arXiv subjects

Qiwei Zhan

Publications and source records attributed to Qiwei Zhan.

3 recordsLinked to original sources

arXiv2025

MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation

In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schr\"{o}dinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matrix exponential integrators, which enables us to derive algebraic preconditioners that rely solely on sparse matrix-vector products. Spectral analysis and a detailed numerical implementation strategy, including performance improvements achieved through complex shifting, are discussed. Finally, numerical experiments on 2D and large-scale 3D homogeneous and inhomogeneous models, including benchmark seismic examples, substantiate the effectiveness and scalability of the proposed methods.

math.NA↗

arXiv2024

On-Chip Vectorial Structured Light Manipulation via Inverse Design

On-chip structured light, with potentially infinite complexity, has emerged as a linchpin in the realm of integrated photonics. However, the realization of arbitrarily tailoring a multitude of light field dimensions in complex media remains a challenge1, Through associating physical light fields and mathematical function spaces by introducing a mapping operator, we proposed a data-driven inverse design method to precisely manipulate between any two structured light fields in the on-chip high-dimensional Hilbert space. To illustrate, light field conversion in on-chip topological photonics was achieved. High-performance topological coupling devices with minimal insertion loss and customizable topological routing devices were designed and realized. Our method provides a new paradigm to enable precise manipulation over the on-chip vectorial structured light and paves the way for the realization of complex photonic functions.

physics.optics↗

arXiv2021

Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

In this paper, we investigate fast algorithms in the small fraction order regime to approximate the Caputo derivative $^C_0D_t^αu(t)$ when $α$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme proposed in [26], and we show that the latter is superior when $α$ is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as $α$ gets small. Hence, only minimal cost for the fast evaluation is required in the small $α$ regime, which matches prevailing protocols in engineering practice. We also present improved stability and error analysis of FIDR for solving linear fractional diffusion equations, which achieves clear dependence of the error bound on the fraction order $α$. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when $α$ is small.

math.NA↗