insight - Numerical analysis

### Unphysical Oscillations and Convergence of MFS Solutions for Two-Dimensional Laplace-Neumann Problems

The method of fundamental solutions (MFS) can generate convergent solutions to Laplace-Neumann problems even when the intermediate auxiliary source currents exhibit unphysical divergence and oscillations.

### Efficient Adaptive Sparse Spectral Method for Solving Multidimensional Spatiotemporal Integrodifferential Equations in Unbounded Domains

The authors develop an adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method to efficiently solve multidimensional spatiotemporal integrodifferential equations in unbounded domains. The AHMJ method uses adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, allowing for effective solution of various spatiotemporal integrodifferential equations with reduced numbers of basis functions. The analysis provides a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, enabling effective error control.

### Convergence Analysis of Iterative One-Shot Inversion Methods for Linear Inverse Problems

The core message of this article is to establish sufficient conditions on the descent step for the convergence of multi-step one-shot inversion methods, where the forward and adjoint problems are solved iteratively with incomplete inner iterations, for general linear inverse problems.

### Optimal Approximation of Snapshot Vectors using Proper Orthogonal Decomposition

The proper orthogonal decomposition (POD) method provides an optimal way to approximate a finite set of snapshot vectors in a Hilbert space using a low-dimensional subspace. The POD basis vectors are obtained as the eigenvectors of a specific linear operator associated with the snapshot data.

### Efficient Numerical Method for Solving Quasiperiodic Elliptic Equations and Computing Quasiperiodic Homogenized Coefficients

This study presents a highly efficient algorithm for solving quasiperiodic elliptic equations and computing homogenized coefficients for quasiperiodic multiscale problems. The key innovations include the use of the projection method to transform the quasiperiodic problem into a higher-dimensional periodic system, a compressed storage format for the stiffness matrix, and a diagonal preconditioner to accelerate the iterative solver.

### Iterative Splitting Methods for Solving Linear Systems

This paper introduces a general class of iterative splitting methods for solving linear systems, which include previously proposed methods like Jacobi, Gauss-Seidel, and some recently introduced variants as special cases. The authors analyze the convergence properties of this general class of methods, establishing connections between the partial order of the splittings and the convergence rates.

### Efficient Parallel-in-Time Preconditioned MINRES Solver for Wave Equations

The authors propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve the all-at-once linear system arising from the discretization of wave equations. The proposed preconditioner is Hermitian positive definite, enabling its use with the MINRES solver, and achieves a matrix-size independent convergence rate.

### Spectral Method for Fractional Integral Equations using Jacobi Fractional Polynomials

The authors present a spectral method for solving one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including those with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis of Jacobi fractional polynomials, which incorporate the algebraic singularities of the solution into the basis functions.

### Preserving Asymptotic and Relaxation Properties in Numerical Discretization of Partially Dissipative Hyperbolic Systems

The central finite-difference scheme preserves both the asymptotic behavior and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems satisfying the Kalman rank condition.

### Optimal Convergence Analysis of Novel Discontinuous Galerkin Methods for Convection-Dominated Problems

The authors propose and analyze a numerically stable and convergent discontinuous Galerkin (DG) scheme for solving convection-dominated convection-diffusion-reaction equations. The method uses a novel DG finite element differential calculus framework and achieves optimal convergence rates in both the diffusion-dominated and convection-dominated regimes.