A fast method for solving the heat equation by layer potentials

Tausch, Johannes

doi:10.1016/j.jcp.2006.11.001

Cited by 52 publications

(68 citation statements)

References 12 publications

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“…On the other hand, clustering techniques, which include the fast multipole method, H-matrices, and adaptive cross approximations, have proved to be extremely successful for solving integral formulations elliptic problems with complicated geometries. For parabolic problems, multipolebased space-time boundary element methods which cluster sources in space and time have become available recently [31,32] and will be used here to evaluate thermal layer potentials efficiently.…”

Section: Introductionmentioning

confidence: 99%

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

Harbrecht¹,

Tausch²

2013

SIAM J. Sci. Comput.

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Abstract. The present paper is concerned with the numerical solution of a shape identification problem for the heat equation. The goal is to determine of the shape of a void or an inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional, (b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations. 1. Introduction. Recently shape optimization for elliptic boundary value problems has become a well-established mathematical and computational tool; see, e.g., [3,15,19,26,30] and the references therein. In this context boundary element methods have proved themselves to be an extremely effective tool for the solution of the associated state equation; see, e.g., [8,13,23,25,27]. The main reason is that no domain triangulation has to be computed. Moreover, fast methods to handle the associated dense matrix problems have reached a mature state and can be readily applied within the iterative solution of the minimization problem.In contrast to elliptic shape optimization, the literature on parabolic shape optimization problems is rather limited. There are theoretical results (see, e.g., [5,16,17,29] and the references therein), but the development of efficient numerical methods for shape optimization problems with a parabolic state equation is still in its beginning stages, especially for three-dimensional geometries. With the goal to develop such efficient methods, we consider in the present paper a shape identification problem for the heat equation. Specifically, we detect an inclusion or void of zero temperature inside a solid or liquid body by measurements of the temperature and the transient heat flux at the accessible outer boundary.This problem is severely ill-posed and was discussed in a two-dimensional setting in [1,2], where the uniqueness of the inclusion and the differentiability with respect to its boundary was established. The approach in these papers is to use Newton's method to solve the nonlinear functional equation that maps the shape of the inclusion to the measured data. Thus each step of the iteration involves one forward problem

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Section: Introductionmentioning

confidence: 99%

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

Harbrecht¹,

Tausch²

2013

SIAM J. Sci. Comput.

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“…The initial potential would require O(N M 2 ) work. Fortunately, fast algorithms have been developed for layer potentials (and volume potentials) in both bounded and unbounded domains [7,8,14,17,16,22,23]. In this paper, we follow the approach of [7,8].…”

Section: γ(τ ) (13)mentioning

confidence: 99%

“…Alternative methods for the rapid evaluation of layer heat potentials have been proposed, for example, in [17,16,22,23]. In [17,16], the authors work with the Laplace transform in time of the heat kernel.…”

Section: γ(τ ) (13)mentioning

confidence: 99%

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High Order Accurate Methods for the Evaluation of Layer Heat Potentials

Li¹,

Greengard²

2009

SIAM J. Sci. Comput.

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Abstract. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as "geometrically induced stiffness," but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the "history part" of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.

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“…For the time discretization, appropriate singularity corrected trapezoidal quadrature rules are applied to handle the singularities of the heat kernel and the solution. A space-time fast multipole method for the rapid evaluation of thermal potentials, developed in [10,11], is employed to solve the discretized boundary integral equations.…”

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confidence: 99%