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15
A Framework for the Construction of Level Set Methods for Shape Optimization and Reconstruction
 Interfaces and Free Boundaries
, 2002
"... The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric ..."
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Cited by 36 (4 self)
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The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case.
Discretization of Dirac Delta Functions in Level Set Methods
 J. Comput. Phys
, 2004
"... Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to ..."
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Cited by 27 (2 self)
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Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations on Cartesian grids and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on the distance to the singularity with a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized onedimensional delta functions.
Numerical Approximations of Singular Source Terms in Differential Equations
 J. Comput. Phys
, 2003
"... Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the ..."
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Cited by 25 (1 self)
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Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi dimensions. New high order multi dimensional techniques for di#erential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions to extend one dimensional regularization to higher dimensions may produce O(1) errors.
A posteriori pointwise error estimation for compressible fluid flows using adjoint parameters and Lagrange remainder
, 2005
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Adjoint Correction and Bounding of Error Using Lagrange Form of Truncation Term
 Computers & Mathematics with Applications
, 2005
"... An International Journal computers & ..."
A field spacebased level set method for computing multivalued solutions to 1D EulerPoisson equations
 J. Comp. Phys
"... Abstract. We present a field space based level set method for computing multivalued solutions to onedimensional EulerPoisson equations. The system of these equations has many applications, and in particular arises in semiclassical approximations of the SchrödingerPoisson equation. The proposed a ..."
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Cited by 3 (1 self)
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Abstract. We present a field space based level set method for computing multivalued solutions to onedimensional EulerPoisson equations. The system of these equations has many applications, and in particular arises in semiclassical approximations of the SchrödingerPoisson equation. The proposed approach involves an implicit Eulerian formulation in an augmented space — called field space, which incorporates both velocity and electric fields into the configuration. Both velocity and electric fields are captured through common zeros of two level set functions, which are governed by a field transport equation. Simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data. The averaged density is then resolved by the integration of the obtained f against the Dirac deltafunction of two level set functions in the field space. Moreover, we prove that such obtained averaged density is simply a linear superposition of all multivalued densities; and the averaged field quantities are weighted superposition of corresponding multivalued ones. Computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method.
On a posteriori pointwise error estimation using adjoint temperature and Lagrange remainder
, 2005
"... ..."
AnnaKarin Tornberg a,*
, 2004
"... www.elsevier.com/locate/jcp Numerical approximations of singular source terms in differential equations ..."
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www.elsevier.com/locate/jcp Numerical approximations of singular source terms in differential equations
ARTICLE IN PRESS 2 On a posteriori pointwise error estimation using
, 2004
"... 10 The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is 11 addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The con12 tribution of the local error to the total pointwise err ..."
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10 The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is 11 addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The con12 tribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that 13 the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error 14 correction and that an asymptotic error bound may be found.