On the recovering of acoustic attenuation in 2D acoustic tomography

The inverse problem of recovering the acoustic attenuation in the inclusions inside the human tissue is considered. The coefficient inverse problem is formulated for the first-order system of PDE. We reduce the inverse problem to the optimization of the cost functional by gradient method. The gradient of the functional is determined by solving a direct and conjugate problem. Numerical results are presented.

METHODS OF OPTIMIZATION OF PARAMETRIC SYSTEMS

The paper considers methods of parametric optimization of a dynamical system, which is described by a parametric system of differential equations. The gradient of the functional in the form of Boltz is found, which is the basis of methods such as gradient descent. Another method is based on the application of the sensitivity function.

Loaded equations, arising during synthesis of control of rod heating by moving sources

The article proposes an approach to solving the problem of synthesis of motion and power control of lumped sources. For concreteness, the problem of linear feedback control of moving heat sources during heating of the rod is considered. The powers and motion of point sources, participating in the right-hand side of the differential equation of parabolic type, are determined depending on the measured values of the process state at the points of measurement. As a result, the right-hand side of the differential equation linearly depends on the values of the process state at the given points of the bar. Formulas for the components of the gradient of the functional with respect to the parameters of linear feedback are obtained, which make it possible to use first-order optimization methods for the numerical solution of synthesis problems.

Space-time finite element method for determination of a source in parabolic equations from boundary observations

A novel inverse source problem concerning the determination of a term in the right-hand side of parabolic equations from boundary observation is investigated. The observation is given by an imprecise Dirichlet data on some part of the boundary. The unknown heat source is sought as a function depending on both space and time variables with an a priori information. The problem is reformulated as an optimal control problem with a Tikhonov regularization term. The gradient of the functional is derived via an adjoint problem. The space-time discretization approach is employed which allows the use of general space-time finite elements. The convergence of the approach is proved. Some numerical examples are presented for showing the efficiency of the approach.

Numerics of acoustical 2D tomography based on the conservation laws

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.

NUMERICAL SOLUTION OF THE INVERSE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS

The article considers direct and inverse problems of a system of nonlinear differential equations. Such problems are often found in various fields of science, especially in medicine, chemistry and economics. One of the main methods for solving nonlinear differential equations is the numerical method. The initial direct problem is solved by the Rune-Kutta method with second accuracy and graphs of the numerical solution are shown. The inverse problem of finding the coefficients of a system of nonlinear differential equations with additional information on solving the direct problem is posed. The numerical solution of this inverse problem is reduced to minimizing the objective functional. One of the methods that is applicable to nonsmooth and noisy functionals, unconditional optimization of the functional of several variables, which does not use the gradient of the functional, is the Nelder-Mead method. The article presents the NellerMead algorithm. And also a numerical solution of the inverse problem is shown.

On Determining Initial Conditions of Equations Flexural-Torsional Vibrations of a Bar

The problem of finding the initial conditions in the boundary-value problem for the system of flexural-torsional vibrations of a bar with additional conditions on the straight line is reduced to an optimal control problem and studied by the methods of optimal control theory. The gradient of the functional is calculated and using the gradient expression a necessary and sufficient optimality condition are proved.

On the data completion problem and the inverse obstacle problem with partial Cauchy data for Laplace’s equation

We study the inverse problem of obstacle detection for Laplace’s equation with partial Cauchy data. The strategy used is to reduce the inverse problem into the minimization of a cost-type functional: the Kohn–Vogelius functional. Since the boundary conditions are unknown on an inaccessible part of the boundary, the variables of the functional are the shape of the inclusion but also the Cauchy data on the inaccessible part. Hence we first focus on recovering these boundary conditions, i.e. on the data completion problem. Due to the ill-posedness of this problem, we regularize the functional through a Tikhonov regularization. Then we obtain several theoretical properties for this data completion problem, as convergence properties, in particular when data are corrupted by noise. Finally we propose an algorithm to solve the inverse obstacle problem with partial Cauchy data by minimizing the Kohn–Vogelius functional. Thus we obtain the gradient of the functional computing both the derivatives with respect to the missing data and to the shape. Several numerical experiences are shown to discuss the performance of the algorithm.

On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions

This paper presents theoretical results about control of the coefficient function in a hyperbolic problem with Dirichlet conditions. The existence and uniqueness of the optimal solution for optimal control problem are proved and adjoint problem is used to obtain gradient of the functional. However, a second adjoint problem is given to calculate the gradient on the space W210,l. After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.

Determination of the initial condition in parabolic equations from boundary observations

The problem of determining the initial condition in parabolic equations from boundary observations is studied. It is reformulated as a variational problem and then a formula for the gradient of the functional to be minimized is derived via an adjoint problem. The variational problem is discretized by finite difference splitting methods and solved by the conjugate gradient method. Some numerical examples are presented to show the efficiency of the method. Also as a by-product of the variational method, we propose a numerical scheme for numerically estimating singular values of the solution operator in the inverse problem.