Heuristic regularization methods for numerical differentiation

AbstractIn this paper, we use smoothing splines to deal with numerical differentiation. Some heuristic methods for choosing regularization parameters are proposed, including the L-curve method and the de Boor method. Numerical experiments are performed to illustrate the efficiency of these methods in comparison with other procedures

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable

Boundary-integral approach to the numerical solution of the Cauchy problem for the Laplace equation

We present a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the input data may completely destroy the procedure of finding the approximate solution. We describe and present the results for a procedure of regularization aimed at the stable determination of the required quantities based on the representation of the solution to the Cauchy problem in the form a single-layer potential. For given data, this representation yields a system of boundary integral equations with two unknown densities. We establish the existence and uniqueness of these densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider the cases of simply connected domains of the solution and unbounded domains. Numerical examples are presented both for two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy with relatively small amount of computations

Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

In many practical situations concerned with high temperatures/pressures/loads and/or hostile environments, certain properties of the physical medium, geometry, boundary and/or initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible. In such situations, one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available. However, the simultaneous identification of several non-constant physical properties along with initial and/or boundary conditions is very challenging, especially when it cannot be decoupled, as it combines both nonlinear as well as ill-posedness features. One such new inverse problem concerning the identification of the space-dependent reaction coefficient, the initial temperature and the source term from measured temperatures at two instants t1, t2 and at the final time T , where 0 < t1 < t2 < T , is investigated in this paper. Insight into the uniqueness of solution is gained by considering various particular cases. Moreover, as in practice the input temperature data are usually noise polluted due to the errors that are inherently present, their influence on the solution of inversion has to be assessed. As such, the least-squares objective functional modelling the gap between the measured and computed data is minimized to obtain the quasi-solution to the inverse problem, and the Fréchet gradients are obtained. The conjugate gradient method (CGM) with the Fletcher–Reeves formula is applied to estimate the three unknown coefficients numerically. Numerical examples are illustrated to show that accurate and stable numerical solutions are obtained using the CGM regularized by the discrepancy principle

Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type

Identification of the forcing term in hyperbolic equations

We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite-difference methods. Numerical results are presented and discussed for one and two-dimensional problems

Determination of the ambient temperature in transient heat conduction.

The restoration of the space- or time-dependent ambient temperature entering a third-kind convective Robin boundary condition in transient heat conduction is investigated. The temperature inside the solution domain together with the ambient temperature are determined from additional boundary measurements. In both cases of the space- or time-dependent unknown ambient temperature the inverse problems are linear and ill-posed. Least-squares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the conjugate gradient method combined with a boundary element direct solver are presented and discussed

Determination of a source in the heat equation from integral observations

A novel inverse problem which consists of the simultaneous determination of a source together with the temperature in the heat equation from integral observations is investigated. These integral observations are weighted averages of the temperature over the space domain and over the time interval. The heat source is sought in the form of a sum of two space- and time-dependent unknown components in order to ensure the uniqueness of a solution. The local existence and uniqueness of the solution in classical Hölder spaces are proved. The inverse problem is linear, but it is ill-posed because small errors in the input integral observations cause large errors in the output source. For a stable reconstruction a variational least-squares method with or without penalization is employed. The gradient of the functional which is minimized is calculated explicitly and the conjugate gradient method is applied. Numerical results obtained for several benchmark test examples show accurate and stable numerical reconstructions of the heat source