An inverse problem for the acoustics equations: A multilevel adaptive algorithm

A numerical solution to an inverse problem for the acoustic equations using an optimization method for a stratified medium is presented. With the distribution of an acoustic wave field on the medium’s surface, the 1D distributions of medium’s density, as well as the velocity and absorption coefficient of the acoustic wave, are determined. Absorption in a Voigt body model is considered. The conjugate gradients and the Newton method are used for minimization. To increase the efficiency of the numerical method, a multilevel adaptive algorithm is proposed. The algorithm is based on a division of the whole procedure of solving the inverse problem into a series of consecutive levels. Each level is characterized by the number of parameters to be determined at the level. In moving from one level to another, the number of parameters changes adaptively according to the functional minimized and the convergence rate. The minimization parameters are chosen as illustrated by results of solving the inverse problem in a spectral domain, where the desired quantities are presented as Chebyshev polynomial series and minimization is carried out with respect to the coefficients of these series. The method is compared in efficiency with a nonadaptive method. The optimal parameters of the multilevel method are chosen. It is shown that the multilevel algorithm offers several advantages over the one without partitioning into levels. The algorithm produces primarily a more accurate solution to the inverse problem.     

An Algorithm for Solving Inverse Geoelectrics Problems Based on the Neural Network Approximation

A neural network approximation algorithm for solving inverse geoelectrics problems in the class of grid (block) models of media is presented. The algorithm is based on using neural networks for constructing an approximate inverse operator and enables formalized construction of solutions of inverse geoelectrics problem with a total number of sought-for medium parameters of ~ n · 10³. The correctness of the problem of constructing neural network inverse operators is considered. A posteriori estimates of the degree of ambiguity of solutions of the resulting inverse problem are calculated. The operation of the algorithm is illustrated by examples of 2D and 3D inversions of synthetic and field geoelectric data obtained by the MTS method.     

Numerical solution of an inverse electrocardiography problem for a medium with piecewise constant electrical conductivity

A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.     

On an optimal method for solving an inverse Stefan problem

An algorithm optimal in order is proposed for solving an inverse Stefan problem. We also give some exact estimates of accuracy of this method.     

Radon Transform for Solving an Inverse Scattering Problem in a Planar Layered Acoustic Medium

A two-dimensional inverse scattering problem in a layered acoustic medium occupying a half-plane is considered. Data is the scattered wavefield from a surface point source measured on the boundary of the half-plane. On the basis of the Radon transform, an algorithm is constructed that recovers the velocity and the acoustic impedance of the medium from the scattering data. An analytical solution is presented for an inverse scattering problem, and several inverse scattering problems are solved numerically.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

The numerical solution of an inverse scattering problem for acoustic waves

In a previous paper, the authors presented a dual space methodfor the numerical solution of the two-dimensional inverse scatteringproblem for acoustic waves in an inhomogeneous medium. Here,by making major modifications to the dual space method, a dramaticimprovement in the numerical performance of this method is achievedfor solving the inverse scattering problem.     

On the dynamic solution of an operator inverse problem

We exhibit an algorithm for solving an operator inverse problem that is stable under informational noise and computational errors. The algorithm is based on the constructions of the theory of positional control and Tikhonov's regularization method (the smoothing functional method). Bibliography: 15 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 28–35.     

Optimization of the heating of a heat-sensitive layer under constraints on the stresses in the plastic region of deformation of the material

We study the problem of optimal control for rapidity of the heating of a heat-sensitive layer under constraints on the control (the temperature of the heating medium or the heat flux) and maximal values of the stress intensity in the plastic region of deformation of the material. We propose an algorithm for solving the problem that presumes it has been reduced to the inverse problem of thermoplasticity. For the case of one-sided heating we give a numerical analysis of the direct and inverse problems of thermoplasticity. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Numerical realization of an algorithm for reconstructing an inhomogeneous medium in an X-ray tomography problem

Under study is the X-ray tomography problem that is an inverse problem for the transport differential equation. We take into account the absorption of particles by the medium and their single scattering. The statement of the problem corresponds to multiple probing. The medium is unknown; while the densities of the outcoming flux averaged over energy are given. The object in question is the discontinuity surfaces of the coefficients of the equation. This corresponds to searching for the boundaries between various substances contained in the medium that we probe. The solution is constructive, and a numerical realization of the obtained algorithm is presented.     

An Iterative Algorithm for the Coefficient Inverse Problem of Differential Equations

In this paper, an iterative algorithm for solving a coefficient inverse problem is submitted. The key of the method is to project an unknown coefficient function on a finite dimensional function space. Thus, the inverse problem can be changed into a nonlinear algebraic system of equations.     

The numerical solution of an inverse periodic transmission problem

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We discuss convergence and implementation of an optimization method for solving the inverse TE transmission problem, following an approach first developed by Kirsch and Kress for acoustic obstacle scattering. The convergence analysis includes the case of Lipschitz grating profiles and relies on variational methods and solvability properties of periodic boundary integral equations. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Integral Equation Method for Modeling of Two-Dimensional Geoelectricity Problems

The article computes the electromagnetic field on the surface of a layered medium with a local nonhomogeneity. The problem is transformed from three- to two-dimensional and the singulari-ties are investigated using the integral equation method. The proposed algorithm efficiently sim-ulates two-dimensional H-polarization fields by solving a system of integral equations. The method is particularly effective for solving inverse problems. __________ Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 5–16, 2004.     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.     

Quasi-one-dimensional method for the two-dimensional inverse problem of magnetotelluric sounding

The article presents a quasi-one-dimensional method for solving the inverse problem of electromagnetic sounding. The quasi-one-dimensional method is an iteration process that in each iteration solves a parametric one-dimensional inverse problem and a two-dimensional direct problem. The solution results of these problems are applied to update the input values for the parametric one-dimensional inverse problem in the next iteration. The method has been implemented for a two-dimensional inverse problem of magnetotelluric sounding in a quasi-layered medium.     

二维恒定各向同性介质渗透系数反演的遗传算法

给出了利用遗传算法求解二维恒定各项同性介质渗透系数反演的一种新方法,该方法把参数反演问题转化为优化问题通过遗传算法求解.数值模拟结果表明:该方法具有精度高、收敛速度快、编程简单、易于计算机实现等优点,值得在实际工作采用.