Spacetime finite element methods for control problems subject to the wave equation

Burman, Erik; Feizmohammadi, Ali; Münch, Arnaud; Oksanen, Lauri

doi:10.48550/arxiv.2109.07890

Cited by 3 publications

(4 citation statements)

References 44 publications

(112 reference statements)

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“…see [14]. Hence the control Y that we obtained as the limit coincides with that given by [18, theorem 5.1], and…”

Section: Integral Identity and Controllabilitysupporting

confidence: 83%

Linearized boundary control method for an acoustic inverse boundary value problem

Oksanen¹,

Yang²,

Yang³

2022

Inverse Problems

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We develop a linearized boundary control method for the inverse boundary value problem of determining a potential in the acoustic wave equation from the Neumann-to-Dirichlet map. When the linearization is at the zero potential, we derive a reconstruction formula based on the boundary control method and prove that it is of Lipschitz-type stability. When the linearization is at a nonzero potential, we prove that the problem is of H"{o}lder-type stability in two and higher dimensions. The proposed reconstruction formula is implemented and evaluated using several numerical experiments to validate its feasibility.

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“…see [14]. Hence the control Y that we obtained as the limit coincides with that given by [18, theorem 5.1], and…”

Section: Integral Identity and Controllabilitysupporting

confidence: 83%

Linearized boundary control method for an acoustic inverse boundary value problem

Oksanen¹,

Yang²,

Yang³

2022

Inverse Problems

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“…see [14]. Hence the control Y that we obtained as the limit coincides with that given by [18, Theorem 5.1], and G(φ) = (Y0, Y1).…”

Section: Integral Identity and Controllabilitysupporting

confidence: 75%

A Linearized Boundary Control Method for the Acoustic Inverse Boundary Value Problem

Oksanen,

Yang,

Yang

2021

Preprint

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We develop a linearized boundary control method for the inverse boundary value problem of determining a potential in the acoustic wave equation from the Neumann-to-Dirichlet map. When the linearization is at the zero potential, we derive a reconstruction formula based on the boundary control method and prove that it is of Lipschitz-type stability. When the linearization is at a nonzero potential, we prove that the problem is of Hölder-type stability. The proposed reconstruction formula is implemented and evaluated using several numerical experiments to validate its feasibility.

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“…‖Γ 𝐼×𝜔 𝑧‖ 𝐿2(𝐼×𝜔) for functions 𝑧 that satisfy 𝑧 = 0 and 𝛾 𝐼×𝜕Ω 𝑧 = 0, see e.g., [30], and the Energy Estimate [18], which builds on Theorem 2.1 and 2.3 of [29]. From both these estimates, one easily derives (3.3) (cf.…”

Section: Applicationsmentioning

confidence: 97%

“…The use of conditional stability estimates for the numerical solution of various ill-posed PDEs has been advocated in series of papers [11][12][13][14][15][16][17][18][19]. In those works a control functional is minimized under the constraint that the state satisfies the PDE.…”

Section: Introductionmentioning

confidence: 99%

Least squares solvers for ill-posed PDEs that are conditionally stable

2023

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ABSTRACT. This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.

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Spacetime finite element methods for control problems subject to the wave equation

Cited by 3 publications

References 44 publications

Linearized boundary control method for an acoustic inverse boundary value problem

Linearized boundary control method for an acoustic inverse boundary value problem

A Linearized Boundary Control Method for the Acoustic Inverse Boundary Value Problem

Least squares solvers for ill-posed PDEs that are conditionally stable

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