Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

Korpela, Jussi; Lassas, Matti; Oksanen, Lauri

doi:10.3934/ipi.2019027

Cited by 9 publications

(5 citation statements)

References 57 publications

(122 reference statements)

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“…holds for every x ∈ X. Regularization strategies have been found for other inverse problems including, for example, electrical impedance tomography (EIT) [16] and inverse problem for the 1 + 1 dimensional wave equation [18,19]. We will next prove that the regularized inversion operator P s α I * obtained in theorem 2 actually provides an admissible regularization strategy with a quantitative stability estimate.…”

Section: 3mentioning

confidence: 91%

Torus Computed Tomography

Ilmavirta¹,

Koskela²,

Railo³

2020

SIAM J. Appl. Math.

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We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have implemented the Torus CT method using Matlab and tested it with simulated data with promising results. The inversion method is meshless in the sense that it gives out a closed form function that can be evaluated at any point of interest.

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Section: 3mentioning

confidence: 91%

Torus Computed Tomography

Ilmavirta¹,

Koskela²,

Railo³

2020

SIAM J. Appl. Math.

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“…This control problem can be solved via regularized minimization problems. In [40] the problem was solved using Tikhonov regularization, while in this paper we consider sparse regularization techniques that are closely related to neural networks.…”

Section: Boundary Controlmentioning

confidence: 99%

Deep learning architectures for nonlinear operator functions and nonlinear inverse problems

Hoop¹,

Lassas²,

Wong³

2019

Preprint

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We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating highly nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse problems for the wave equation. Traditional neural networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with the linear operators that correspond to the measurement data in such inverse problems. We therefore introduce a new parametric family that resembles a standard neural network architecture, but where the input data acts multiplicatively on vectors. Motivated by compact operators appearing in boundary control and the analysis of inverse boundary value problems for the wave equation, we promote structure and sparsity in selected weight matrices in the network. After describing this architecture, we study its representation properties as well as its approximation properties. We furthermore show that an explicit regularization can be introduced that can be derived from the mathematical analysis of the mentioned inverse problems, and which leads to some guarantees on the generalization properties. We observe that the sparsity of the weight matrices improves the generalization estimates. Lastly, we discuss how operator recurrent networks can be viewed as a deep learning analogue to deterministic algorithms such as boundary control for reconstructing the unknown wavespeed in the acoustic wave equation from boundary measurements.

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“…On the other hand, the linearized approach introduced in the present paper is stable. It should be mentioned that the BC method can be implemented in a stable way in the one-dimensional case, see [22]. For an interesting application of a variant of the method in the one-dimensional case, see [11] on detection of blockage in networks.…”

Section: Introductionmentioning

confidence: 99%

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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Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

Cited by 9 publications

References 57 publications

Torus Computed Tomography

Torus Computed Tomography

Deep learning architectures for nonlinear operator functions and nonlinear inverse problems

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

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