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The Factorization Method for Inverse Problems

Andreas Kirsch and Natalia Grinberg

Published in print:
2007
Published Online:
September 2008
ISBN:
9780199213535
eISBN:
9780191707629
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213535.001.0001
Subject:
Mathematics, Applied Mathematics

This book is devoted to problems of shape identification in the context of (inverse) scattering problems and problems of impedance tomography. In contrast to traditional methods which are based on ... More


Boundaries and Geometries

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691141725
eISBN:
9781400848904
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691141725.003.0010
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude ... More


The factorization method for other types of inverse obstacle scattering problems

Andreas Kirsch and Natalia Grinberg

in The Factorization Method for Inverse Problems

Published in print:
2007
Published Online:
September 2008
ISBN:
9780199213535
eISBN:
9780191707629
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213535.003.0002
Subject:
Mathematics, Applied Mathematics

This chapter studies some examples of inverse scattering problems in which the data operator fails to be normal. The impedance boundary condition, the reconstruction from limited-angle data, and the ... More


Nonconforming finite element methods

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577040
eISBN:
9780191595172
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577040.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of ... More


Algorithmic Development

Peter Monk

in Finite Element Methods for Maxwell's Equations

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198508885
eISBN:
9780191708633
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198508885.003.0013
Subject:
Mathematics, Numerical Analysis

This chapter examines the phase error problem and also shows, via a dispersion analysis, that higher order methods can significantly improve phase accuracy. Once a solution is computed it is ... More


The mixed boundary value problem

Andreas Kirsch and Natalia Grinberg

in The Factorization Method for Inverse Problems

Published in print:
2007
Published Online:
September 2008
ISBN:
9780199213535
eISBN:
9780191707629
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213535.003.0003
Subject:
Mathematics, Applied Mathematics

For the mixed boundary value problem, the scattering domain D consists of several components with different types of boundary conditions. The chapter begins by reviewing results on uniqueness and ... More


HYPERBOLIC REDUCTIONS OF THE FIELD DQUATIONS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199205677
eISBN:
9780191709371
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205677.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

From the early 1990s, a large number of alternative formulations of the 3+1 evolution equations have been proposed. There are currently more formulations than there are numerical groups capable of ... More


Equation-Free Computation: An Overview of Patch Dynamics

G. Samaey, A. J. Roberts, and I. G. Kevrekidis

in Multiscale Methods: Bridging the Scales in Science and Engineering

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199233854
eISBN:
9780191715532
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233854.003.0008
Subject:
Mathematics, Applied Mathematics

This chapter overviews recent progress in the development of patch dynamics, an essential ingredient of the equation-free framework. In many applications we have a given detailed microscopic ... More


THE HOMOGENEOUS IBVP

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0007
Subject:
Mathematics, Applied Mathematics

When the boundary condition is homogeneous, one may think that there is no data at the boundary and that there is no need of boundary estimates in the maximal estimates. This is reminiscent of the ... More


Nonlinear Finite-Amplitude Dynamics

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691141725
eISBN:
9781400848904
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691141725.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After ... More


BOUNDARY CONDITIONS FOR EULER EQUATIONS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0014
Subject:
Mathematics, Applied Mathematics

The nature of an IBVP for a gas depends on the sign of the normal velocity, and on the sub- and super-sonicity. This chapter proceeds to a classification, describing which boundary conditions are ... More


THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.003.0011
Subject:
Mathematics, Mathematical Physics

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann ... More


FRIEDRICHS-SYMMETRIC DISSIPATIVE IBVPs

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0003
Subject:
Mathematics, Applied Mathematics

This chapter provides an elementary introduction to Initial Boundary Value Problems (IBVP). It introduces the notion of dissipative boundary condition, here in a classical sense. Such IBVPs can be ... More


Variational Theory for the Cavity Problem

Peter Monk

in Finite Element Methods for Maxwell's Equations

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198508885
eISBN:
9780191708633
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198508885.003.0004
Subject:
Mathematics, Numerical Analysis

This chapter presents a standard variational method based on the electric field for the cavity problem to prepare for the finite element approximation of this problem. Mixed boundary conditions ... More


NUMERICAL METHODS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199205677
eISBN:
9780191709371
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205677.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

There are many ways to solve partial differential equations numerically. The most popular methods are: finite differencing, finite elements, and spectral methods. This chapter describes the main ... More


Alternative sampling and probe methods

Andreas Kirsch and Natalia Grinberg

in The Factorization Method for Inverse Problems

Published in print:
2007
Published Online:
September 2008
ISBN:
9780199213535
eISBN:
9780191707629
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213535.003.0007
Subject:
Mathematics, Applied Mathematics

The Factorization Method is not the only sampling method. This final chapter introduces three alternative methods which belong to this class but, in contrast to the Factorization Method, are based on ... More


Discontinuous Galerkin methods (DCGMs), with V. Dolejší

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577040
eISBN:
9780191595172
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577040.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 7 presents with V. Dolejší presents discontinuous Galerkin methods (DCGMs): violated boundary conditions and continuity of the piecewise polynomials are compensated by additional penalty ... More


Parameterization and implementation issues

Bijan Mohammadi and Olivier Pironneau

in Applied Shape Optimization for Fluids

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199546909
eISBN:
9780191720482
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199546909.003.0006
Subject:
Mathematics, Mathematical Physics

This chapter presents parameterization and geometrical issues. This is also one of key points for an efficient OSD platform. It describes different strategies for shape deformation within and without ... More


Field Quantization and Vacuum Energy in the Presence of Boundaries

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko

in Advances in the Casimir Effect

Published in print:
2009
Published Online:
September 2009
ISBN:
9780199238743
eISBN:
9780191716461
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199238743.003.0003
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

From the standpoint of quantum field theory, the Casimir effect is related to the vacuum polarization that arises in quantization volumes restricted by boundaries or in spaces with nontrivial ... More


The Casimir Effect For Two Ideal-Metal Planes

M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko

in Advances in the Casimir Effect

Published in print:
2009
Published Online:
September 2009
ISBN:
9780199238743
eISBN:
9780191716461
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199238743.003.0007
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter considers the simple but most important configuration of two parallel ideal-metal planes. First, the theory of the scalar and electromagnetic Casimir effects between parallel planes is ... More


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