A new method of solving the coefficient inverse problem

This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.     

Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination

The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem.     

Existence and uniqueness of an inverse problem for nonlinear Klein‐Gordon equation

In this paper, an initial boundary value problem for nonlinear Klein‐Gordon equation is considered. Giving an additional condition, a time‐dependent coefficient multiplying nonlinear term is determined, and existence and uniqueness theorem for small times is proved. The finite difference method is proposed for solving the inverse problem.     

一类二次规划逆问题的交替方向数值方法

考虑求解一类二次规划逆问题的交替方向数值算法．首先给出矩阵变量子问题解的显示表达式,而后构造了两个求解向量变量子问题近似解的数值算法,其中一个算法基于不动点原理,另一算法则应用半光滑牛顿法．数值实验表明,所提出的算法能够快速高效地求解二次规划逆问题．     

Globally strongly convex cost functional for a coefficient inverse problem

A Carleman Weight Function (CWF) is used to construct a new cost functional for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of an arbitrary size in a certain Sobolev space, one can choose the parameter of the CWF in such a way that the constructed cost functional will be strongly convex on that set. Next, convergence of the gradient method, which starts from an arbitrary point of that set, is established. Since restrictions on the size of that set are not imposed, then this is the global convergence.     

The enclosure method for an inverse problem arising from a spot welding

A two dimensional version of a reconstruction problem of an unknown weld on the interface between two electric conductive plates is considered. It is assumed that the two plates have a same known isotropic homogeneous conductivity, and the line where the welding area is located is known. Under these assumptions, an explicit extraction formula of the location of the tips of the welding area on the line from a single set of an electric current density and the corresponding voltage potential on the boundary of the material formed by the plates is given. This result may have possibility of application to quality evaluation of spot welding fixation strength of a lamina. Copyright © 2015 John Wiley & Sons, Ltd.     

Determination of the Robin coefficient in a nonlinear boundary condition for a steady‐state problem

In this paper, a constant heat transfer coefficient present in a nonlinear Robin‐type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one‐dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Uniqueness and stability in an inverse problem for a Poisson's equation

Consider the Poisson's equation(?)″(x)=-e~(v-(?)) e~((?)-v)-N(x)with the Diriehlet boundary data,and we mainly investigate the inverse problem of determining the unknown function N(x)from a parameter function family.Some uniqueness and stability results in the inverse problem are obtained.     

A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients q(s) and p(s) in the equation u _tt = a ² u _xx + q(u)u _t ? p(u)u _x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.     

Direct problem and inverse problem for the supersonic plane flow past a curved wedge

In this paper, we consider the isentropic irrotational steady plane flow past a curved wedge. First, for a uniform supersonic oncoming flow, we study the direct problem: For a given curved wedge y = f(x), how to globally determine the corresponding shock y = g(x) and the solution behind the shock? Then, we solve the corresponding inverse problem: How to globally determine the curved wedge y = f(x) under the hypothesis that the position of the shock y = g(x) and the uniform supersonic oncoming flow are given? This kind of problems plays an important role in the aviation industry. Under suitable assumptions, we obtain the global existence and uniqueness for both problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method

Inverse (time-reverse) simulation of three-dimensional thermoconvective flows is considered for a highly viscous incompressible fluid with temperature-dependent density and viscosity. The model of the fluid dynamics is described by the Stokes equations, the incompressibility and heat balance equations subject to the appropriate initial and boundary conditions. To solve the problem backward in time, the quasi-reversibility method is applied to the heat balance equation. The numerical solution is based on the introduction of a two-component vector potential for the velocity of the medium, on the application of the finite element method with a special tricubic spline basis for computing this potential, and on the application of the splitting method and the method of characteristics for computing the temperature. The numerical algorithm is designed to be executed on parallel computers. The proposed numerical algorithm is used to reconstruct the evolution of diapiric structures in the Earth’s upper mantle. The computational efficiency of the algorithm is analyzed on the basis of the appropriate functionals of residuals.     

Determination of an unknown function in a parabolic inverse problem by Sinc‐collocation method

In this article, an inverse problem of determining an unknown time‐dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution‐type. By using Sinc‐collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1584–1598, 2010     

An inverse backward problem for degenerate parabolic equations

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017     

Stability estimate for an inverse problem for the wave equation in a magnetic field

In this article, we prove stability estimate of the inverse problem of determining the magnetic field entering the magnetic wave equation in a bounded smooth domain in ? ^d from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the magnetic wave equation. We prove in dimension d ≥ 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic wave equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.     

A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity

We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K₀(t)+ϵxK₁(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K₀,K₁ up to a correction of the order of O(ϵ²).     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016     

Uniqueness and stability in an inverse problem for a Poisson’s equation

Consider the Poisson's equation ψ" (x) ＝ -ev-ψ eψ-v-N(x) with the Dirichlet boundary data, and we mainly investigate the inverse problem of determining the unknown function N(x) from a parameter function family. Some uniqueness and stability results in the inverse problem are obtained.     

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

Direct numerical method for an inverse problem of a parabolic partial differential equation

A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function u(x,t) and the unknown coefficient a(t) in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.