scholarly journals Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jone Apraiz ◽  
Jin Cheng ◽  
Anna Doubova ◽  
Enrique Fernández-Cara ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Spatial Dimension ◽  
Numerical Reconstruction ◽  
Spatial Interval ◽  
Boundary Information ◽  
Size Estimates

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

Regularization by Denoising for simultaneous source separation

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Breno Bahia ◽  
Rongzhi Lin ◽  
Mauricio Sacchi
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Data Processing ◽  
Spectrum Analysis ◽  
Numerical Experiments ◽  
Source Separation ◽  
Singular Spectrum Analysis ◽  
Source Data ◽  
Blended Data ◽  
Simultaneous Source

Denoisers can help solve inverse problems via a recently proposed framework known as regularization by denoising (RED). The RED approach defines the regularization term of the inverse problem via explicit denoising engines. Simultaneous source separation techniques, being themselves a combination of inversion and denoising methods, provide a formidable field to explore RED. We investigate the applicability of RED to simultaneous-source data processing and introduce a deblending algorithm named REDeblending (RDB). The formulation permits developing deblending algorithms where the user can select any denoising engine that satisfies RED conditions. Two popular denoisers are tested, but the method is not limited to them: frequency-wavenumber thresholding and singular spectrum analysis. We offer numerical blended data examples to showcase the performance of RDB via numerical experiments.

scholarly journals Recovery of dipolar sources and stability estimates

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4095-4114
Author(s):  
Ridha Mdimagh
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Time Dependent ◽  
Stability Estimates ◽  
Lateral Boundary ◽  
Lipschitz Stability ◽  
Spatial Locations ◽  
Stability Results

The inverse problem of identifying dipolar sources with time-dependent moments, located in a bounded domain, via the heat equation is investigated, by applying a heat flux, and from a single lateral boundary measurement of temperature. An uniqueness, and local Lipschitz stability results for this inverse problem are established which are the main contributions of this work. A non-iterative algebraic algorithm based on the reciprocity gap concept is proposed, which permits to determine the number, the spatial locations, and the time-dependent moments of the dipolar sources, Some numerical experiments are given in order to test the efficiency and the robustness of this method.

A NEW METHOD FOR THE SOLUTION OF AN INVERSE PROBLEM FOR THE WAVE EQUATION

1994 ◽  
Vol 04 (05) ◽  
pp. 741-754
Author(s):  
V.I. AGOSHKOV ◽  
F. MAGGIO
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Approximate Solution ◽  
Iterative Process ◽  
Numerical Experiments ◽  
New Method

In this paper we consider an inverse problem for the wave equation. We prove the solvability of the problem and derive the iterative process for the computation of the approximate solution. We present the results of two numerical experiments.

Travel time(like) variables and the solution of velocity inverse problems

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 758-766 ◽  
Author(s):  
Frank G. Hagin ◽  
Samuel H. Gray
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Travel Time ◽  
Arbitrary Point ◽  
Three Dimensions ◽  
Critical Position ◽  
Straight Line ◽  
Definition Of ◽  
Inversion Techniques

It was demonstrated earlier that, in the solution of 1-D inverse problems, there is great advantage to changing the independent variable from, say, z (depth) to travel time [Formula: see text]. Essentially the advantage comes from the fact that the acoustic wave equation in travel time has the unknown c(z) appearing in a less critical position. The current paper takes a step toward applying these ideas to the much harder, but more interesting, inverse problem in three dimensions. There is no simple 3-D analog of the above definition of τ. However, it is shown that a surprisingly effective way of decomposing travel time into x, y, z components is straightforward. These are defined via line integrals from, say, (0, 0, 0) to an arbitrary point (x, y, z) along the straight line connecting the points, thus approximating the more natural integrals along the unknown raypath. These line integrals define the new coordinates, and the associated wave equation is derived and then simplified by dropping less important terms. The inverse problem is then attacked in this setting using the 3-D inversion techniques of Cohen and Bleistein (1979). The resulting algorithm is demonstrated to be very similar to those earlier results; however, it is shown that for a single reflecting plane the new results are of “second‐order” accuracy as oppose to first order (when the change in c is small relative to c itself). The algorithm also has some similarity to that of Raz (1982), and some comparisons between these two results are made.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

Variational-hemivariational inverse problem for electro-elastic unilateral frictional contact problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Othmane Baiz ◽  
Hicham Benaissa ◽  
Zakaria Faiz ◽  
Driss El Moutawakil
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Contact Problem ◽  
Existence And Uniqueness ◽  
Frictional Contact ◽  
Hemivariational Inequalities ◽  
Frictional Contact Problem ◽  
Uniqueness Of A Solution

AbstractIn the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results.

scholarly journals Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

2020 ◽  
Vol 26 ◽  
pp. 7
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Spectral Properties ◽  
Seismic Waves ◽  
Forced Vibrations ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Spatial Interval ◽  
Rational Multiple ◽  
Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

Sign in / Sign up

Export Citation Format

Share Document