Advances in Computational Mathematics最新文献

The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Its decay has extensively been studied in approximation theory and for the solution of partial differential equations (PDEs). Particular interest has occurred within model order reduction (MOR) of parameterized PDEs, e.g., by the reduced basis method (RBM). While it is known that the N-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to (N^{-1/2}). On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the N-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the N-width seems to be lacking. In this paper, we state that the optimal linear space is a direct sum of shift-isometric eigenspaces corresponding to the largest eigenvalues, yielding an exact representation of the N-width as their sum. For the linear transport problem, which is modeled by half-wave symmetric initial and boundary conditions g, we obtain such an optimal decomposition by sorted trigonometric functions with eigenvalues that match the Fourier coefficients of g. Further, for normalized g in the Sobolev space (H^r) of broken order (r>0), the sorted eigenfunctions give the sharp upper bound of the N-width, which is a reciprocal of a certain power sum. Yet, for ease, we also provide the decay ((pi N)^{-r}), obtained by the non-optimal space of ordering the trigonometric functions by frequency rather than by eigenvalue. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.

Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the approximation and generalization capabilities of the proposed neural networks for simulating the regularized pseudo-inverses of the linearized forward operators in direct scattering problems is provided. The results show that, with sufficient training data and parameters, the proposed neural networks can effectively approximate the inverse process with desirable generalization. Preliminary numerical results show the feasibility of the proposed neural networks for recovering two types of isotropic inhomogeneous media. Furthermore, the trained neural network is capable of reconstructing the isotropic representation of certain types of anisotropic media.

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time-dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables. The focus of the study is to analyze stability results and to establish optimal error estimates, up to a logarithmic factor, for both the solution and the flux in (L^2)-norm when the initial data (u_0in H_0^1(Omega )cap H^2(Omega )). Additionally, an error estimate in (L^infty )-norm is derived for 2D problems. All the derived estimates and bounds in this article remain valid as (alpha rightarrow 1^{-}), where (alpha ) is the order of the Caputo fractional derivative. Finally, the results of several numerical experiments conducted at the end of this paper are confirming our theoretical findings.

We study numerical integration over bounded regions in (mathbb {R}^s), (s ge 1), with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions which aim to fill the space more evenly than random points. Ordinarily, such quasi-Monte Carlo point sets are designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next, we consider target densities which can be approximated with such mixture distributions. In order to be able to use an approximation of the target density, we require the approximation to be a sum of coordinate-wise products and that the approximation is positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate this mixture distribution with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us (at least to some degree) to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital (t, s)-sequences over the finite field (mathbb {F}_2) and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.

Parametric model order reduction techniques often struggle to accurately represent transport-dominated phenomena due to a slowly decaying Kolmogorov n-width. To address this challenge, we propose a non-intrusive, data-driven methodology that combines the shifted proper orthogonal decomposition (POD) with deep learning. Specifically, the shifted POD technique is utilized to derive a high-fidelity, low-dimensional model of the flow, which is subsequently utilized as input to a deep learning framework to forecast the flow dynamics under various temporal and parameter conditions. The efficacy of the proposed approach is demonstrated through the analysis of one- and two-dimensional wildland fire models with varying reaction rates, and its error is compared with the error of other similar methods. The results indicate that the proposed approach yields reliable results within the percent range, while also enabling rapid prediction of system states within seconds.

In this paper, we propose a Scaling Fractional Asymptotical Regularization (S-FAR) method for solving linear ill-posed operator equations in Hilbert spaces, inspired by the work of (2019 Fract. Calc. Appl. Anal. 22(3) 699-721). Our method is incorporated into the general framework of linear regularization and demonstrates that, under both Hölder and logarithmic source conditions, the S-FAR with fractional orders in the range (1, 2] offers accelerated convergence compared to comparable order optimal regularization methods. Additionally, we introduce a de-biasing strategy that significantly outperforms previous approaches, alongside a thresholding technique for achieving sparse solutions, which greatly enhances the accuracy of approximations. A variety of numerical examples, including one- and two-dimensional model problems, are provided to validate the accuracy and acceleration benefits of the FAR method.

In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on (P_1)-element in the z-direction and a finite-element (FE) discretization based on (P_1^{NC})-nonconforming element in the (x, y)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of (P_1^{NC})-nonconforming finite element methods in the 2D elliptic equation and the error estimation of the ({H^1})-norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the (P_1) element in the z-direction and the FE method based on the Crouzeix-Raviart element, the (P_1) linear element, the Park-Sheen element, and the (Q_1) bilinear element, respectively, in the (x, y)-plane.

We introduce a new system of surface integral equations for Maxwell’s transmission problem in three dimensions (3-D). This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well-posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self-adjoint, constraints-free linear system—expressed in the desirable form of an identity plus a compact weakly-singular operator—is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.

Problems with dominant advection, discontinuities, travelling features, or shape variations are widespread in computational mechanics. However, classical linear model reduction and interpolation methods typically fail to reproduce even relatively small parameter variations, making the reduced models inefficient and inaccurate. This work proposes a model order reduction approach based on the Radon Cumulative Distribution Transform (RCDT). We demonstrate numerically that this non-linear transformation can overcome some limitations of standard proper orthogonal decomposition (POD) reconstructions and is capable of interpolating accurately some advection-dominated phenomena, although it may introduce artefacts due to the discrete forward and inverse transform. The method is tested on various test cases coming from both manufactured examples and fluid dynamics problems.

The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). In Jia and Wang, SIAM J. Optim., 31, 887–914 2021. Z. Jia et al. considered the convergence of this method and established some a priori error bounds on the residual and the Lagrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we present a non-asymptotic bound for the convergence of the Lagrange multiplier and define a factor that plays an important role in the convergence of the Lagrange multiplier. Third, we revisit the convergence of the Krylov subspace method for the cubic regularization variant of the trust-region subproblem and substantially improve the convergence result established in Jia et al., SIAM J. Matrix Anal. Appl. 43 (2022), pp. 812–839 2022 on the multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.