arXiv subjects

Michelle Michelle

Publications and source records attributed to Michelle Michelle.

10 recordsLinked to original sources

arXiv2026

Arbitrarily High-Order Convergence of Wavelet-based Galerkin Scheme for 1D Elliptic Interface Problems

The solution $u$ of an elliptic interface problem in a domain $Ω$ is often smooth away from the interface $Γ\subset Ω$, but its gradient is discontinuous across $Γ$. Consequently, $u$ has low global regularity and generally does not belong to $H^{3/2}(Ω)$. This paper studies 1D elliptic interface problems using wavelet methods. We propose a Galerkin method based on compactly supported biorthogonal wavelet bases on bounded intervals with approximation order $m$, for any integer $m \ge 2$. Our approach involves incorporating wavelet basis functions from higher scale levels to capture the singularity in the neighbourhood of the interface $Γ$. A principal contribution of this paper is a rigorous convergence analysis establishing the optimal orders $m-1$ in the $H^1(Ω)$-norm and $m$ in the $L^2(Ω)$-norm. The proof combines careful decay estimates for dual wavelet coefficients associated with wavelets supported away from the interface $Γ$ and with those supported in its neighborhood, wavelet characterizations of Sobolev spaces, and standard convergence arguments from the finite element method (FEM). Various numerical experiments are provided to verify the theoretical findings, including a special two-dimensional elliptic interface problem with vertical interface lines.

math.NA↗

arXiv2026

Galerkin Scheme Using Biorthogonal Wavelets on Intervals for Elliptic Interface Problems

This paper presents a wavelet Galerkin method for solving elliptic interface problems of the form $-\nabla\cdot(a\nabla u)=f$ in $Ω\backslash Γ$, where $Γ$ is a smooth interface within $Ω$. Since the scalar variable coefficient $a>0$ and source term $f$ are often discontinuous across $Γ$, the solution $u$ typically has discontinuous gradient $\nabla u$ across $Γ$ and hence $u\not\in H^{1.5}(Ω)$, posing significant challenges for traditional numerical methods. By utilizing a compactly supported biorthogonal wavelet for $H^1_0(Ω)$, we develop a strategy that incorporates additional wavelet elements (or basis functions) along the interface to resolve the complex geometry of the interface $Γ$ and the resulting gradient discontinuities. For the two-dimensional (2D) elliptic interface problem, the proposed method achieves near-optimal convergence rates: $\mathcal{O}(h |\log(h)|)$ in the $H^1(Ω)$-norm and $\mathcal(h^2 |\log(h)|^2)$ in the $L^{2}$-norm with respect to the approximation order. A key theoretical contribution is the use of the dual biorthogonal wavelet basis to establish the $H^1(Ω)$ convergence results. This is supported by the development of weighted Bessel properties for wavelets and several inequalities in fractional Sobolev spaces. To maintain high accuracy and robustness against high-contrast coefficients, our method leverages an augmented set of wavelet elements, similar to meshfree approaches, thereby eliminating the need for the complex re-meshing required by finite element methods. Unlike existing techniques, this wavelet Riesz basis framework captures the geometry of $Γ$ seamlessly while ensuring that the condition numbers of the coefficient matrices remain small and uniformly bounded, independent of the problem size.

math.NA↗

arXiv2025

High-order, Compact, and Symmetric Finite Difference Methods for $d$-Dimensional Elliptic Equations

This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a $d$-dimensional hypercube. Our schemes produce symmetric linear systems: an important property that does not immediately hold for a high-order FDM. This symmetry, combined with the stencil's minimal support, keeps the storage requirements to a minimum. For the model problem considered here, the resulting linear systems are, in fact, symmetric positive definite, allowing a wide range of efficient solvers to be applied. Designing compact, symmetric, and high-order FDMs is challenging, because all overlapping stencils have to satisfy highly specific relations and central differences alone are not enough. We prove that a compact 3-point, symmetric 1D FDM on a uniform grid can achieve arbitrary consistency order. On the other hand, in the $d$-dimensional setting, where $d \ge 2$, the maximum consistency order that a compact $3^d$-point, symmetric FDM on a uniform grid can achieve is 4. If $d=2$ and the diffusion coefficient satisfies a certain derivative condition, the maximum consistency order is 6. Moreover, the compact $3^d$-point, symmetric, 4th-order FDMs for $d\ge 3$, can be conveniently expressed as a linear combination of two types of FDMs: one that depends on partial derivatives along one axis, and the other along two axes. All finite difference stencils are explicitly provided for ease of reproducibility.

math.NA↗

arXiv2025

Optimal Neural Network Approximation for High-Dimensional Continuous Functions

Recently, the authors of \cite{SYZ22} developed a neural network with width $36d(2d + 1)$ and depth $11$, which utilizes a special activation function called the elementary universal activation function, to achieve the super approximation property for functions in $C([a,b]^d)$. That is, the constructed network only requires a fixed number of neurons (and thus parameters) to approximate a $d$-variate continuous function on a $d$-dimensional hypercube with arbitrary accuracy. More specifically, only $\mathcal{O}(d^2)$ neurons or parameters are used. One natural question is whether we can reduce the number of these neurons or parameters in such a network. By leveraging a variant of the Kolmogorov Superposition Theorem, \textcolor{black}{we show that there is a composition of networks generated by the elementary universal activation function with at most $10889d + 10887$ nonzero parameters such that this super approximation property is attained. The composed network consists of repeated evaluations of two neural networks: one with width $36(2d+1)$ and the other with width 36, both having 5 layers.} Furthermore, we present a family of continuous functions that requires at least width $d$, and thus at least $d$ neurons or parameters, to achieve arbitrary accuracy in its approximation. This suggests that the number of nonzero parameters is optimal in the sense that it grows linearly with the input dimension $d$, unlike some approximation methods where parameters may grow exponentially with $d$.

cs.LG↗

arXiv2025

Wavelet Galerkin Method for an Electromagnetic Scattering Problem

The Helmholtz equation with variable wavenumbers is challenging to solve numerically due to the pollution effect, which often results in a huge ill-conditioned linear system. In this paper, we present a high-order wavelet Galerkin method to numerically solve an electromagnetic scattering from a large cavity problem modeled by the 2D Helmholtz equation with variable wavenumbers. The high approximation order and the sparse linear system with uniformly bounded condition numbers offered by wavelets are useful in dealing with the pollution effect. Using the direct approach in [B. Han and M. Michelle, Appl. Comp. Harmon. Anal., 53 (2021), 270-331], we present various optimized spline biorthogonal wavelets on a bounded interval. We provide a self-contained proof to show that the tensor product of such wavelets forms a 2D Riesz wavelet in the appropriate Sobolev space. Compared to the coefficient matrix of the finite element method (FEM), when an iterative scheme is applied to the coefficient matrix of our wavelet Galerkin method, much fewer iterations are needed for the relative residuals to be within a tolerance level. Furthermore, for a given bounded variable wavenumber, the number of required iterations is practically independent of the size of the wavelet coefficient matrix, due to the small uniformly bounded condition numbers of such wavelets. In contrast, when an iterative scheme is applied to the FEM coefficient matrix, the number of required iterations doubles as the mesh size for each axis is halved. The implementation can also be done conveniently thanks to the simple structure, the refinability property, and the analytic expression of our wavelet bases.

math.NA↗

arXiv2023

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method achieves a sixth order consistency for a constant wavenumber, and a fifth order consistency for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $\textsf{k} h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size.

math.NA↗

arXiv2022

Sharp Stability Wavenumber-explicit Bounds for 2D Helmholtz Equations

Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution, which brings about the pollution effect. A very fine mesh size is necessary to deal with a large wavenumber leading to a severely ill-conditioned huge coefficient matrix. To understand and tackle such challenges, it is crucial to analyze how the solution of the 2D Helmholtz equation depends on (perturbed) boundary and source data for large wavenumbers. In fact, this stability analysis is critical in the error analysis and development of effective numerical schemes. Therefore, in this paper, we analyze and derive several new sharp wavenumber-explicit stability bounds for the 2D Helmholtz equation with inhomogeneous mixed boundary conditions: Dirichlet, Neumann, and impedance. We use Fourier techniques, the Rellich's identity, and a lifting strategy to establish these stability bounds. Some examples are given to show the optimality of our derived wavenumber-explicit stability bounds.

math.AP↗

arXiv2021

Dirac Assisted Tree Method for 1D Heterogeneous Helmholtz Equations with Arbitrary Variable Wave Numbers

In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT, we propose a compact finite difference method (FDM) with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. This compact FDM is particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. DAT with such compact FDMs can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - $4 \times 4$ matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion. Several numerical examples are provided to illustrate the effectiveness of DAT using the $M$th order compact FDMs with $M=6,8$ for numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special 2D Helmholtz equations using DAT.

math.NA↗

arXiv2021

Wavelets on Intervals Derived from Arbitrary Compactly Supported Biorthogonal Multiwavelets

(Bi)orthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. A lot of problems in applications are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. From any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on $[0,\infty)$ or $[0,1]$ with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. (Multi)wavelets on intervals satisfying homogeneous boundary conditions will also be addressed. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. Several examples of orthogonal and biorthogonal multiwavelets on the interval $[0,1]$ will be provided to illustrate our construction approaches and proposed algorithms.

math.NA↗

arXiv2019

Wavelet-based Methods for Numerical Solutions of Differential Equations

Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing and computational mathematics. This paper primarily intends to shed some light on the advantages of using wavelets in the context of numerical differential equations. We shall identify a few prominent problems in this field and recapitulate some important results along these directions. Wavelet-based methods for numerical differential equations offer the advantages of sparse matrices with uniformly bounded small condition numbers. We shall demonstrate wavelets' ability in solving some one-dimensional differential equations: the biharmonic equation and the Helmholtz equation with high wave numbers (of magnitude $O(10^4)$ or larger).

math.NA↗