High Order Accurate Methods for the Evaluation of Layer Heat Potentials

Li, Jing-Rebecca; Greengard, Leslie

doi:10.1137/080732389

Cited by 32 publications

(46 citation statements)

References 22 publications

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“…It was shown in [11] that Fourier techniques can be used to ameliorate the high computational cost of evaluating the time convolution in thermal layer potentials. An application of this approach is reported in [22]. 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.…”

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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“…There is recent work by Li and Greengard in [6,7] on developing fast solvers for the heat equation. Their methods use an integral equation formulation based on the heat kernel and rely on a non-uniform FFT.…”

Section: Introductionmentioning

confidence: 99%

Fast integral equation methods for Rothe’s method applied to the isotropic heat equation

Kropinski

Quaife

2011

Computers & Mathematics with Applications

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a b s t r a c tWe present an efficient integral equation approach to solve the forced heat equation, x, u, t), in a two-dimensional, multiply-connected domain, with Dirichlet boundary conditions. Instead of using an integral equation formulation based on the heat kernel, we discretize in time, first. This approach, known as Rothe's method, leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. We formulate the solution to this equation as a volume potential plus a double layer potential, and both of these potentials are calculated with available tools accelerated by the fast multipole method. For a total of N points in the discretization of the boundary and the domain, the total computational cost per time step is O(N). We demonstrate our approach on the heat equation and the Allen-Cahn equation.

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“…Discretization and evaluation of the single layer potential. We first consider the evaluation of the single layer potential, following the treatment of the heat equation in [31,35,53]. There are three fundamental observations to be made.…”

Section: Discretization and Numerical Evaluation Of Layer Potentialsmentioning

confidence: 99%

“…Third, when τ is bounded away from the current time t, as in the history part, then the kernels (and resulting fields) are smooth and admit a variety of simpler approximations. In order to overcome the singular quadrature issues, we design special product integration-based schemes, following the treatment in [53]. In that paper, it is shown that for robustness, it is essential to interchange the order of integration in the layer potentials and to carry out integration in time analytically for polynomial approximations of the layer potential density φ(y, τ ).…”

Section: Discretization and Numerical Evaluation Of Layer Potentialsmentioning

confidence: 99%

Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

Jiang¹,

Veerapaneni²,

Greengard³

2012

SIAM J. Sci. Comput.

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Abstract. We present an integral equation formulation for the unsteady Stokes equations in two dimensions. This problem is of interest in its own right, as a model for slow viscous flow, but perhaps more importantly, as an ingredient in the solution of the full, incompressible Navier-Stokes equations. Using the unsteady Green's function, the velocity evolves analytically as a divergence-free vector field. This avoids the need for either the solution of coupled field equations (as in fully implicit PDE-based marching schemes) or the projection of the velocity field onto a divergence free field at each time step (as in operator splitting methods). In addition to discussing the analytic properties of the operators that arise in the integral formulation, we describe a family of high-order accurate numerical schemes and illustrate their performance with several examples.

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High Order Accurate Methods for the Evaluation of Layer Heat Potentials

Cited by 32 publications

References 22 publications

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

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

Fast integral equation methods for Rothe’s method applied to the isotropic heat equation

Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

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