry of the Daubechies's filters, this error are much bigger at boundary x = 1 than at x = 0. This is a known drawback [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF].

Remark 1.

As can be seen, adding terms that not depending on the wavelets coefficients cannot affect the steps of the resolution of the systems (34), ( 35) and (42).

Then our approach works for equations with a second member of type:

∂ 2 t u(t, x) = ν∂ 2 x u(t, x) + f (t, x) and ∂ t u(t, x) = ν∂ 2 x u(t, x) + g(t, x).
In this case, we will have a time discretization error of the used numerical method to computed the integral of the added new terms.

Finally, we tested the method on noisy data. Precisely, we added to our data an artificial white Gaussian noise with an amplitude 1000 times greater than the MATLAB floating-point relative accuracy which is around 2 × 10 -16 for the double precision floating point format. Similarly, Figure 6 and Figure 7 show these errors in the case of wave equation and Figure 8 and Figure 6 show these errors in the case of heat equation. The residual errors at grid point are plotted on Figure 7 and Figure 9, respectively. As the numerical experiments point out, these errors are much larger with high amplitude noise, especially for the reversion of the diffusion problem. Then, the method converges only if the noise amplitude is of the order the machine floating-point precision as predicted 310 by [START_REF] Seidman | Optimal Filtering for the Backward Heat Equation[END_REF].

Conclusion

We have presented a numerical method to reconstruct the initial conditions of systems arising from the Galerkin method applied to the wave equation and the heat equation. Singular value decomposition is applied to diagonalize this 315 system and thus solve the equation without time discretization errors. The approach does not generate more errors than a classical Galerkin method and it is simple and easy to implement. The extension to higher dimensions is immediate. However, the method is limited only to the linear case. In the general non linear case, we do not have the exact expression of the solutions.

The method assumes that the data for the reconstruction are given on all the grid points. It would then be interesting to see if we can do the reconstruction with sparse measurements at selected positions or using the non-linear approximation of dense measurements. This will be the subject of future work.

Figure 1 :

 1 Figure 1: Norms of the relative errors on the exact solution (57) (upper two figures) and on the exact solution (58) (bottom two figures). Daubechies orthogonal generator with r = 4.

Figure 2 :

 2 Figure 2: Relative errors norm on the reconstructed initial conditions u0 and u1 associated to (57). Daubechies orthogonal generator with r = 4.

Figure 3 :

 3 Figure 3: Residual errors at grid points for the reconstructed initial conditions u0 and u1 associated to (57). Daubechies orthogonal generator with r = 4.

Figure 4 :

 4 Figure 4: Relative errors norms on the reconstructed initial conditions u0 associated to (58). Daubechies orthogonal generator with r = 4.

Figure 5 :

 5 Figure 5: Residual errors at grid points for the reconstructed initial conditions u0 associated to (58). Daubechies orthogonal generator with r = 4.

Figure 6 :

 6 Figure 6: Relative errors norm on the reconstructed initial conditions u0 and u1 associated to (57) and from noisy data. Daubechies orthogonal generator with r = 4.

Figure 7 :

 7 Figure 7: Residual errors at grid points on the reconstructed initial conditions u0 and u1 associated to (57) and from noisy data. Daubechies orthogonal generator with r = 4.

Figure 8 : 7 Figure 9 :

 879 Figure 8: Relative errors norms on the reconstructed initial conditions u0 associated to (58) and from noisy data. Daubechies orthogonal generator with r = 4.