On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

2021 ◽  
Author(s):  
Jiaqing Yang ◽  
Meng Ding ◽  
Keji Liu
Keyword(s):  
Wave Equation ◽  
Inverse Problems ◽  
Bounded Domain ◽  
Open Subset ◽  
Wave Equations ◽  
A Priori ◽  
Singularity Analysis ◽  
Piecewise Constant ◽  
Small Domain ◽  
Dirichlet To Neumann Map

Abstract In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain Ω belongs to R^N (N >= 2) for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary \partial Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

scholarly journals On the method of calculating the modulus of continuity of the inverse operator and its modifications with application to non-linear problems of geoelectrics

2020 ◽  
pp. 350-372
Author(s):  
М.И. Шимелевич
Keyword(s):  
Inverse Problems ◽  
Modulus Of Continuity ◽  
A Priori Estimates ◽  
A Priori ◽  
Inverse Operator ◽  
Approximate Solutions ◽  
Piecewise Constant ◽  
Nonlinear Inverse Problems ◽  
Monte Carlo Algorithms ◽  
Priori Estimates

Рассматриваются априорные оценки неоднозначности (погрешности) приближенных решений условно-корректных нелинейных обратных задач, основанные на модуле непрерывности обратного оператора и его модификациях. Установлена связь модуля непрерывности обратного оператора с разрешающей способностью геофизического метода. Показано, что в классе кусочно-постоянных решений, определенных на заданной сетке параметризации, модуль непрерывности обратного оператора и его модификации монотонно возрастают с увеличением размерности сетки. Предложен метод построения оптимальной сетки параметризации, которая имеет максимальную размерность при условии, что модуль непрерывности обратного оператора не превышает заданной величины. Представлен численный алгоритм расчета модуля непрерывности обратного оператора и его модификаций с использованием алгоритмов Монте-Карло, исследуются вопросы сходимости алгоритма. Предлагаемый метод применим также для расчета классических апостериорных оценок погрешности. Приводятся численные примеры для нелинейных обратных задач геоэлектрики. The article considers a priori estimates of the ambiguity (error) of approximate solutions of conditionally correct nonlinear inverse problems based on the modulus of continuity of the inverse operator and its modifications. It is shown that in the class of piecewise constant solutions defined on a given parametrization grid, the modulus of continuity of the inverse operator and its modifications monotonously increase with increasing mesh dimension. A method is proposed for constructing an optimal parameterization grid that has a maximum dimension provided that the modulus of continuity of the inverse operator does not exceed a given value. A numerical algorithm for calculating the modulus of continuity of the inverse operator and its modifications using Monte Carlo algorithms is presented; questions of convergence of the algorithm are investigated. The proposed method is also applicable for calculating classical posterior error estimates. Numerical examples are given for nonlinear inverse problems of geoelectrics.

scholarly journals Counterexamples to inverse problems for the wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tony Liimatainen ◽  
Lauri Oksanen
Keyword(s):  
Wave Equation ◽  
Inverse Problems ◽  
Wave Operator ◽  
Time Dependent ◽  
Lorentzian Manifolds ◽  
Minkowski Metric ◽  
Lorentzian Metrics ◽  
Partial Data ◽  
Dirichlet To Neumann Map

<p style='text-indent:20px;'>We construct counterexamples to inverse problems for the wave operator on domains in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n \ge 2 $\end{document}</tex-math></inline-formula>, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula> the metrics are conformal to the Minkowski metric.</p>

scholarly journals Factorized One-Way Wave Equations

Acoustics ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 717-722
Author(s):  
Oskar Bschorr ◽  
Hans-Joachim Raida
Keyword(s):  
Wave Equation ◽  
Longitudinal Wave ◽  
Wave Equations ◽  
A Priori ◽  
Order Partial Differential Equation ◽  
Cauchy Equation ◽  
Huge Number ◽  
Wave Propagation Direction ◽  
Numerical Finite Element ◽  
Definition Of

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver.

scholarly journals Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers

2021 ◽  
Vol 9 ◽  
Author(s):  
Ali Feizmohammadi ◽  
Matti Lassas ◽  
Lauri Oksanen
Keyword(s):  
Wave Equation ◽  
Inverse Problems ◽  
Wave Equations ◽  
Hyperbolic Equations ◽  
Lorentzian Manifold ◽  
Linear Wave ◽  
Nonlinear Hyperbolic Equations ◽  
Solution Map ◽  
Finite Speed Of Propagation ◽  
Hyperbolic Spacetimes

Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension $n+1\geqslant 3$ and with partial data. We consider the case when the set $\Omega _{\mathrm{in}}$ , where the sources are supported, and the set $\Omega _{\mathrm{out}}$ , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\in \Omega _{\mathrm{in}}$ and the past of the point $p_{out}\in \Omega _{\mathrm{out}}$ . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in $\Omega _{\mathrm{in}}$ and observations in $\Omega _{\mathrm{out}}$ , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

On the existence of compressional MHD oscillations in an inhomogeneous magnetoplasma

1990 ◽  
Vol 44 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Andrew N. Wright
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Density Distribution ◽  
Cold Plasma ◽  
Wave Equations ◽  
Perturbation Field ◽  
Plasma Density Distribution ◽  
Background Magnetic Field ◽  
Transverse Fields ◽  
The Magnetic Field

In a cold plasma the wave equation for solely compressional magnetic field perturbations appears to decouple in any surface orthogonal to the background magnetic field. However, the compressional fields in any two of these surfaces are related to each other by the condition that the perturbation field b be divergence-free. Hence the wave equations in these surfaces are not truly decoupled from one another. If the two solutions happen to be ‘matched’ (i.e. V.b = 0) then the medium may execute a solely compressional oscillation. If the two solutions are unmatched then transverse fields must evolve. We consider two classes of compressional solutions and derive a set of criteria for when the medium will be able to support pure compressional field oscillations. These criteria relate to the geometry of the magnetic field and the plasma density distribution. We present the conditions in such a manner that it is easy to see if a given magnetoplasma is able to executive either of the compressional solutions we investigate.

scholarly journals Partial inverse problems for quadratic differential pencils on a graph with a loop

2020 ◽  
Vol 28 (3) ◽  
pp. 449-463 ◽  
Author(s):  
Natalia P. Bondarenko ◽  
Chung-Tsun Shieh
Keyword(s):  
Inverse Problems ◽  
A Priori ◽  
Spectral Characteristics ◽  
The Other ◽  
Boundary Edge ◽  
Partial Inverse ◽  
Quadratic Pencil ◽  
Sturm Liouville ◽  
Differential Pencils ◽  
Liouville Operators

AbstractIn this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

scholarly journals Partial inverse problems for Sturm–Liouville operators on trees

2017 ◽  
Vol 147 (5) ◽  
pp. 917-933 ◽  
Author(s):  
Natalia Bondarenko ◽  
Chung-Tsun Shieh
Keyword(s):  
Inverse Problems ◽  
A Priori ◽  
Potential Functions ◽  
Inverse Spectral Problems ◽  
Spectral Sets ◽  
Spectral Problems ◽  
Partial Inverse ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Liouville Operators

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

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