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Finite elements in computational electromagnetism
, 2002
"... This article discusses finite element Galerkin schemes for a number of lin-ear model problems in electromagnetism. The finite element schemes are in-troduced as discrete differential forms, matching the coordinate-independent statement of Maxwell’s equations in the calculus of differential forms. Th ..."
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Cited by 128 (8 self)
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This article discusses finite element Galerkin schemes for a number of lin-ear model problems in electromagnetism. The finite element schemes are in-troduced as discrete differential forms, matching the coordinate-independent statement of Maxwell’s equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete dif-
A Level-Set Approach for Inverse Problems Involving Obstacles
, 1996
"... . An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry. We develop two computational metho ..."
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Cited by 97 (2 self)
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. An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry. We develop two computational methods based on this idea. One method results in a nonlinear time-dependent partial differential equation for the level-set function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of levelset functions that reduces the residual. The methods are illustrated in two applications: a deconvolution problem and a diffraction screen reconstruction problem. Keywords: Inverse problems, level-set method, Hamilton-Jacobi equations, surface evolution, optimization, deconvolution, diffraction. 1. Inverse problems involving obstacles There is a host of inverse problems wherein the desired unknown is a region in IR 2 or IR 3 . ...
Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Differential Equations
, 1998
"... Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that ..."
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Cited by 85 (12 self)
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Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multidimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations. Key words. PDEs, adjoint equations, error analysis, superconvergence AMS subject classifications. 65G99, 76N15 PII. S0036144598349423
Electromagnetic wave scattering by small bodies
, 2005
"... A reduction of the Maxwell's system to a Fredholm second-kind integral equation withweakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic sizeof the bodies tends to zero. The technique ..."
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Cited by 73 (44 self)
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A reduction of the Maxwell's system to a Fredholm second-kind integral equation withweakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic sizeof the bodies tends to zero. The technique developed is used for solving the manybody EM wave scattering problem by rigorously reducing it to solving linear algebraic systems, completely bypassing the usage of integral equations. An equation is derived for the effective field in the medium, in which many small particles are embedded. Amethod for creating a desired refraction coefficient is outlined.
The Factorization Method for Inverse Problems
, 2007
"... since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hül ..."
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Cited by 63 (4 self)
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since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hülsmeyer devised a way to determine the distance to the object. At that rate of improvement one might have extrapolated to unimaginable twenty-first-century capabilities. Unfortunately, what has proved to be unimaginable is the difficulty of doing much more than the original device had already accomplished. It would seem that some forms of bionic vision have gone the way of rocket backpacks – that is, until recently. The newest book by Andreas Kirsch with coauthor Natalia Grinberg, The Factorization Method for Inverse Problems, collects over a decade of work by Kirsch and collaborators on a simple method for shape identification in inverse scattering. This book belongs to the next generation of monographs on inverse scattering following the now standard works of Colton and Kress [2] (Inverse Acoustic and Electromagnetic Scattering Theory (1998)) and Isakov [7] (Inverse Problems for Partial Differential Equations). Kirsch’s factorization method arose from experimentation with noniterative inverse scattering methods that avoid the computational expense of calculating the solution to the forward problem at each itera-tion. Noniterative methods attack head-on the inverse problem of determining the scatterer from measured
A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering
- NUMERISCHE MATHEMATIK
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A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
- SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 48 (16 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Solving time-harmonic scattering problems based on the pole condition: Convergence of the PML method
, 2001
"... In this paper we study the PML method for Helmholtz-type scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a Perfectly Matched sponge Layer. We prove that the approximate solution obtained by the PML method converges exponential ..."
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Cited by 46 (9 self)
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In this paper we study the PML method for Helmholtz-type scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a Perfectly Matched sponge Layer. We prove that the approximate solution obtained by the PML method converges exponentially fast to the true solution in the computational domain as the thickness of the sponge layer tends to infinity. This is a generalization of results by Lassas and Somersalo based on boundary integral equation techniques. Here we use techniques based on the pole condition instead. This makes it possible to treat problems without an explicitly known fundamental solution
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 functional-analytic 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 46 (6 self)
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The aim of this paper is to develop a functional-analytic 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.
