An inversion method for parabolic equations based on quasireversibility

AbstractThis paper is concerned with a new method to solve a linearized inverse problem for one-dimensional parabolic equations. The inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary. The method considers a temporal interval during which time dependent measurements are provided. It linearizes the working equation around the system response for a background medium. It is then possible to relate the inverse problem of interest to an ill-posed boundary value problem for a differential-integral equation, whose solution is obtained by the method of quasireversibility. This approach leads to an iterative method. A number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only

The method of fundamental solutions for some direct and inverse problems

We propose and investigate applications of the method of fundamental solutions (MFS) to several parabolic time-dependent direct and inverse heat conduction problems (IHCP). In particular, the two-dimensional heat conduction problem, the backward heat conduction problem (BHCP), the two-dimensional Cauchy problem, radially symmetric and axisymmetric BHCPs, the radially symmetric IHCP, inverse one and two-phase linear Stefan problems, the inverse Cauchy-Stefan problem, and the inverse two-phase one-dimensional nonlinear Stefan problem. The MFS is a collocation method therefore it does not require mesh generation or integration over the solution boundary, making it suitable for solving inverse problems, like the BHCP, an ill-posed problem. We extend the MFS proposed in Johansson and Lesnic (2008) for the direct one-dimensional heat equation, and Johansson and Lesnic (2009) for the direct one-phase one-dimensional Stefan problem, with source points placed outside the space domain of interest and in time. Theoretical properties, including linear independence and denseness, the placement of source points, and numerical investigations are included showing that accurate results can be efficiently obtained with small computational cost. Regularization techniques, in particular, Tikhonov regularization, in conjunction with the L-curve criterion, are used to solve the illconditioned systems generated by this method. In Chapters 6 and 8, investigating the linear and nonlinear Stefan problems, the MATLAB toolbox lsqnonlin, which is designed to minimize a sum of squares, is used

An approximation theory for the identification of linear thermoelastic systems

An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures

Determination of time-dependent coefficients for a weakly degenerate heat equation

In this paper, we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation, which vanishes at the initial moment of time, and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation, from some additional information about the process (the so-called over-determination conditions). Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution. The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed. The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. For both exact and noisy input data, accurate and stable numerical results are obtained

Numerical identification of initial temperatures in heat equation with dynamic boundary conditions

We investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data is provided after a final time. This is a backward parabolic problem which is severely ill-posed. As a first step, the problem is reformulated as an optimization problem with an associated cost functional. Using the weak solution approach, an explicit formula for the Fr\'echet gradient of the cost functional is derived from the corresponding sensitivity and adjoint problems. Then the Lipschitz continuity of the gradient is proved. Next, further spectral properties of the input-output operator are established. Finally, the numerical results for noisy measured data are performed using the regularization framework and the conjugate gradient method. We consider both one- and two-dimensional numerical experiments using finite difference discretization to illustrate the efficiency of the designed algorithm. Aside from dealing with a time derivative on the boundary, the presence of a boundary diffusion makes the analysis more complicated. This issue is handled in the 2-D case by considering the polar coordinate system. The presented method implies fast numerical results