Boundary element methods for Dirichlet boundary control problems

Phan, Thanh Xuan; Steinbach, Olaf

doi:10.1002/mma.1356

Cited by 17 publications

(18 citation statements)

References 31 publications

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“…We observe an estimated order of convergence (eoc) close to 2 as predicted by the error estimates (3.26) and (3.27) for u and p. For the control z the observed order of convergence is larger than 3/2 which is predicted by the error estimate (3.25). So far we are not able to prove this higher order of convergence for a finite element discretization but for the boundary element discretization presented in [32] when assuming z ∈ H 2 pw (Γ). In this case the second order of convergence reflects the optimal approximation property when using piecewise linear basis functions on the boundary.…”

Section: Numerical Results and Concluding Remarksmentioning

confidence: 87%

“…The stability and error analysis of the resulting boundary element approach for the solution of Dirichlet control problems is discussed in [32], see also [33].…”

Section: Numerical Results and Concluding Remarksmentioning

confidence: 99%

“…A possible choice is to consider as in [32] the so-called hypersingular boundary integral operator D which does not require an inversion as in the definition of S h . But the use of the hypersingular integral operator may result in a lower regularity of the control z, and therefore in a lower order of convergence of the finite element approximation, see [32,33].…”

Section: Numerical Results and Concluding Remarksmentioning

confidence: 99%

See 2 more Smart Citations

An energy space finite element approach for elliptic Dirichlet boundary control problems

Phan

Steinbach

2014

Numer. Math.

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In this paper we present a finite element analysis for a Dirichlet boundary control problem where the Dirichlet control is considered in a convex closed subspace of the energy space H 1/2 (Γ). As an equivalent norm in H 1/2 (Γ) we use the energy norm induced by the so-called Steklov-Poincaré operator which realizes the Dirichlet to Neumann map, and which can be implemented by using standard finite element methods. The presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems, and their finite element approximations. Some numerical results are given, which confirm on one hand the theoretical estimates, but on the other hand indicate the differences when modelling the control in L 2 (Γ).

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Section: Numerical Results and Concluding Remarksmentioning

confidence: 87%

“…The stability and error analysis of the resulting boundary element approach for the solution of Dirichlet control problems is discussed in [32], see also [33].…”

Section: Numerical Results and Concluding Remarksmentioning

confidence: 99%

Section: Numerical Results and Concluding Remarksmentioning

confidence: 99%

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An energy space finite element approach for elliptic Dirichlet boundary control problems

Phan

Steinbach

2014

Numer. Math.

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“…In [24], a proof of the energy error estimate in H 1/2 (Γ) is given for the particular case of a smooth boundary of a bounded domain in two dimensions, and numerical examples are given. For numerical results in the case of contact problems in linear elasticity, see for example [8], and for optimal Dirichlet control problems [22].…”

Section: Signorini Problemmentioning

confidence: 99%

“…Variational inequalities of the form (1.2) occur, for example, when considering the variational formulation of second order partial differential equations with boundary conditions of Signorini type, e.g., [12,19,24], or when considering contact problems in elasticity without friction, e.g., [4,7,8,11]. Other applications involve Dirichlet boundary control problems with control constraints, e.g., [17,22].…”

Section: Introductionmentioning

confidence: 99%

Boundary element methods for variational inequalities

Steinbach

2013

Numer. Math.

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In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in H 1/2 (Γ). In addition to error estimates in the energy norm we also provide, by applying the Aubin-Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including L 2 (Γ). The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.

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Space–time shape uncertainties in the forward and inverse problem of electrocardiography

Gander

Krause

Multerer

et al. 2021

Numer Methods Biomed Eng

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In electrocardiography, the "classic" inverse problem is the reconstruction of electric potentials at a surface enclosing the heart from remote recordings at the body surface and an accurate description of the anatomy. The latter being affected by noise and obtained with limited resolution due to clinical constraints, a possibly large uncertainty may be perpetuated in the inverse reconstruction. The purpose of this work is to study the effect of shape uncertainty on the forward and the inverse problem of electrocardiography. To this aim, the problem is first recast into a boundary integral formulation and then discretised with a collocation method to achieve high convergence rates and a fast time to solution. The shape uncertainty of the domain is represented by a random deformation field defined on a reference configuration. We propose a periodic-in-time covariance kernel for the random field and approximate the Karhunen-Loève expansion using low-rank techniques for fast sampling. The space-time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi-Monte Carlo method. We present several numerical experiments on a simplified but physiologically grounded two-dimensional geometry to illustrate the validity of the approach. The tested parametric dimension ranged from 100 up to 600. For the forward problem, the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi-Monte Carlo method perform as expected, except for the total variation regularisation, where convergence is limited by lack of regularity. We finally investigate an H 1=2 regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.

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Boundary element methods for Dirichlet boundary control problems

Cited by 17 publications

References 31 publications

An energy space finite element approach for elliptic Dirichlet boundary control problems

An energy space finite element approach for elliptic Dirichlet boundary control problems

Boundary element methods for variational inequalities

Space–time shape uncertainties in the forward and inverse problem of electrocardiography

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