Calcolo最新文献

The proposed work discusses discrete collocation and discrete Galerkin methods for second kind Fredholm–Hammerstein integral equations on half line ([0,infty )) using Kumar and Sloan technique. In addition, the finite section approximation method is applied to transform the domain of integration from ([0, infty )) to ([0,alpha ],~ alpha >0). In contrast to previous studies in which the optimal order of convergence is achieved for projection methods, we attained superconvergence rates in uniform norm using piecewise polynomial basis function. Moreover, these superconvergence rates are further enhanced by using discrete multi-projection (collocation and Galerkin) methods. In order to support the provided theoretical framework, numerical examples are included as well.

Abstract

In this article, band structure calculations of two dimensional (2D) anisotropic photonic-crystal fibers (PhCFs) are considered. In 2D PhCFs, Maxwell’s equations for the transversal electric and magnetic mode become decoupled, but the difficulty, arising from the anisotropic permittivity ({{varvec{varepsilon }}}) and/or permeability ({{varvec{mu }}},) plaguing the frequency-domain finite difference method, especially the original Yee’s scheme, is our top concern. To resolve this difficulty, we re-establish the connection between the lowest order finite element method with the quasi-periodic condition and Yee’s scheme using 2D non-orthogonal mesh, whereby the decoupled Maxwell’s equations in 2D anisotropic PhCFs are readily discretized into a generalized eigenvalue problem (GEP). Moreover, we spell out the nullspace of the resulting GEP, if it exists, and explicitly construct the Moore–Penrose pseudoinverse of the singular coefficient matrix, whose smallest positive eigenvalues can be solved by the inverse Lanczos method. Extensive band structures of 2D PhCFs are calculated and benchmarked against reliable results to demonstrate the accuracy and efficiency of our method.

This paper addresses the tensor completion problem, whose task is to estimate missing values with limited information. However, the crux of this problem is how to reasonably represent the low-rank structure embedded in the underlying data. In this work, we consider a new low-rank tensor completion model combined with the multi-directional partial tensor nuclear norm and the total variation (TV) regularization. Specifically, the partial sum of the tensor nuclear norm (PSTNN) is used to narrow the gap between the tensor tubal rank and its lower convex envelop [i.e. tensor nuclear norm (TNN)], and the TV regularization is adopted to maintain the smooth structure along the spatial dimension. In addition, the weighted sum of the tensor nuclear norm (WSTNN) is introduced to replace the traditional TNN to extend the PSTNN to the high-order tensor, which also can flexibly handle different correlations along different modes, resulting in an improved low d-tubal rank approximation. To tackle this new model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the proposed optimization problem. Theoretical analysis of the ADMM is conducted to prove the Karush–Kuhn–Tucker (KKT) conditions. Numerical examples demonstrate the proposed method outperforms some state-of-the-art methods in qualitative and quantitative aspects.

The approximate linear equation (Axapprox b) with some columns of A error-free can be solved via mixed least squares-total least squares (MTLS) model by minimizing a nonlinear function. This paper is devoted to the Gauss–Newton iteration for the MTLS problem. With an appropriately chosen initial vector, each iteration step of the standard Gauss–Newton method requires to solve a smaller-size least squares problem, in which the QR of the coefficient matrix needs a rank-one modification. To improve the convergence, we devise a relaxed Gauss–Newton (RGN) method by introducing a relaxation factor and provide the convergence results as well. The convergence is shown to be closely related to the ratio of the square of subspace-restricted singular values of [A, b]. The RGN can also be modified to solve the total least squares (TLS) problem. Applying the RGN method to a Bursa–Wolf model in parameter estimation, numerical results show that the RGN-based MTLS method behaves much better than the RGN-based TLS method. Theoretical convergence properties of the RGN-MTLS algorithm are also illustrated by numerical tests.

Abstract

Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost (k+1) order in an energy norm. Here (kge 1) is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter (varepsilon ) or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm.

In this paper, we consider the numerical methods for both the forward and backward problems of a time-space fractional diffusion equation. For the two-dimensional forward problem, we propose a finite difference method. The stability of the scheme and the corresponding Fast Preconditioned Conjugated Gradient algorithm are given. For the backward problem, since it is ill-posed, we use a quasi-boundary-value method to deal with it. Based on the Fourier transform, we obtain two kinds of order optimal convergence rates by using an a-priori and an a-posteriori regularization parameter choice rules. Numerical examples for both forward and backward problems show that the proposed numerical methods work well.

This paper studies the problem of learning an unknown function f from given data about f. The learning problem is to give an approximation ({hat{f}}) to f that predicts the values of f away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about f (known as a model class assumption), (ii) how we measure the accuracy of how well ({hat{f}}) predicts f, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal ({hat{f}}) can be found by solving a certain finite-dimensional over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation ({hat{f}}) of the function f from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of f. An extension of these results to the case where the data is polluted by additive deterministic noise is also given.

In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver.

In this work, three discontinuous Galerkin (DG) methods are formulated and analysed to solve Stokes equations with power law slip boundary condition. Numerical examples exhibited confirm the theoretical findings, moreover we also test the methods on the lid Driven cavity and compare the three DG methods.

Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called ((varvec{b},varepsilon ))-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.