Quasi-Two-Dimensional Coefficient Inverse Problem for the Wave Equation in a Weakly Horizontally Inhomogeneous Medium with Memory

В работе представлена обратная задача последовательного определения двух неизвестных - коэффициента, характеризующего свойства среды со слабо горизонтальной неоднородностью, и ядра интегрального оператора, описывающего память среды. Прямая начально-краевая задача содержит нулевые данные и граничное условие Неймана. В качестве дополнительной информации задается след на границе среды Фурье-образа решения прямой задачи. Для исследования обратных задач предполагается, что искомый коэффициент разлагается в асимптотический ряд по степеням малого параметра. В статье построен метод нахождения (с учетом памяти среды) коэффициента с точностью до поправки, имеющей порядок $O(\epsilon^2)$. На первом этапе одновременно определяется решение прямой задачи в нулевом приближении и ядро интегрального оператора, при этом обратная задача сводится к эквивалентной задаче решения системы нелинейных интегральных уравнений Вольтерра второго рода. На втором этапе ядро считается заданным, и одновременно определяется решение прямой задачи в первом приближении и искомый коэффициент. В этом случае решение эквивалентной обратной задачи будет решением линейной системы интегральных уравнений Вольтерра второго рода. Доказаны теоремы однозначной локальной разрешимости поставленных обратных задач. Приведены результаты численных расчетов функции ядра и коэффциента.

On the existence and uniqueness of the solution of the coefficient inverse problem for a semilinear parabolic equation

In this paper, we consider the problem of determining the source function and the coefficient by the derivative with respect to time in a semilinear parabolic equation with overdetermination conditions defined on two different hyperplanes. The existence and uniqueness theorems of the classical solution of the posed coefficient inverse problem in the class of smooth bounded functions were proved. An example of input data satisfying the conditions of the proved theorems is given.

On the modeling of ultrasound wave propagation in the frame of inverse problem solution

In this paper we consider the inverse problem of detecting the inclusions inside the human tissue by using the acoustic sounding wave. The problem is considered in the form of coefficient inverse problem for first-order system of PDE and we use the gradient descent approach to recover the coefficients of that system. The important part of the sceme is the solution of the direct and adjoint problem on each iteration of the descent. We consider two finite-volume methods of solving the direct problem and study their Influence on the performance of recovering the coefficients.

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.

THEORETICAL AND EXPERIMENTAL STUDY OF POROPERM PROPERTIES ANISOTROPY USING ARTIFICIAL GEOMATERIALS

The reservoir properties (reservoir properties) of reservoir rocks depend not only on the petrophysical characteristics of the formation, but also on the stresses acting in the rock mass. The redistribution of the latter during drilling causes the occurrence of permeability anisotropy in the near-wellbore zone, which affects not only the flow characteristics of the wells, but also the process of penetration of the drilling mud filtrate into the formation. This should be taken into account when building a geomechanical-electrohydrodynamic model of the near-wellbore space. The results of filtration tests of regularly inhomogeneous cylindrical samples made of artificial geomaterial are presented and a method for synthesizing the dependence of the effective permeability on the polar angle based on the solution of the coefficient inverse problem is proposed. The obtained dependencies added to the database of reservoir properties of rocks used for inversion of GIS data for quantitative assessment of reservoir properties, an example of which is presented in this article.

Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations

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.