Solving a Cauchy problem for the heat equation using cubic smoothing splines

Abstract

The Cauchy problem for the heat equation is a model of situation where one seeks to compute the temperature, or heat-flux, at the surface of a body by using interior measurements. The problem is well-known to be ill-posed, in the sense that measurement errors can be magnified and destroy the solution, and thus regularization is needed. In previous work it has been found that a method based on approximating the time derivative by a Fourier series works well [Berntsson F. A spectral method for solving the sideways heat equation. Inverse Probl. 1999;15:891-906; Eldén L, Berntsson F, Regińska T. Wavelet and Fourier methods for solving the sideways heat equation. SIAM J Sci Comput. 2000;21(6):2187-2205]. However, in our situation it is not resonable to assume that the temperature is periodic which means that additional techniques are needed to reduce the errors introduced by implicitly making the assumption that the solution is periodic in time. Thus, as an alternative approach, we instead approximate the time derivative by using a cubic smoothing spline. This means avoiding a periodicity assumption which leads to slightly smaller errors at the end points of the measurement interval. The spline method is also shown to satisfy similar stability estimates as the Fourier series method. Numerical simulations shows that both methods work well, and provide comparable accuracy, and also that the spline method gives slightly better results at the ends of the measurement interval.

Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines

“…A consequence of the fact that u(x) solves the least squares problem (3.10), is that h(y − u ) T u = λ|u| 2 2 , see [18,Lemma 3.7]. Inserting into (3.15), we obtain the estimate…”

“…The matrix D 2 λ is real and symmetric. Proof The matrix D 2 λ can be represented as D 2 λ = E W A T , where the matrix [18]. A direct computation shows that…”

“…In this paper, we develop a novel regularization method for the Cauchy problem for the Helmholtz equation. In our previous work [18] we have shown that approximating a derivative by a bounded approximation based on Cubic smoothing splines has a stabilizing effect for the inverse heat conduction problem. Here we apply the same technique to the Helmholtz equation and are able to derive stability estimates.…”