Analysis of orthogonal multi-matching pursuit under restricted isometry property

Orthogonal multi-matching pursuit(OMMP)is a natural extension of orthogonal matching pursuit(OMP)in the sense that N(N≥1)indices are selected per iteration instead of 1.In this paper,the theoretical performance of OMMP under the restricted isometry property(RIP)is presented.We demonstrate that OMMP can exactly recover any K-sparse signal from fewer observations y=φx,provided that the sampling matrixφsatisfiesδKN-N+1+(K/N)~(1/2)θKN-N+1,N1.Moreover,the performance of OMMP for support recovery from noisy observations is also discussed.It is shown that,for l_2 bounded and l_∞bounded noisy cases,OMMP can recover the true support of any K-sparse signal under conditions on the restricted isometry property of the sampling matrixφand the minimum magnitude of the nonzero components of the signal.     

Reconstruction of biologic tissue conductivity from noisy magnetic field by integral equation method

Magnetic resonance electrical impedance tomography (MREIT) is a new technique to recover the conductivity of biologic tissue from the induced magnetic flux density. This paper proposes an inversion scheme for recovering the conductivity from one component of the magnetic field based on the nonlinear integral equation method. To apply magnetic fields corresponding to two incoherent injected currents, an alternative iteration scheme is proposed to update the conductivity. For each magnetic field, the regularizing technique on the finite dimensional space is applied to solve an ill-posed linear system. Compared with the well-developed harmonic B_z method, the advantage of this inversion scheme is its stability, since no differential operation is required on the noisy magnetic field. Numerical implementations are given to show the convergence of the iteration and its validity for noisy input data.     

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ +N ^?2 ln² N) ln(j +1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.     

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N ₁ ^-1 N ₂ ^-1 ? ε, where N ₁ and N ₂ are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε, N ₁,N ₂, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N ₁,N ₂ → ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N ₁,N ₂ → ∞ for fixed ε, ε ∈ (0,1, is called a computer difference scheme. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)—Gaussians, sech's and Inverse Quadratics—the condition number κ(α,N) rapidly asymptotes to a limit κasymp(α) that is independent of N and depends only on α, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed α grows as N². For all four, there is growth proportional to an exponential of 1/α (1/α² for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N—a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.     

General algorithm for the numerical integration of functions of several variables

An algorithm is proposed for the numerical integration of an arbitrary function represent-able as a sum of an absolutely converging multiple trigonometric Fourier series. The resulting quadrature formulas have identical weights, and the nodes form a Korobov grid that is completely defined by two positive integers, of which one is the number of nodes. In the case of classes of functions with dominant mixed smoothness, it is shown that the algorithm is almost optimal in the sense that the construction of a grid of N nodes requires far fewer elementary arithmetic operations than NlnlnN. Solutions of related problems are also given.     

A bound on the treewidth of planar even-hole-free graphs

The class of planar graphs has unbounded treewidth, since the k×k grid, ∀k∈N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.     

Toward a classification of quasirational solutions of the nonlinear Schrödinger equation

Based on a representation in terms of determinants of the order 2N, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrödinger equation and also formulate several conjectures about the structure of the solutions. These solutions can be written as a product of a t-dependent exponential times a quotient of two N(N+1)th degree polynomials in x and t depending on 2N?2 parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the P_N breathers.     

Bregman iterative model using the G-norm

In this paper, we analyze the Bregman iterative model using the G-norm. Firstly, we show the convergence of the iterative model. Secondly, using the source condition and the symmetric Bregman distance, we consider the error estimations between the iterates and the exact image both in the case of clean and noisy data. The results show that the Bregman iterative model using the G-norm has the similar good properties as the Bregman iterative model using the L2-norm.     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.     

An approximation problem of noisy data by cubic and bicubic splines

Smoothing of noisy data has always been a topic of interest in many areas where computer simulations have been performed, including natural sciences as well as economics and social sciences. In this paper we present an approximation method of explicit curves or surfaces from exact and noisy data by fairness cubic or bicubic splines. A variational problem of explicit curves or surfaces is obtained by minimizing a quadratic functional in a space of cubic or bicubic splines from noisy data. We show the existence and uniqueness of this problem as long as a convergence result especially for noisy data is carefully established. We analyze some numerical and graphical examples using fictional noisy data in order to prove the validity of our method.     

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

Quantized CP approximation and sparse tensor interpolation of function‐generated data

In this article, we consider the iterative schemes to compute the canonical polyadic (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the quantics‐tensor train (QTT) method (“O(d log N)‐quantics approximation of N‐d tensors in high‐dimensional numerical modeling” in Constructive Approximation, 2011) developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach, the target vector of length 2^L is reshaped to a Lth‐order tensor with two entries in each mode (quantized representation) and then approximated by the QTT tensor including 2r²L parameters, where r is the maximal TT rank. In what follows, we consider the alternating least squares (ALS) iterative scheme to compute the rank‐r CP approximation of the quantized vectors, which requires only 2rL?2^L parameters for storage. In the earlier papers (“Tensors‐structured numerical methods in scientific computing: survey on recent advances” in Chemom Intell Lab Syst, 2012), such a representation was called Q_Can format, whereas in this paper, we abbreviate it as the QCP (quantized canonical polyadic) representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases, we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank‐r QCP format using the reduced number of the functional samples, calculated only at O(2rL) grid points, is presented. The special version of the ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP‐interpolation method that allows to recover all 2rL representation parameters of the rank‐r QCP tensor. Numerical examples show the efficiency of the QCP‐ALS‐type iteration and indicate the exponential convergence rate in r.     

Construction Algorithms for Good Extensible Lattice Rules

Extensible (polynomial) lattice rules have the property that the number N of points in the node set may be increased while retaining the existing points. It was shown by Hickernell and Niederreiter in a nonconstructive manner that there exist generating vectors for extensible integration lattices of excellent quality for N=b,b ²,b ³,…, where b is a given integer greater than 1. Similar results were proved by Niederreiter for polynomial lattices. In this paper we provide construction algorithms for good extensible lattice rules. We treat the classical as well as the polynomial case.     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T(N?N^×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1?β?2, the unique KMSβ-state is of type III₁. We prove this by reducing the type classification from T(N?N^×) to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N?N^× in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N?N^× on the Nica spectrum, we can also recover all the KMS-states of T(N?N^×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβ-states for β>2 that does not ostensibly come from an action of T on the C^?-algebra.     

On the Error Estimate of the Harmonic B_z Algorithm in MREIT from Noisy Magnetic Flux Field

Magnetic resonance electrical impedance tomography(MREIT, for short) is a new medical imaging technique developed recently to visualize the cross-section conductivity of biologic tissues. A new MREIT image reconstruction method called harmonic Bz algorithm was proposed in 2002 with the measurement of Bz that is a single component of an induced magnetic flux density subject to an injection current. The key idea is to solve a nonlinear integral equation by some iteration process. This paper deals with the convergence analysis as well as the error estimate for noisy input data Bz, which is the practical situation for MREIT. By analyzing the iteration process containing the Laplacian operation on the input magnetic field rigorously, the authors give the error estimate for the iterative solution in terms of the noisy level δ and the regularizing scheme for determiningΔBz approximately from the noisy input data. The regularizing scheme for computing the Laplacian from noisy input data is proposed with error analysis. Our results provide both the theoretical basis and the implementable scheme for evaluating the reconstruction accuracy using harmonic Bz algorithm with practical measurement data containing noise.     

Multidimensional pseudo-spectral methods on lattice grids

When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L²-integrable, the contributions from many of the Ns terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.