Multi-parameters identification problem for a degenerate parabolic equation

<abstract><p>In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.</p></abstract>

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.

Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method

The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

Identification of time-dependent convection coefficient in a time-fractional diffusion equation

We consider the inverse coefficient problem of identifying the principal coefficient in the damped wave equation , subject to the boundary conditions , , from the Neumann boundary output f (t): = r(0)ux(0,t), . We propose detailed microlocal analysis of the regularity of the solution of the wave equation in each subdomain defined by the characteristics of the wave equation. This analysis allows us to derive sufficient conditions for the regularity of the weak solution of the direct problem, especially along the characteristic lines, as well as to prove necessary energy estimates including also stability estimate for auxiliary hyperbolic problem. Based on this analysis we prove the compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator , corresponding to the inverse problem. The last property allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional and also Fréchet differentiability of this functional. For the case when and , a uniqueness theorem is derived. An explicit formula for the Fréchet gradient of the Tikhonov functional and its justification are derived by making use of the unique solution to corresponding adjoint problem. The approach proposed in this paper is expected to lead to very effective computational identification algorithms.

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

Temperature dependent thermal conductivity determination and source identification for nonlinear heat conduction by means of the Trefftz and homotopy perturbation methods

An inverse coefficient problem is considered for a stationary nonlinear convection- diffusion-reaction equation, in which reaction coefficient has a rather common dependence on substance concentration and on spacial variable. The solvability of the considered nonlinear boundary value problem is proved in a general case. The existence of solutions of the inverse problem is proved for the reaction coefficients, which are defined by the product of two functions. The first function depends on a spatial variable, the second one depends nonlinearly on the solution of the boundary value problem. The mentioned inverse problem consists in reconstructing the first function with the help of additional information provided by the solution of the boundary value problem.

We establish conditions for the unique existence of a solution of the inverse problem of simultaneous determination of two unknown coefficients in a parabolic equation. One of these coefficients is the leading coefficient that depends on time, and the other coefficient depends on a space variable.

An inverse boundary value problem for a third-order pseudo-hyperbolic equation with integral condition of the first kind is investigated.To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary problem and prove its equivalence (in a certain sense) to the original problem.Then using the Banach fixed point principle, the existence and uniqueness of a solution to this problem is shown.Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

: Coefficient Inverse Problems (CIPs) for Partial Differential Equations (PDEs) represent a very important tool for such needs of the Army as imaging of unknown targets hidden in cluttered heterogeneous backgrounds. The goal of this project is the development of globally convergent numerical methods for a wide class of CIPs. These methods are tested on mathematical models of the interest to the Army such as imaging of antipersonnel land mines and targets on battlefields covered by smogs and flames. In our definition global entails: (1) a rigorous convergence analysis that does not depend on the quality of the initial guess, and (2) numerical simulations that confirm the advertised convergence property. A conventional way to solve a CIP is via the minimization of a least squares objective functional. This functional characterizes misfit between the data and the solution of that PDE with a for the unknown coefficient. However, it is well known to researchers working on computations of inverse problems that the phenomenon of multiple local minima of these functionals represents the major obstacle for the development of reliable numerical methods for multidimensional CIPs. This phenomenon in turn is caused by the above mentioned non-linearity and ill-posedness. Because of local minima, one should somehow guess in advance about a good approximation for the solution. Without the availability of a first good guess, however, there is no guarantee that the calculated coefficient is indeed close to the correct one. In our terminology these are locally convergent numerical methods. In other words, their convergence to the correct solution can be guaranteed only if the starting point is located in a small neighborhood of this solution. Because of local minima, conventional numerical methods for multidimensional CIPs are locally convergent ones. However, in many important applications the first good guess is unavailable.

Identifying an unknown coefficient in the fractional parabolic differential equation