Advances in Computational Mathematics最新文献

In this paper, we propose a discontinuous plane wave neural network (DPWNN) method with (hp-)refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with (h-)refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.

Exponential integrators are an efficient alternative to implicit schemes for the time integration of stiff system of differential equations. In this paper, low-rank exponential integrators of orders one and two for stiff differential Riccati equations are proposed and investigated. The error estimates of the proposed schemes are established. The proposed approach allows to overcome the main difficulties that lay in the interplay of time integration and low-rank approximation in the numerical schemes, which is uncommon in standard discretization of differential equations. Results of numerical experiments demonstrate the validity of the convergence analysis and show the performance of the proposed low-rank approximations with different settings.

In this paper, we adopt a quasi-boundary-value method to solve the nonlinear space-fractional backward problem with perturbed both final value and variable diffusion coefficient in general dimensional space, which is a severely ill-posed problem. The existence, uniqueness and stability of the solution for the quasi-boundary-value problem are proved. Convergence estimates are presented under an a-priori bound assumption of the exact solution. Finally, several numerical examples are given by the finite difference scheme and the fixed-point iteration method to show the effectiveness of the theoretical results.

We investigate numerical approximations for the stochastic Burgers equation driven by an additive cylindrical fractional Brownian motion with Hurst parameter (H in (frac{1}{2}, 1)). To discretize the continuous problem in space, a spectral Galerkin method is employed, followed by the presentation of a nonlinear-tamed accelerated exponential Euler method to yield a fully discrete scheme. By showing the exponential integrability of the stochastic convolution of the fractional Brownian motion, we present the boundedness of moments of semidiscrete and full-discrete approximations. Building upon these results and the convergence of the fully discrete scheme in probability proved by a stopping time technique, we derive the strong convergence of the proposed scheme.

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. 68, 337–404 1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.

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.