Wavelet projection methods for solving pseudodifferential inverse problems

2014 ◽  
Vol 12 (03) ◽  
pp. 1450025 ◽  
Author(s):  
E. P. Serrano ◽  
M. I. Troparevsky ◽  
M. A. Fabio
Keyword(s):  
Inverse Problem ◽  
Approximate Solution ◽  
Inverse Problems ◽  
Pseudodifferential Operator ◽  
Projection Method ◽  
Projection Methods ◽  
Wavelet Basis ◽  
Partial Knowledge ◽  
Orthonormal Wavelet ◽  
Ill Posed

We consider the Inverse Problem (IP) associated to an equation of the form Af = g, where A is a pseudodifferential operator with symbol[Formula: see text]. It consists in finding a solution f for given data g. When the operator A is not strongly invertible and the data is perturbed with noise, the IP may be ill-posed and the solution must be approximate carefully. For the present application we regard a particular orthonormal wavelet basis and perform a wavelet projection method to construct a solution to the Forward Problem (FP). The approximate solution to the IP is achieved based on the decomposition of the perturbed data calculating the elementary solutions that are nearly the preimages of the wavelets. Based on properties of both, the basis and the operator, and taking into account the energy of the data, we can handle the error that arises from the partial knowledge of the data and from the non-exact inversion of each element of the wavelet basis. We estimate the error of the approximation and discuss the advantages of the proposed scheme.

scholarly journals Determination of conductivity in a heat equation

2000 ◽  
Vol 24 (9) ◽  
pp. 589-594 ◽  
Author(s):  
Ping Wang ◽  
Kewang Zheng
Keyword(s):  
Numerical Simulation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Heat Equation ◽  
Ill Posed ◽  
Smooth Data

We consider the problem of determining the conductivity in a heat equation from overspecified non-smooth data. It is an ill-posed inverse problem. We apply a regularization approach to define and construct a stable approximate solution. We also conduct numerical simulation to demonstrate the accuracy of our approximation.

scholarly journals О постановках обратных задач

2020 ◽  
pp. 203-206
Author(s):  
Yurii Menshikov
Keyword(s):  
Mathematical Model ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Inverse Problems ◽  
Solution Algorithms ◽  
The Mathematical Model

Some possible options for the formulation of inverse problems are considered. The ultimate research goals in these cases determine the algorithms for the approximate solution of the inverse problem and allow one to correctly interpret these solutions. Two main statements of inverse problems considered: inverse problems of synthesis and inverse problems of measurement. It is shown that in inverse synthesis problems one should not take into account the error of the mathematical model. In addition, it is possible in these cases to synthesize approximate solution algorithms that do not have a regularizing property. Examples of practical problems considered.

A linear filter approach to designing off‐diagonal damping matrices for least‐squares inverse problems

Geophysics ◽  
1992 ◽  
Vol 57 (7) ◽  
pp. 948-951 ◽  
Author(s):  
Livia J. Squires ◽  
Guillaume Cambois
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Linear Filter ◽  
Transmission Tomography ◽  
Seismic Wavelet ◽  
Consistent Model ◽  
Ill Posed ◽  
Band Limited ◽  
Filter Approach ◽  
Geometrical Relationships

Many geophysical data processing applications are formulated as either linear or linearized‐nonlinear inverse problems. Usually, the linear systems that describe these problems are ill‐posed. For example, seismic deconvolution is an ill‐posed inverse problem because the seismic wavelet is band limited. Borehole transmission tomography is ill‐posed because of insufficient ray coverage around the tomographic model (Worthington, 1984). Surface‐consistent statics’ estimation is intrinsically ill‐posed because of the geometrical relationships between the parameters defined by the surface-consistent model (Taner et al., 1974).

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

2018 ◽  
Vol 26 (2) ◽  
pp. 277-286 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Multiple Solutions ◽  
Convergence Rates ◽  
Nonlinear Inverse Problems ◽  
Range Type ◽  
Ill Posed ◽  
Tikhonov’S Regularization

AbstractVariational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

scholarly journals ON THE SELF-REGULARIZATION OF ILL-POSED PROBLEMS BY THE LEAST ERROR PROJECTION METHOD

2014 ◽  
Vol 19 (3) ◽  
pp. 299-308 ◽  
Author(s):  
Alina Ganina ◽  
Uno Hamarik ◽  
Urve Kangro
Keyword(s):  
Approximate Solution ◽  
Projection Method ◽  
Regularization Method ◽  
The Self ◽  
Noise Levels ◽  
Right Hand ◽  
Ill Posed ◽  
The Right

We consider linear ill-posed problems where both the operator and the right hand side are given approximately. For approximate solution of this equation we use the least error projection method. This method occurs to be a regularization method if the dimension of the projected equation is chosen properly depending on the noise levels of the operator and the right hand side. We formulate the monotone error rule for choice of the dimension of the projected equation and prove the regularization properties.

Inverse problems in Pareto’s demand theory and their applications to analysis of stock market crises

2018 ◽  
Vol 26 (1) ◽  
pp. 95-108
Author(s):  
Nikolay I. Klemashev ◽  
Alexander A. Shananin ◽  
Shuhua Zhang
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Stock Market ◽  
Nonparametric Method ◽  
Chinese Stock Market ◽  
Stock Market Crash ◽  
Demand Theory ◽  
Ill Posed

AbstractWe develop an approach to the analysis of stock market crises based on the generalized nonparametric method. The generalized nonparametric method is based on solvability and regularization of ill-posed inverse problem in Pareto’s demand theory. Our approach allows one to select a few companies that may be considered as the main reason for the crisis. We apply this approach to study the Chinese stock market crash in 2015.

scholarly journals On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction–Diffusion–Advection-Type with Data on the Position of a Reaction Front

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2894
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Dmitry Lukyanenko ◽  
Maxim Shishlenin
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Inverse Problems ◽  
Reaction Front ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Singularly Perturbed Equations ◽  
Coefficient Inverse Problems

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction–diffusion–advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

Numerical solution of an inverse problem of coefficient recovering for a wave equation by a stochastic projection methods

2015 ◽  
Vol 21 (3) ◽  
Author(s):  
Sergey I. Kabanikhin ◽  
Karl K. Sabelfeld ◽  
Nikita S. Novikov ◽  
Maxim A. Shishlenin
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Projection Method ◽  
Low Cost ◽  
Initial Approximation ◽  
Projection Methods ◽  
Smoothing Spline ◽  
Random Errors ◽  
Linear Integral ◽  
Linear Integral Equations

AbstractAn inverse problem of reconstructing the two-dimensional coefficient of the wave equation is solved by a stochastic projection method. We apply the Gel'fand–Levitan approach to reduce the nonlinear inverse problem to a family of linear integral equations. The stochastic projection method is applied to solve the relevant linear system. We analyze the structure of the problem to increase the efficiency of the method by constructing an improved initial approximation. A smoothing spline is used to treat the random errors of the method. The method has low cost and memory requirements. Results of numerical calculations are presented.

Methods of solving ill-posed inverse problems

1983 ◽  
Vol 45 (5) ◽  
pp. 1237-1245 ◽  
Author(s):  
O. M. Alifanov
Keyword(s):  
Inverse Problems ◽  
Ill Posed

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.

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