two sided generalized hyperbolic qr factorization and its perturbation analysis abstract we first propose the two sided generalized hyperbolic qr factorization of a matrix pair which is a generalization of the generalized qr factorization more explicitly two orthogonal matrices in the generalized qr factorization are replaced with two corresponding j orthogonal matrices where j is a signature matrix then we consider the perturbation analysis of the new matrix factorization some first order normwise perturbation bounds are derived using the refined matrix equation approach and the updated matrix vector equation approach the corresponding perturbation bounds for the generalized qr factorization are also obtained as the special case in comparison these bounds can be much tighter than the previous ones derived from barrlund 1994 a barrlund perturbation bounds for the generalized qr factorization linear algebra appl 207 1994 251 271

powers of the euler class abstract let g be the topological group of orientation preserving homeomorphisms of the circle s 1 the classical euler class and its powers generate the cohomology of the classifying space b g in the transition from continuous g to discrete g the behavior of the powers of the euler class is altered significantly there are subgroups k of g for which the powers of the discrete euler class e n h 2 n b g z are non vanishing only up to a certain dimension in this work we determine the threshold dimension for certain groups of homeomorphisms of the circle our results are applied to the based mapping class groups m g and we show e n m g 0 n g but e n m g 0 n g at the threshold dimension n g the discrete euler class has torsion of order 2 g 2 g 1

lupod collocation in pod via lu decomposition a collocation method is developed for the truncated pod of a set of snapshots in other words pod computations are performed using only a set of collocation points whose number is comparable to the number of retained modes in a similar fashion as in collocation spectral methods intending to rely on simple ideas which moreover are consistent with the essence of pod collocation points are computed via the lu decomposition with pivoting of the snapshot matrix the new method is illustrated in simple applications in which pod is used as a data processing method the performance of the method is tested in the computationally efficient construction of reduced order models based on pod plus galerkin projection for the complex ginzburg landau equation in one and two space dimensions

a tutorial history of least squares with applications to astronomy and geodesy this article surveys the history development and applications of least squares including ordinary constrained weighted and total least squares the presentation includes proofs of the basic theory in particular unitary factorizations and singular value decompositions of matrices numerical examples with real data demonstrate how to set up and solve several types of problems of least squares the bibliography lists comprehensive sources for more specialized aspects of least squares

optimal three ball inequalities and quantitative uniqueness for the lame system with lipschitz coefficients in this article we study the local behavior of a solution to the lame system with lipschitz coefficients in dimension n 2 our main result is the bound on the vanishing order of a nontrivial solution which immediately implies the strong unique continuation property sucp we solve the open problem of the sucp for the lame system with lipschitz coefficients in any dimension

the smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems let t left begin array 20 c a b c d end array right be a partitioned rectangular matrix we exhibit perturbations of d of smallest norm which lower the rank of t we apply this result to generalize the total least squares problem sx b when only a rectangular subset of the data s b is considered uncertain

unique continuation for the two dimensional anisotropic elasticity system and its applications to inverse problems under some generic assumptions we prove the unique continuation property for the two dimensional inhomogeneous anisotropic elasticity system having established the unique continuation property we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

the smale conjecture for hyperbolic 3 manifolds isom m 3 diff m 3 the main result of this paper is if m is a closed hyperbolic 3 manifold then the inclusion of the isometry group isom m into the diffeomorphism group diff m is a homotopy equivalence

real perturbation values and real quadratic forms in a complex vector space abstract a sequence of real numbers connected to a complex matrix is introduced it is shown how these real perturbation values can be computed and that they have several properties similar to the singular values the so called real pseudospectra and real stability radii can be computed using the real perturbation values the main result concerns the signature of real quadratic forms in complex vector spaces

strong uniqueness for second order differential operators abstract we prove a strong unique continuation result for differential inequalities of the form p x d u c1 x 2 u c2 x 1 u wherep x d nj k 1 ajk x djdkis an elliptic second order differential operator with lipschitz coefficients such thatajk 0 is real c1andc2are positive constants such thatc2is sufficiently small our assumption on the constantc2is justified by counterexamples due to alinhac and baouendi 2 and wolff 6 showing that the strong unique continuation fails ifc2is not small

two sided hyperbolic svd abstract in this paper we propose the two sided hyperbolic svd 2hsvd for square matrices i e a u v where u and v are j unitary j diag 1 and is a real diagonal matrix of double hyperbolic singular values we show that with some natural conditions such decomposition exists without the use of hyperexchange matrices in other words u and v are really j unitary with regard to j and not some matrix j which is permutationally similar to matrix j we provide full characterization of 2hsvd and completely relate it to the semidefinite j polar decomposition

the hyperbolic schur decomposition we propose a hyperbolic counterpart of the schur decomposition with the emphasis on the preservation of structures related to some given hyperbolic scalar product we give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices

topological transversality holds in all dimensions on suppose m et x des sous varietes propres de y x a un microfibre normal il y a des sous ensembles fermes c d y et m est transversable a pres de c alors il y a une isotopie de m a support dans un voisinage donne de d c m x vers une sous variete transversale a pres de d

an adaptive parameterized block based singular value decomposition for image de noising and compression abstract we propose an adaptive parametrized block based singular value decomposition apbsvd for preserving the edge structure and avoiding blurred image after the compressing process we also exploit a proper selection of the peak signal to noise ratio psnr parameter at each block to appropriately eliminate high frequency noise our numerical results show that the proposed methods are promising which effectively remove the noise and preserves the edge information well during the block based svd image compression

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

on doubling inequalities for elliptic systems abstract we prove doubling inequalities for solutions of elliptic systems with an iterated laplacian as diagonal principal part and for solutions of the lame system of isotropic linearized elasticity these inequalities depend on global properties of the solutions

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

strong unique continuation for the lame system with less regular coefficients we study the strong unique continuation property sucp for the lame system in the plane the main contribution of our work is to prove that the sucp holds when lame coefficients mu lambda in w 2 s omega times l infty omega for some s 4 3 in other words we establish the sucp for the lame system in the plane when lambda is bounded and mu belongs to certain holder classes

a differential equation approach to the singular value decomposition of bidiagonal matrices we consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one parameter family of differentiable matrix flows it is shown that this approach can be fully expressed as an autonomous homogeneous and cubic dynamical system asymptotic behavior is justified by the established theory of the toda lattice

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

optimal regularized low rank inverse approximation abstract in this paper we consider the problem of finding approximate inverses of a given matrix we give an explicit solution to the rank constrained regularized inverse approximation problem and obtain an inverse regularized eckart young like theorem

adaptive pod based low dimensional modeling supported by residual estimates summary an adaptive low dimensional model is considered to simulate time dependent dynamics in nonlinear dissipative systems governed by pdes the method combines an inexpensive pod based galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the pod modes switching between the numerical solver and the galerkin system is decided on the fly by monitoring i a truncation error estimate and ii a residual estimate the latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental galerkin system based on a larger number of pod modes the most computationally expensive run of the numerical solver occurs at the outset when the whole set of pod modes is calculated this step is improved by using mode libraries which may either be generic or result from former applications of the method the outcome is a flexible robust computationally inexpensive procedure that adapts itself to the local dynamics by using the faster galerkin system for the majority of the time and few on demand short runs of a numerical solver the method is illustrated considering the complex ginzburg landau equation in one and two space dimensions copyright 2015 john wiley sons ltd

stratified generalized procrustes analysis generalized procrustes analysis computes the best set of transformations that relate matched shape data in shape analysis the transformations are usually chosen as similarities while in general statistical data analysis other types of transformation groups such as the affine group may be used generalized procrustes analysis has a nonlinear and nonconvex formulation the classical approach alternates the computation of a so called reference shape and the computation of transformations relating this reference shape to each shape datum in turn we propose the stratified approach to generalized procrustes analysis it first uses the affine transformation group to analyze the data and then upgrades the solution to the sought after group whether euclidean or similarity we derive a convex formulation for each of these two steps and efficient practical algorithms that gracefully handle missing data incomplete shapes extensive experimental results show that our approaches perform well on simulated and real data in particular our closed form solution gives very accurate results for generalized procrustes analysis of euclidean data

a note on the existence of the hyperbolic singular value decomposition abstract in this note we present a new matrix decomposition for a matrix pair a b with a hermitian the hyperbolic svd comes as a special case of the decomposition with a set to be the signature matrix

nonlocal image restoration with bilateral variance estimation a low rank approach simultaneous sparse coding ssc or nonlocal image representation has shown great potential in various low level vision tasks leading to several state of the art image restoration techniques including bm3d and lssc however it still lacks a physically plausible explanation about why ssc is a better model than conventional sparse coding for the class of natural images meanwhile the problem of sparsity optimization especially when tangled with dictionary learning is computationally difficult to solve in this paper we take a low rank approach toward ssc and provide a conceptually simple interpretation from a bilateral variance estimation perspective namely that singular value decomposition of similar packed patches can be viewed as pooling both local and nonlocal information for estimating signal variances such perspective inspires us to develop a new class of image restoration algorithms called spatially adaptive iterative singular value thresholding saist for noise data saist generalizes the celebrated bayesshrink from local to nonlocal models for incomplete data saist extends previous deterministic annealing based solution to sparsity optimization through incorporating the idea of dictionary learning in addition to conceptual simplicity and computational efficiency saist has achieved highly competent often better objective performance compared to several state of the art methods in image denoising and completion experiments our subjective quality results compare favorably with those obtained by existing techniques especially at high noise levels and with a large amount of missing data

deep restricted kernel machines using conjugate feature duality the aim of this letter is to propose a theory of deep restricted kernel machines offering new foundations for deep learning with kernel machines from the viewpoint of deep learning it is partially related to restricted boltzmann machines which are characterized by visible and hidden units in a bipartite graph without hidden to hidden connections and deep learning extensions as deep belief networks and deep boltzmann machines from the viewpoint of kernel machines it includes least squares support vector machines for classification and regression kernel principal component analysis pca matrix singular value decomposition and parzen type models a key element is to first characterize these kernel machines in terms of so called conjugate feature duality yielding a representation with visible and hidden units it is shown how this is related to the energy form in restricted boltzmann machines with continuous variables in a nonprobabilistic setting in this new framework of so called restricted kernel machine rkm representations the dual variables correspond to hidden features deep rkm are obtained by coupling the rkms the method is illustrated for deep rkm consisting of three levels with a least squares support vector machine regression level and two kernel pca levels in its primal form also deep feedforward neural networks can be trained within this framework

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

existence of the hyperbolic singular value decomposition abstract the hyperbolic singular value decomposition is defined on a general n m matrix and an m m signature matrix pair it is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products such differences arise in signal processing applications in the context of the covariance differencing the hyperbolic svd applies in problems where the conventional svd cannot be employed the existence of the hyperbolic singular value decomposition is here extended to the most general case where neither the general matrix nor the matrix product is assumed full rank

blind carrier frequency offset estimation for interleaved ofdma uplink in this paper we develop two novel blind carrier frequency offset cfo estimators for interleaved orthogonal frequency division multiple access ofdma uplink transmission in a multiantenna system the first estimator is the subspace based one and could blindly estimate multiple cfos from a rank reduction approach the second estimator is based on the maximum likelihood ml approach and improves the performance as compared to the first one the higher computational complexity of the ml estimator is alleviated by the alternating projection ap method both the proposed estimators support fully loaded data transmissions i e all subcarriers being occupied which provides higher bandwidth efficiency as compared to the existing schemes the numerical results are then provided to corroborate the proposed studies

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

efficient gaussian process regression for large datasets gaussian processes are widely used in nonparametric regression classification and spatiotemporal modelling facilitated in part by a rich literature on their theoretical properties however one of their practical limitations is expensive computation typically on the order of n3 where n is the number of data points in performing the necessary matrix inversions for large datasets storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with increasing n various methods have been proposed to address these problems including predictive processes in spatial data analysis and the subset of regressors technique in machine learning the idea underlying these approaches is to use a subset of the data but this raises questions concerning sensitivity to the choice of subset and limitations in estimating fine scale structure in regions that are not well covered by the subset motivated by the literature on compressive sensing we propose an alternative approach that involves linear projection of all the data points onto a lower dimensional subspace we demonstrate the superiority of this approach from a theoretical perspective and through simulated and real data examples

strong unique continuation for the lame system with lipschitz coefficients in this paper we prove the strong unique continuation property for a lame system with lipschitz coefficients in the plane the proof relies on reducing the lame system to a first order elliptic system and suitable carleman estimates with polynomial weights

homological stability and stable moduli of flat manifold bundles abstract we prove that the group homology of the diffeomorphism group of g s n s n int d 2 n as a discrete group is independent of g in a range provided that n 2 this answers the high dimensional version of a question posed by morita about surface diffeomorphism groups made discrete the stable homology is isomorphic to the homology of a certain infinite loop space related to the haefliger s classifying space of foliations one geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized mumford morita miller classes lift to a secondary r z invariants for flat g s n s n bundles provided g 0

a proof of the smale conjecture diff s3 0 4 the smale conjecture 9 is the assertion that the inclusion of the orthogonal group 0 4 into diff s3 the diffeomorphism group of the 3 sphere with the cw topology is a homotopy equivalence there are many equivalent forms of this conjecture some of which are listed in the appendix to this paper we shall prove

global analytic solution of fully observed variational bayesian matrix factorization the variational bayesian vb approximation is known to be a promising approach to bayesian estimation when the rigorous calculation of the bayes posterior is intractable the vb approximation has been successfully applied to matrix factorization mf offering automatic dimensionality selection for principal component analysis generally finding the vb solution is a nonconvex problem and most methods rely on a local search algorithm derived through a standard procedure for the vb approximation in this paper we show that a better option is available for fully observed vbmf the global solution can be analytically computed more specifically the global solution is a reweighted svd of the observed matrix and each weight can be obtained by solving a quartic equation with its coefficients being functions of the observed singular value we further show that the global optimal solution of empirical vbmf where hyperparameters are also learned from data can also be analytically computed we illustrate the usefulness of our results through experiments in multi variate analysis

the failure of rank one connections this article is concerned with interface problems for lipschitz mappings f n n and f n n in the half spaces which agree on the common boundary n 1 n n these naturally occur in mathematical models for material microstructures and crystals the task is to determine the relationship between the sets of values of the differentials df and df for some time it has been thought that the polyconvex hulls df pc and df pc satisfy hadamard s jump condition or are at least rank one connected our examples here refute this idea the estimates of the jacobians we obtain in the course of solving the so called monge ampere inequalities seem also to be of independent interest as an application we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar cauchy riemann equations for which the unique continuation property fails

parametric models for samples of random functions a new class of parametric models referred to as sample parametric models is developed for random elements that match sample rather than the first two moments and or other global properties of these elements the models can be used to characterize e g material properties at small scale in which case their samples represent microstructures of material specimens selected at random from a population the samples of the proposed models are elements of finite dimensional vector spaces spanned by samples eigenfunctions of karhunen loeve kl representations or modes of singular value decompositions svds the implementation of sample parametric models requires knowledge of the probability laws of target random elements numerical examples including stochastic processes and random fields are used to demonstrate the construction of sample parametric models assess their accuracy and illustrate how these models can be used to solve efficiently stochastic equations

on best rank n matrix approximations abstract we consider the problem of approximating matrices by matrices of rank n in the p q norm among other results we prove that if a is a totally positive matrix and a 1 1 is the norm given by the sum of the absolute values of the entries of the matrix then a best rank n approximation to a in this norm is given by a matrix that agrees with a on n rows and n columns

the analysis of an ill posed problem using multi scale resolution and second order adjoint techniques abstract a wavelet regularization approach is presented for dealing with an ill posed problem of adjoint parameter estimation applied to estimating inflow parameters from down flow data in an inverse convection case applied to the two dimensional parabolized navier stokes equations the wavelet method provides a decomposition into two subspaces by identifying both a well posed as well as an ill posed subspace the scale of which is determined by finding the minimal eigenvalues of the hessian of a cost functional measuring the lack of fit between model prediction and observed parameters the control space is transformed into a wavelet space the hessian of the cost is obtained either by a discrete differentiation of the gradients of the cost derived from the first order adjoint or by using the full second order adjoint the minimum eigenvalues of the hessian are obtained either by employing a shifted iteration method x zou i m navon f x le dimet tellus 44a 4 1992 273 or by using the rayleigh quotient the numerical results obtained show the usefulness and applicability of this algorithm if the hessian minimal eigenvalue is greater or equal to the square of the data error dispersion in which case the problem can be considered as well posed i e regularized if the regularization fails i e the minimal hessian eigenvalue is less than the square of the data error dispersion of the problem the following wavelet scale should be neglected followed by another algorithm iteration the use of wavelets also allowed computational efficiency due to reduction of the control dimension obtained by neglecting the small scale wavelet coefficients

a new data processing inequality and its applications in distributed source and channel coding in the distributed coding of correlated sources the problem of characterizing the joint probability distribution of a pair of random variables satisfying an n letter markov chain arises the exact solution of this problem is intractable in this paper we seek a single letter necessary condition for this n letter markov chain to this end we propose a new data processing inequality on a new measure of correlation through a spectral method based on this new data processing inequality we provide a single letter necessary condition for the required joint probability distribution we apply our results to two specific examples involving the distributed coding of correlated sources multiple access channel with correlated sources and multiterminal rate distortion region and propose new necessary conditions for these two problems

the uncovering of hidden structures by latent semantic analysis latent semantic analysis lsa is a well known method for information retrieval it has also been applied as a model of cognitive processing and word meaning acquisition this dual importance of lsa derives from its capacity to modulate the meaning of words by contexts dealing successfully with polysemy and synonymy the underlying reasons that make the method work are not clear enough we propose that the method works because it detects an underlying block structure the blocks corresponding to topics in the term by document matrix in real cases this block structure is hidden because of perturbations we propose that the correct explanation for lsa must be searched in the structure of singular vectors rather than in the profile of singular values using the perron frobenius theory we show that the presence of disjoint blocks of documents is marked by sign homogeneous entries in the vectors corresponding to the documents of one block and zeros elsewhere in the case of nearly disjoint blocks perturbation theory shows that if the perturbations are small the zeros in the leading vectors are replaced by small numbers pseudo zeros since the singular values of each block might be very different in magnitude their order does not mirror the order of blocks when the norms of the blocks are similar lsa works fine but we propose that when the topics have different sizes the usual procedure of selecting the first k singular triplets k being the number of blocks should be replaced by a method that selects the perturbed perron vectors for each block

three spheres inequalities for a two dimensional elliptic system and its application in this paper we prove three spheres inequalities for a two dimensional strongly elliptic system we then give an application of these three spheres inequalities to the inverse problem of identifying cavities by partial boundary measurements

hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least squares abstract for each finite set of points in a euclidean space of any dimension the algorithm presented here determines all the algebraically best fitting circles or lines spheres or planes or hyperspheres or hyperplanes in a seamless manner from spherical through affine manifolds in particular affine submanifolds of any dimensions are not singularities of the algorithm to this end the algorithm combines projective geometry coope s and gander et al s layouts of the equations and golub et al s generalization of the schmidt mirsky matrix approximation theorem to solve the equations the resulting best fitting manifolds remain invariant under rigid transformations moreover if the best fitting manifold is affine then it coincides with golub and van loan s affine manifold of total least squares thus the algorithm can also fit hyperspheres in a manner that remains robust with data lying near a hyperplane furthermore an analysis with a theorem of wedin s shows that the fitted hyperspheres sensitivity to perturbations of the data increases as the colinearity of the data increases

singular value decomposition in minkowski space abstract the well known singular value decomposition theorem states that every matrix m can be written in the form m qdr where d is diagonal and q and r are orthogonal we explore the analogous decomposition where q and r are required to be orthogonal with respect to the minkowski rather than the euclidean metric

error bounds for pod expansions of parameterized transient temperatures abstract we focus on the convergence analysis of the pod expansion for the parameterized solution of transient heat equations the parameter of interest is the conductivity coefficient we prove that this expansion converges with exponential accuracy uniformly if the conductivity coefficient remains within a compact set of positive numbers this convergence result is independent of the regularity of the temperature with respect to the space and time variables we present some numerical experiments to show that a reduced number of modes allows to represent with high accuracy the family of solutions corresponding to parameters that lie in the compact set under study