On Shape Optimization with Parabolic State Equation

  • Helmut HarbrechtEmail author
  • Johannes Tausch
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


The present paper intends to summarize the main results of Harbrecht and Tausch (Inverse Probl 27:065013, 2011; SIAM J Sci Comput 35:A104–A121, 2013) on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization problem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equation 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. To demonstrate the similarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.


Shape optimization Heat equation Boundary integral equation Multipole method 

Mathematics Subject Classification (2010)

35K05 58J35 65K10 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsSouthern Methodist UniversityDallasUSA

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