An inverse potential problem for subdiffusion: stability and reconstruction*

Jin, Bangti

doi:10.1088/1361-6420/abb61e

Cited by 21 publications

(9 citation statements)

References 41 publications

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“…Choulli and Yamamoto proved a generic well-posedness result in a Hölder space [4], and then proved a conditional stability result in a Hilbert space setting [5] for sufficiently small T . By using refined properties of two-parameter Mittag-Leffler functions, e.g., complete monotonicity and asymptotics, a similar result was proved in [16] for the case that α ∈ (0, 1). Kaltenbacher and Rundell [18] proved the invertibility of the linearized map (of the direct problem) from the space L 2 (Ω) to H 2 (Ω) under the condition u 0 > 0 in Ω and q ∈ L ∞ (Ω) using a Paley-Wiener type result and a type of strong maximum principle.…”

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confidence: 61%

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Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Zhang¹,

Zhang²,

Zhou³

2022

Preprint

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The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived a priori error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.

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confidence: 61%

“…[37, Lemma 2.2] and [32]), which is not applicable in high dimensional cases. We refer interested readers to [4,5,16] for some conditional stability results for sufficiently small T .…”

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confidence: 99%

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Zhang¹,

Zhang²,

Zhou³

2022

Preprint

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“…It has been shown that Anderson acceleration improves the convergence rate of contractive fixed-point iterations in the vicinity of a fixed-point, [4], but will actually slow the rate of quadratically convergent schemes such as those based on Newton methods. The use of this acceleration method has so far been quite rare in the inverse problems literature, but see [8].…”

Section: Andersen Accelerationmentioning

confidence: 99%

Determining the nonlinearity in an acoustic wave equation

Kaltenbacher,

Rundell

2021

Preprint

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We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form pp t . However, this should be considered as a low order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f (p) p t and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consists of time trace observations of the acoustic pressure at a single point or on a one dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of wellposedness of the resulting PDE, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f . Numerical simulations will also be shown to illustrate the efficiency of the methods.

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“…There are two closely related inverse problems to the concerned one. (i) is to recover the spatially dependent potential q from the terminal data u(T ) [20,23,47], which enjoys much better stability estimates (e.g. local Lipschitz stability) and effective iterative algorithms for numerical recovery, e.g.…”

Section: Introductionmentioning

confidence: 99%

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source ^*

Jin

2021

Inverse Problems

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This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order α ∈ (0, 1) in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatiallydependent potential coefficient and the order α of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from t = 0. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.

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An inverse potential problem for subdiffusion: stability and reconstruction*

Cited by 21 publications

References 41 publications

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Determining the nonlinearity in an acoustic wave equation

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source ^*

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An inverse potential problem for subdiffusion: stability and reconstruction*

Cited by 21 publications

References 41 publications

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

Determining the nonlinearity in an acoustic wave equation

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source *

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Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source ^*