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Let X be a Banach space with norm ∥ ⋅ ∥ . Let A : D ⁢ (A) ⊂ X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X satisfying u t + A ⁢ u = 0 , 0 < t < T , ∥ u ⁢ (T) - φ ∥ ⩽ ε , for φ in X, is regularized by the generalized Sobolev equation u α ⁢ t + A α ⁢ u α = 0 , 0 < t < T , u α ⁢ (T) = φ , where 0 < α < 1 and A α = A ⁢ (I + α ⁢ A b ) - 1 with b ⩾ 1 . Error estimates of the method with respect to the noise level are proved. [ABSTRACT FROM AUTHOR]

The article aimed to show the fundamental possibility of constructing a computational digital twin of the acoustic tomograph within the framework of a unified physics–mathematical model based on the Navier–Stokes equations. The authors suggested that the size of the modeling area is quite small, sound waves are waves of "small" disturbance, and given that a person consists of more than 60% water, human organs can be modeled using a liquid model, taking into account their density. During numerical experiments, we obtained the pressure registered in the receivers that are located on the side walls of the tomograph. The differences in pressure values are shown depending on the configuration of inclusions in the mannequin imitating internal organs. The results show that the developed technology can be used to probe the human body in medical acoustic tomographs and determine the acoustic parameters of the human body to detect neoplasms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations. [ABSTRACT FROM AUTHOR]

This paper considers a model for the accumulation of mutations in a population of mice with a weakened function of polymerases responsible for correcting DNA copying errors during cell division. The model uses the results of the experiment published by Japanese scientists, which contain data on the accumulation of phenotypic differences in three isolated groups of laboratory mice. We have developed a model for the accumulation of negative mutations. Since the accumulation of phenotypic differences in each of the three groups of mice occurred in its own way, we assumed that these differences were associated with genotypic differences in the zeroth generation and set the inverse problem of determining the initial distribution of these differences. Additional information for solving the inverse problem was a set of experimental data on the number of mutant lines and the number of individuals in each group of mice. The results obtained confirmed our assumption. [ABSTRACT FROM AUTHOR]

We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach. [ABSTRACT FROM AUTHOR]

The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 is significantly more efficient than Landweber iteration. [ABSTRACT FROM AUTHOR]

A problem of modeling radiation patterns of wave sources in two-dimensional acoustic tomography is considered. Each source has its own radiation patterns, and their modeling will be used to improve the solvability of inverse problems of recovering the acoustic parameters of human soft tissues and come closer to building a digital twin of acoustic tomography. The problem is considered as a control problem of the right side for the velocities by spatial variables. Two statements are investigated—control by time or space functions. A numerical solution method is implemented. The results of numerical calculations are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers' type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

We investigate the mathematical modeling of the 2D acoustic waves propagation, based on the conservation laws. The hyperbolic first-order system of partial differential equations is considered and solved by the method of S. K. Godunov. The inverse problem of reconstructing the density and the speed of sound of the medium is considered. We apply the gradient method to reconstruct the parameters of the medium. The gradient of the functional is obtained. Numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method. [ABSTRACT FROM AUTHOR]

The coefficient inverse problem for the acoustic equation is considered. We propose the method for reconstructing the density based on the N-approximation by the finite system of one-dimensional problems and the two-dimensional M. G. Krein approach. The two-dimensional analogue of the M. G. Krein approach is applied to reduce the non-linear inverse problem to a family of linear integral equations. We consider the fast algorithm for solving the relevant linear system, based on using the block-Toeplitz structure of the matrix. The algorithm applied to the M. G. Krein equation allows to obtain the solution of the whole family of the integral equations by solving only one linear system. Results of numerical calculations are presented. [ABSTRACT FROM AUTHOR]

The coefficient inverse problem for the two-dimensional wave equation is solved. We apply the Gelfand-Levitan approach to transform the nonlinear inverse problem to a family of linear integral equations. We consider the Monte Carlo method for solving the Gelfand-Levitan equation. We obtain the estimation of the solution of the Gelfand-Levitan equation in one specific point, due to the properties of the method. That allows the Monte Carlo method to be more effective in terms of span cost, compared with regular methods of solving linear system. Results of numerical simulations are presented. [ABSTRACT FROM AUTHOR]

An inverse problem of reconstructing the two-dimensional coefficient of thewave 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 andmemory requirements. Results of numerical calculations are presented. [ABSTRACT FROM AUTHOR]

This minisymposium, held at the 5th International Conference ''Inverse Problems: Modeling and Simulation'', May 24--29, 2010 in Antalya, Turkey, was dedicated to the 60th birthday of Michael V. Klibanov. [ABSTRACT FROM AUTHOR]

The inverse problem for the acoustic equation is considered. We propose a method of reconstruction of the density approximating 2D inverse acoustic problem by a finite system of one dimensional inverse acoustic problems. The 2D analogy of the Gel'fand--Levitan--Krein method is established. The inverse acoustic problem is formulated and the short outline of the history and development in this field are given in Section 1. In Section 2 we consider the 2D analogy of the Gel'fand--Levitan--Krein equation. The N-approximation of the Gel'fand--Levitan--Krein equation is obtained for inverse acoustic problem in Section 3. The numerical results are presented in Section 4. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems. [ABSTRACT FROM AUTHOR]

We consider the coefficient inverse problem for the first-order hyperbolic system, which describes the propagation of the 2D acoustic waves in a heterogeneous medium. We recover both the denstity of the medium and the speed of sound by using a finite number of data measurements. We use the second-order MUSCL-Hancock scheme to solve the direct and adjoint problems, and apply optimization scheme to the coefficient inverse problem. The obtained functional is minimized by using the gradient-based approach. We consider different variations of the method in order to obtain the better accuracy and stability of the appoach and present the results of numerical experiments. [ABSTRACT FROM AUTHOR]

We investigate the mathematical model of the 2D acoustic waves propagation in a heterogeneous domain. The hyperbolic first order system of partial differential equations is considered and solved by the Godunov method of the first order of approximation. This is a direct problem with appropriate initial and boundary conditions. We solve the coefficient inverse problem (IP) of recovering density. IP is reduced to an optimization problem, which is solved by the gradient descent method. The quality of the IP solution highly depends on the quantity of IP data and positions of receivers. We introduce a new approach for computing a gradient in the descent method in order to use as much IP data as possible on each iteration of descent. [ABSTRACT FROM AUTHOR]