In this article we study inverse problems of recovering a space-time dependent source component from the lateral boundary observation in a subidffusion model. The mathematical model involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time, and a second-order elliptic operator with time-dependent coefficients. We establish a well-posedness and a conditional stability result for the inverse problems using a novel perturbation argument and refined regularity estimates of the associated direct problem. Further, we present an algorithm for efficiently and accurately reconstructing the source component, and provide several two-dimensional numerical results showing the feasibility of the recovery.
Some Unexplored Topics in The Reconstruction of Scalar Parameters of Subdiffusion
Istituto Italiano di Tecnologia（意大利理工学院）；S.P. Timoshenko Institute of Mechanics of NAS of Ukraine（乌克兰国家科学院谢尔盖·P·蒂莫申科力学研究所）；Institute of Applied Mathematics and Mechanics of NAS of Ukraine（乌克兰国家科学院应用数学与力学研究所）
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 develo…
Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation
Taras Shevchenko National University（塔拉斯·舍甫琴科国立大学）；Johann Radon Institute（约翰·拉德恩研究所）；Istituto Italiano di Tecnologia（意大利理工学院）；Institute of Applied Mathematics and Mechanics of NAS of Ukraine（乌克兰国家科学院应用数学与力学研究所）；S.P. Timoshenko Institute of Mechanics of NAS of Ukraine（乌克兰国家科学院S.P. 什捷潘诺夫力学研究所）
This article concerns second-order time discretization of subdiffusion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with time-dependent coefficients. Using these regularity results and a perturbation argument of freezing the diffusion coefficient, we prove that the convolution quadrature generated by the second-order backward differentiation formula, with proper correction at the first time step, can achieve second-order convergence for both nonsmooth initial data and incompatible source t…
Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a perturbation argument of freezing the diffusion coefficients, we show a stability estimate in Sobolev spaces, under some smallness/largeness condition on the terminal time. Moreover, in case of noisy observation, we apply a quasi-boundary value method to regularize the problem and then show the convergence of the regularization scheme. Finally, to numerically reconst…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Columbia University（哥伦比亚大学）
In this paper, we develop novel perturbation bounds for the high-order orthogonal iteration (HOOI) [DLDMV00b]. Under mild regularity conditions, we establish blockwise tensor perturbation bounds for HOOI with guarantees for both tensor reconstruction in Hilbert-Schmidt norm $\|\widehat{\bcT} - \bcT \|_{\tHS}$ and mode-$k$ singular subspace estimation in Schatten-$q$ norm $\| \sin Θ(\widehat{\U}_k, \U_k) \|_q$ for any $q \geq 1$. We show the upper bounds of mode-$k$ singular subspace estimation are unilateral and converge linearly to a quantity characterized by blockwise errors of the perturbat…
