Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2009.03516

An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction

Bangti Jin·Zhi Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work, we study the inverse problem of recovering a potential coefficient in the subdiffusion model, which involves a Djrbashian-Caputo derivative of order $α\in(0,1)$ in time, from the terminal data. We prove that the inverse problem is locally Lipschitz for small terminal time, under certain conditions on the initial data. This result extends the result in Choulli and Yamamoto (1997) for the standard parabolic case to the fractional case. The analysis relies on refined properties of two-parameter Mittag-Leffler functions, e.g., complete monotonicity and asymptotics. Further, we develop an efficient and easy-to-implement algorithm for numerically recovering the coefficient based on (preconditioned) fixed point iteration and Anderson acceleration. The efficiency and accuracy of the algorithm is illustrated with several numerical examples.

Explore related subjects

math.NAcs.NAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bangti Jin, Zhi Zhou. 2020-09-08. An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction. https://arxiv.org/abs/2009.03516

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise

We establish sharp, uniform-in-time strong error estimates for a nonlinearity-explicit tamed finite element method (FEM) applied to a class of superlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise, including the stochastic Allen--Cahn equation with a moderately thick interface. This tamed-FEM was first introduced in {\it Z. Liu and J. Shen, J. Sci. Comput., 109:Paper No. 61, 26, 2026} to ensure long-time unconditional stability and to preserve the Lyapunov structure of this class of SPDEs. We further prove that the scheme is exponentially ergodic and derive the convergence rate of the numerical invariant measure to the exact one in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity, sharpness, and time-independence of the strong convergence rates for this tamed-FEM.

math.NA↗
arXiv2026

Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product

Tensor equations involving both left and right tensor operators arise naturally in applications where multidimensional data are coupled through transformations acting from both sides. Motivated by the need to analyze such equations, this manuscript investigates a detailed solution analysis of $\mathcal{A}\ltimes \mathcal{X}\ltimes \mathcal{B} = \mathcal{C}$ via the semi-tensor product with the $t$-product. The manuscript provides a more general framework that includes, as a special case, the tensor equation $\mathcal{A}\ltimes\mathcal{X}=\mathcal{B}$ considered in \cite{J.FathitensorequationAX=BunderSTP}. Necessary and sufficient conditions are derived for the existence of vector and matrix valued solutions. Explicit compatibility conditions are established, and constructive Moore-Penrose-inverse-based algorithms are provided for the vector and matrix valued cases. The computational complexity and execution times are compared for Discrete Fourier Transform(DFT) and Fast Fourier Transform(FFT) based \(t\)-product implementations. The practical relevance of the proposed framework is further demonstrated through a color image deblurring application. Examples throughout illustrate the theoretical results.

math.NA↗
arXiv2026

A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems

We design and analyze a nonoverlapping additive Schwarz preconditioner for interior penalty discontinuous Galerkin discretizations of anisotropic elliptic problems. The preconditioned method coupled with a Krylov subspace iteration is shown to be independent of the highly discontinuous (and anisotropic) jump coefficients as well as the subdomain size. To increase efficacy, various auxiliary spaces are considered to reduce the size of the coarse grid operator. We demonstrate how to modify the additive Schwarz preconditioner such that it is applicable to the nonsymmetric IPDG schemes. Several numerical experiments verify the theory and validate the robustness of the preconditioner.

math.NA↗