global uniqueness and stability for an inverse wave source problem for less regular data abstract by means of a carleman estimate we obtain the global uniqueness and stability of the solution of an inverse problem of determining a wave source within the h 1 space

the differentiation matrix for daubechies based wavelets on an interval the differentiation matrix for a daubechies based wavelet basis defined on an interval is constructed it is shown that the differentiation matrix based on the currently available boundary constructions does not maintain the superconvergence encountered under periodic boundary conditions

inductive robust principal component analysis in this paper we address the error correction problem that is to uncover the low dimensional subspace structure from high dimensional observations which are possibly corrupted by errors when the errors are of gaussian distribution principal component analysis pca can find the optimal in terms of least square error low rank approximation to high dimensional data however the canonical pca method is known to be extremely fragile to the presence of gross corruptions recently wright established a so called robust principal component analysis rpca method which can well handle the grossly corrupted data however rpca is a transductive method and does not handle well the new samples which are not involved in the training procedure given a new datum rpca essentially needs to recalculate over all the data resulting in high computational cost so rpca is inappropriate for the applications that require fast online computation to overcome this limitation in this paper we propose an inductive robust principal component analysis irpca method given a set of training data unlike rpca that targets on recovering the original data matrix irpca aims at learning the underlying projection matrix which can be used to efficiently remove the possible corruptions in any datum the learning is done by solving a nuclear norm regularized minimization problem which is convex and can be solved in polynomial time extensive experiments on a benchmark human face dataset and two video surveillance datasets show that irpca cannot only be robust to gross corruptions but also handle the new data well and in an efficient way

quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions this paper deals with typical problems that arise when using wavelets in numerical analysis applications the first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions several types of quadratures are discussed and compared for different classes of wavelets since their construction using monomials is ill conditioned also a modified well conditioned construction using chebyshev polynomials is presented the second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions they are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

algebraic feature extraction for image recognition based on an optimal discriminant criterion abstract a novel algebraic feature extraction method for image recognition is presented the main difference between the present method and any other existing algebraic feature extraction method is that the present method uses an optimal discriminant criterion to extract algebraic features for the given training image samples a set of optimal discriminant projection vectors is calculated according to a generalized fisher criterion function on the basis of the optimal discriminant projection vector the projective feature vectors of an image i e algebraic features are extracted by projecting the image onto all optimal discriminant projection vectors the method of calculating the set of optimal discriminant projection vectors is discussed a minimum distance classifier is also proposed for classifying projective feature vectors experimental results showed that the present method has good recognition performance an important conclusion about the present method is that the foley sammon optimal set of discriminant vectors is a special case of the set of optimal discriminant projection vectors

an efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the daubechies wavelet basis an efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with daubechies wavelet and scaling function basis sets our quadrature is also applicable in the case of adaptive spatial resolution our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wavefunction of the harmonic oscillator in one dimension with and without adaptive resolution as a byproduct we derive a filter which upon application on the scaling function coefficients of a smooth function renders the approximate grid values of this function this also allows for a fast calculation of the charge density from the wavefunction

coifman wavelets in 3 d scattering from very rough random surfaces the coifman wavelets are employed to perform galerkin s procedure in the method of moments mom the coiflets are continuous smooth overlapping yet orthogonal which permit significant reduction of the sampling rate and dramatic compression of the matrix size with respect to the pulse based mom in addition the wavelet based impedance matrix is sparse more importantly the vanishing moments of the coiflets of order l 4 provide us with high precision o h sup 5 one point quadrature which knocks down the computational effort in filling the matrix entries from o n sup 2 to o n our approach is different from the conventional wavelet technique which employs wavelets to sparsify an existing impedance matrix under similarity transformation with computational cost in o n sup 2 excellent agreement between this coifman wavelet method and the previous publications is observed

a multiple direction trefftz method for solving the multi dimensional wave equation in an arbitrary spatial domain in this paper we first express the wave equation in terms of the minkowskian polar coordinates and generate a set of complete hyperbolic type trefftz bases r k cosh k and r k sinh k which are further transformed to wave polynomials as the trial solution bases for the one dimensional wave equation in order to stably solve the wave propagation problems long term we develop a multiple scale trefftz method mstm of which the scales are determined a priori by the collocation points then we derive a very simple method of multi dimensional wave polynomials equipped with different spatial directions which being the normalized wavenumber vectors as the polynomial trefftz bases for solving the multi dimensional wave equations which is named a multiple direction trefftz method mdtm several numerical examples of two and three dimensional wave equations demonstrate that the present method is efficient and stable

general subspace learning with corrupted training data via graph embedding we address the following subspace learning problem supposing we are given a set of labeled corrupted training data points how to learn the underlying subspace which contains three components an intrinsic subspace that captures certain desired properties of a data set a penalty subspace that fits the undesired properties of the data and an error container that models the gross corruptions possibly existing in the data given a set of data points these three components can be learned by solving a nuclear norm regularized optimization problem which is convex and can be efficiently solved in polynomial time using the method as a tool we propose a new discriminant analysis i e supervised subspace learning algorithm called corruptions tolerant discriminant analysis ctda in which the intrinsic subspace is used to capture the features with high within class similarity the penalty subspace takes the role of modeling the undesired features with high between class similarity and the error container takes charge of fitting the possible corruptions in the data we show that ctda can well handle the gross corruptions possibly existing in the training data whereas previous linear discriminant analysis algorithms arguably fail in such a setting extensive experiments conducted on two benchmark human face data sets and one object recognition data set show that ctda outperforms the related algorithms

quadrature prefilters for the discrete wavelet transform discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data prefilters based on lagrange interpolants are derived here for biorthogonal compact support wavelet systems providing exact subspace projection in cases of local polynomial smoothness the resulting divergence acceleration in a nonpolynomial test case is examined irregular sampling rates are also accommodated

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

shooting methods for numerical solutions of exact controllability problems constrained by linear and semilinear wave equations with local distributed controls numerical solutions of exact controllability problems for linear and semilinear wave equations with distributed controls are studied exact controllability problems can be solved by the corresponding optimal control problems the optimal control problem is reformulated as a system of equations an optimality system that consists of an initial value problem for the underlying linear or semilinear wave equation and a terminal value problem for the adjoint wave equation the discretized optimality system is solved by a shooting method the convergence properties of the numerical shooting method in the context of exact controllability are illustrated through computational experiments

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

the decomposition method for forward and backward time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat wave and beam equations is investigated for both forward direct and backward inverse problems it is shown that for forward problems the convergence is faster than for backward problems

optimally scaled vector regularization method to solve ill posed linear problems instead of the tikhonov regularization method which with a scalar being the regularization parameter liu et al 1 have proposed a novel regularization method with a vector as being the regularization parameter as a continuation we further propose an optimally scaled vector regularization method osvrm to solve the ill posed linear problems which is better than the tikhonov regularization method the presently proposed vector regularization method is shown to be well conditioning to overcome the ill posedness of the linear equations system the osvrm causes a significant improvement of stability and accuracy in the numerical solution of ill posed linear problem and its convergence speed is as fast as by solving the well posed linear problem some tests of the linear inverse problems confirm the efficiency and accuracy of the osvrm

robust pca via outlier pursuit singular value decomposition svd and principal component analysis pca is one of the most widely used techniques for dimensionality reduction successful and efficiently computable it is nevertheless plagued by a well known well documented sensitivity to outliers recent work has considered the setting where each point has a few arbitrarily corrupted components yet in applications of svd or pca such as robust collaborative filtering or bioinformatics malicious agents defective genes or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted we present an efficient convex optimization based algorithm that we call outlier pursuit which under some mild assumptions on the uncorrupted points satisfied e g by the standard generative assumption in pca problems recovers the exact optimal low dimensional subspace and identifies the corrupted points such identification of corrupted points that do not conform to the low dimensional approximation is of paramount interest in bioinformatics financial applications and beyond our techniques involve matrix decomposition using nuclear norm minimization however our results setup and approach necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition since we develop an approach to recover the correct column space of the uncorrupted matrix rather than the exact matrix itself in any problem where one seeks to recover a structure rather than the exact initial matrices techniques developed thus far relying on certificates of optimality will fail we present an important extension of these methods which allows the treatment of such problems

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

fourier spectral simulations and gegenbauer reconstructions for electromagnetic waves in the presence of a metal nanoparticle we describe fourier pseudospectral time domain simulations carried out in order to study light interacting with a metallic nanoscale object the difficulty of using fourier methods to accurately predict the electromagnetic scattering in such problems arises from the discontinuity in the dielectric function along the surface of the metallic object standard fourier methods lead to oscillatory behavior in approximating solutions that are nonsmooth or that have steep gradients by applying the gegenbauer reconstruction technique as a postprocessing method to the fourier pseudospectral solution we successfully reduce the oscillations after postprocessing our computational results including comparison with finite difference time domain simulations demonstrate the efficiency and accuracy of the method

higher order wavelet reconstruction differentiation filters and gibbs phenomena an orthogonal wavelet basis is characterized by its approximation order which relates to the ability of the basis to represent general smooth functions on a given scale it is known though perhaps not widely known that there are ways of exceeding the approximation order i e achieving higher order error in the discretized wavelet transform and its inverse the focus here is on the development of a practical formulation to accomplish this first for 1d smooth functions then for 1d functions with discontinuities and then for multidimensional here 2d functions with discontinuities it is shown how to transcend both the wavelet approximation order and the 2d gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet basis grid

fast simulation of multi dimensional wave problems by the sparse scheme of the method of fundamental solutions in this work a meshless scheme is presented for the fast simulation of multi dimensional wave problems the present method is rather simple and straightforward the houbolt method is used to eliminate the time dependence of spatial variables then the original wave problem is converted into equivalent systems of modified helmholtz equations the sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables to demonstrate the effectiveness and simplicity of this new approach three numerical examples have been assessed with excellent performance

on one sided filters for spectral fourier approximations of discontinuous functions in this paper the existence of one sided filters is proved for spectral fourier ap proximations of discontinuous functions which can recover spectral accuracy up to the discontinuity from one side a least square procedure is used to construct such a filter and is tested on several discontinuous functions numerically

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

the polynomial trefftz method for solving backward and inverse source wave problems the trefftz method is a truly meshless boundary type method because the trial solutions automatically satisfy the governing equation in order to stably solve the high dimensional backward wave problems and the one dimensional inverse source problems we develop a multiple scale polynomial trefftz method msptm of which the scales are determined a priori by the collocation points the msptm can retrieve the missing initial data and unknown time varying wave source the present method can also be extended to solve the higher dimensional wave equations long term through the introduction of a director in the two dimensional polynomial trefftz bases several numerical examples reveal that the msptm is efficient and stable for solving severely ill posed inverse problems of wave equations under large noises