We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.
The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.
We consider an inverse problem for the two-dimensional steady-state heat equation. More precisely, the heat equation is valid in a domain O, that is a subset of the unit square. Temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and 1 are assumed to be insulated. From these we wish to determine the temperature in the domain O. Furthermore, a part of the boundary ?O is considered to be unknown, and must also be determined. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least-squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equations, which can be solved using standard numerical methods. The inverse problem that we consider in this paper arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an inverse heat conduction problem.
We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.
The Cauchy problem for the parabolic heat equation, consisting of the reconstruction of the solution from knowledge of the temperature and heat flux on a part of the boundary of the solution domain, is investigated in a planar region containing a cut. This linear inverse ill-posed problem is numerically solved using an iterative regularization procedure, where at each iteration step mixed Dirichlet-Neumann problems for the parabolic heat equation are used. Using the method of Rothe these mixed problems are reduced to a sequence of boundary integral equations. The integral equations have a square root singularity in the densities and logarithmic and hypersingularities in the kernels. Moreover, the mixed parabolic problems have singularities near the endpoints of the cut. Special techniques are employed to handle each of these (four) types of singularities, and analysis is performed in weighted spaces of square integrable functions. Numerical examples are included showing that the proposed regularizing procedure gives stable and accurate approximations.
A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y ≤ 0, and boundary data are for x ≤ 0 and x ≤ 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y ≤ 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle. © 2005 IOP Publishing Ltd.
Kaczmarzs method-sometimes referred to as the algebraic reconstruction technique-is an iterative method that is widely used in tomographic imaging due to its favorable semi-convergence properties. Specifically, when applied to a problem with noisy data, during the early iterations it converges very quickly toward a good approximation of the exact solution, and thus produces a regularized solution. While this property is generally accepted and utilized, there is surprisingly little theoretical justification for it. The purpose of this paper is to present insight into the semi-convergence of Kaczmarzs method as well as its projected counterpart (and their block versions). To do this we study how the data errors propagate into the iteration vectors and we derive upper bounds for this noise propagation. Our bounds are compared with numerical results obtained from tomographic imaging.
An ill-posed Cauchy problem for a 3D elliptic partial differential equation with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability estimates are given. For the numerical implementation, a large sparse system is obtained from discretizing the QBV problem using the finite difference method. A left-preconditioned generalized minimum residual method is used to solve the large system effectively. For the preconditioned system, a fast solver using the fast Fourier transform is given. Numerical results show that the method works well.
Given a measure m on the real line or a finite interval, the cubic string is the third-order ODE - ′′′ ≤ zm where z is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a non-self-adjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper, we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis-Procesi equation. We solve the spectral and inverse spectral problem for the case of m being a finite positive discrete measure. In particular, explicit determinantal formulae for the measure m are given. These formulae generalize Stieltjes' formulae used by Krein in his study of the corresponding second-order ODE -″ ≤ zm. © 2007 IOP Publishing Ltd.
We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.
Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations (equations of the form q¨ = A-1(q)?k(q) where A(q) is a Killing matrix) and with the use of Poisson pencils. A quite general class of integrable perturbations depending on an arbitrary solution of a certain second-order linear PDE is found in the case of harmonic oscillator with equal frequencies. For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. A non-potential generalization of the Korteveg-de Vries integrable case of the Hénon-Heiles system is obtained. In the case when the perturbation is of a driven type (i.e. when one of the equations is autonomous) a method of solution of these systems by separation of variables and quadratures is presented.