Inverse Problems for the Heat Equation Using Conjugate Gradient Methods

In many engineering systems, e.g., in heat exchanges, reflux condensers, combustion chambers, nuclear vessels, etc. concerned with high temperatures/pressures/loads and/or hostile environments, certain properties of the physical medium, geometry, boundary and initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible. In such a situation, one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available. The purpose of this thesis is to determine the unknown space-dependent coefficients and/or initial temperature in inverse problems of heat transfer, especially to simultaneously reconstruct several unknown quantities. These inverse problems are investigated from additional pieces of information, such as internal temperature observations, final measured temperature and time-integral temperature measurement. The main difficulty involved in the solution of these inverse problems is that they are typically ill-posed. Thus, their solutions are unstable under small perturbations of the input data and classical numerical techniques fail to provide accurate and stable numerical results. Throughout this thesis, the inverse problems are transformed into optimization problems, and their minimizers are shown to exist. A variational method is employed to obtain their Fréchet gradients with respect to the unknown quantities. Based on this gradient, the conjugate gradient method (CGM) is established together with the adjoint and sensitivity problems. The stability of the numerical solution is investigated by introducing Gaussian random noise into the input measured data. Accurate and stable numerical solutions are obtained when using the CGM regularized by the discrepancy principle

Simultaneous determination of time and space-dependent coefficients in a parabolic equation

This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the chosen regularization parameters

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.Comment: Inverse source problem, Tikhonov regularization, Crank-Nicolson Galerkin method, Source condition, Convergence rates, Ill-posedness, Parabolic proble

A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science

In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented

Determination of an additive time- and space-dependent coefficient in the heat equation

Purpose: The purpose of this study is to provide an insight and to solve numerically the identification of an unknown coefficient of radiation/absorption/perfusion appearing in the heat equation from additional temperature measurements. Design/methodology/approach: First, the uniqueness of solution of the inverse coefficient problem is briefly discussed in a particular case. However, the problem is still ill-posed as small errors in the input data cause large errors in the output solution. For numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings: Numerical results presented for three examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Research limitations/implications: The mathematical formulation is restricted to identify coefficients which separate additively in unknown components dependent individually on time and space, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data are also limited to single measurements of temperature at a particular time and space location. Practical implications: As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise. Social implications: The identification of the additive time- and space-dependent perfusion coefficient will be of great interest to the bio-heat transfer community and applications. Originality/value: The current investigation advances previous studies which assumed that the coefficient multiplying the lower-order temperature term depends on time or space separately. The knowledge of this physical property coefficient is very important in biomedical engineering for understanding the heat transfer in biological tissues. The originality lies in the insight gained by performing for the first time numerical simulations of inversion to find the coefficient additively dependent on time and space in the heat equation from noisy measurements

Reconstruction of multiplicative space- and time-dependent sources

This paper presents a numerical regularization approach to the simultaneous determination of multiplicative space- and time-dependent source functions in a nonlinear inverse heat conduction problem with homogeneous Neumann boundary conditions together with specified interior and final time temperature measurements. Under these conditions a unique solution is known to exist. However, the inverse prob- lem is still ill-posed since small errors in the input interior temperature data cause large errors in the output heat source solution. For the numerical discretisation, the boundary element method combined with a regularized nonlinear optimization are utilized. Results obtained from several numerical tests are provided in order to illustrate the efficiency of the adopted computational methodology

Coefficient Identification Problems in Heat Transfer

The aim of this thesis is to find the numerical solution for various coefficient identification problems in heat transfer and extend the possibility of simultaneous determination of several physical properties. In particular, the problems of coefficient identification in a fixed or moving domain for one and multiple unknowns are investigated. These inverse problems are solved subject to various types of overdetermination conditions such as non-local, heat flux, Cauchy data, mass/energy specification, general integral type overdetermination, time-average condition, time-average of heat flux, Stefan condition and heat momentum of the first and second order. The difficulty associated with these problems is that they are ill-posed, as their solutions are unstable to inclusion of random noise in input data, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the Crank-Nicolson finite-difference method (FDM) is mainly used as a direct solver except in Chapter 7 where a three-level scheme is employed in order to deal with the nonlinear heat equation. An explicit FDM scheme is also employed in Chapter 10 for the two-dimensional case. The inverse problems investigated are discretised using the FDM and recast as nonlinear least-squares minimization problems with simple bounds on the unknown coefficients. The resulting problem is efficiently solved using the \emph{fmincon} or \emph{lsqnonlin} routines from MATLAB optimization toolbox. The Tikhonov regularization method is included where necessary. The choice of the regularization parameter(s) is thoroughly discussed. The stability of the numerical solution is investigated by introducing Gaussian random noise into the input data. The numerical solutions are compared with their known analytical solution, where available, and with the corresponding direct problem numerical solution where no analytical solution is available

Robust finite volume schemes for 2D shallow water models. Application to flood plain dynamics

This study proposes original combinations of higher order Godunov type finite volume schemes and time discretization schemes for the 2d shallow water equations, leading to fully second-order accuracy with well-balanced property. Also accuracy, positiveness and stability properties in presence of dynamic wet/dry fronts is demonstrated. The test cases are the classical ones plus extra new ones representing the geophysical flow features and difficulties