Andreas Kirsch - Academia.edu (original) (raw)
Papers by Andreas Kirsch
Applied Mathematical Sciences, 2021
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An Introduction to the Mathematical Theory of Inverse Problems, 2021
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SIAM J. Appl. Math., 2021
With this erratum we correct some errors in [A. Kirsch and A. Rieder, Inverse Problems for abstra... more With this erratum we correct some errors in [A. Kirsch and A. Rieder, Inverse Problems for abstract evolution equations II: higher order differentiability for viscoelasticity, SIAM J. Appl. Math. 79-6 (2019), pp. 2639-2662].
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Applied Mathematical Sciences, 2014
In Sect. 2.6 of the previous chapter we have studied the scattering of plane waves by balls. In t... more In Sect. 2.6 of the previous chapter we have studied the scattering of plane waves by balls. In this chapter we investigate the same problem for arbitrary shapes. To treat this boundary value problem we introduce the boundary integral equation method which reformulates the boundary value problem in terms of an integral equation on the boundary of the region. For showing existence of a solution of this integral equation we will apply the Riesz-Fredholm theory.
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Applied Mathematical Sciences, 2014
In this chapter we want to introduce the reader to a second powerful approach for solving boundar... more In this chapter we want to introduce the reader to a second powerful approach for solving boundary value problems for the Maxwell system (or more general partial differential equations) which is the basis of, e.g., the Finite Element technique. We introduce this idea for the cavity problem as our reference problem which has been formulated in the introduction, see (1.21a)–(1.21c).
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Applied Mathematical Sciences, 1996
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Applied Mathematical Sciences, 1996
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Inverse Problems, 2012
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Mathematical Methods in the Applied Sciences, 2017
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Inverse Problems, 2016
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GAMM-Mitteilungen
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The Factorization Method for Inverse Problems, 2007
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The Factorization Method for Inverse Problems, 2007
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Springer Monographs in Mathematics, 2004
Contents Preface 1 Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna ... more Contents Preface 1 Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna Optimization 1.2 Arrays of Point Sources 1.2.1 The Linear Array 1.2.2 Circular Arrays 1.3 Maximization of Directivity and Super-gain 1.3.1 Directivity and Other Measures of Performance 1.3.2 Maximization of Directivity 1.4 Dolph-Tschebyshe. Arrays 1.4.1 Tschebyshe. Polynomials 1.4.2 The Dolph Problem 1.5 Line Sources 1.5.1 The Linear Line Source 1.5.2 The Circular Line Source 1.5.3 Numerical Quadrature 1.6 Conclusion 2 Discussion of Maxwell's Equations 2.1 Introduction 2.2 Geometry of the Radiating Structure 2.3 Maxwell's Equations in Integral Form 2.4 The Constitutive Relations 2.5 Maxwell's Equations in Differential Form 2.6 Energy Flow and the Poynting Vector 2.7 Time Harmonic Fields 2.8 Vector Potentials 2.9 Radiation Condition, Far Field Pattern 2.10 Radiating Dipoles and Line Sources 2.11 Boundary Conditions on Interfaces 2.12 Hertz Potentials and Classes of Solutions 2.13 Radiation Problems in Two Dimensions 3 Optimization Theory for Antennas 3.1 Introductory Remarks 3.2 The General Optimization Problem 3.2.1 Existence and Uniqueness 3.2.2 The Modeling of Constraints 3.2.3 Extreme Points and Optimal Solutions 3.2.4 The Lagrange Multiplier Rule 3.2.5 Methods of Finite Dimensional Approximation 3.3 Far Field Patterns and Far Field Operators 3.4 Measures of Antenna Performance 4 The Synthesis Problem 4.1 Introductory Remarks 4.2 Remarks on Ill-Posed Problems 4.3 Regularization by Constraints 4.4 The Tikhonov Regularization 4.5 The Synthesis Problem for the Finite Linear Line Source 4.5.1 Basic Equations 4.5.2 The Nystrom Method 4.5.3 Numerical Solution of the Normal Equations 4.5.4 Applications of the Regularization Techniques 5Boundary Value Problems for the Two-Dimensional Helmholtz Equation 5.1 Introduction and Formulation of the Problems 5.2 Rellich's Lemma and Uniqueness 5.3 Existence by the Boundary Integral Equation Method 5.4 L2-Boundary Data 5.5 Numerical Methods 5.5.1 Nystrom's Method for Periodic Weakly Singular Kernels 5.5.2 Complete Families of Solutions 5.5.3 Finite Element Methods for Absorbing Boundary Conditions 5.5.4 Hybrid Methods 6 Boundary Value Problems for Maxwell's Equations 6.1 Introduction and Formulation of the Problem 6.2 Uniqueness and Existence 6.3 L2-Boundary Data 7 Some Particular Optimization Problems 7.1 General Assumptions 7.2 Maximization of Power 7.2.1 Input Power Constraints 7.2.2 Pointwise Constraints on Inputs 7.2.3 Numerical Simulations 7.3 The Null-Placement Problem 7.3.1 Maximization of Power with Prescribed Nulls 7.3.2 A Particular Example - The Line Source 7.3.3 Pointwise Constraints 7.3.4 Minimization of Pattern Perturbation 7.4 The Optimization of Signal-to-Noise Ratio and Directivity 7.4.1 The Existence of Optimal Solutions 7.4.2 Necessary Conditions 7.4.3 The Finite Dimensional Problems 8 Conflicting Objectives: The Vector Optimization Problem . 8.1 Introduction 8.2 General Multi-criteria Optimization Problems 8.2.1 Minimal Elements and Pareto Points 8.2.2 The Lagrange Multiplier Rule 8.2.3 Scalarization 8.3 The Multi-criteria Dolph Problem for Arrays 8.3.1 The Weak Dolph Problem 8.3.2 Two Multi-criteria Versions 8.4 Null Placement Problems and Super-gain 8.4.1 Minimal Pattern Deviation 8.4.2 Power and Super-gain 8.5 The Signal-to-noise Ratio Problem 8.5.1 Formulation of the Problem and Existence of Pareto Points 8.5.2 The Lagrange Multiplier Rule 8.5.3 An Example A Appendix A.1 Introduction A.2 Basic Notions and Examples A.3
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Lecture Notes in Economics and Mathematical Systems, 1978
1m Wintersemester 1974/75 hielt der letztgenannte Autor an der Universitat Gottingen eine Vorlesu... more 1m Wintersemester 1974/75 hielt der letztgenannte Autor an der Universitat Gottingen eine Vorlesung uber Optimierung. Diese horte der erstgenannte Autor als Student, der zweitgenannte betreute als Assistent die Ubungen. 1m AnschluB an die nachfolgenden Diskussio nen untereinander entstand der Plan, die vorliegende Arbeit zu schreiben. An Vorkenntnissen sollten dabei nur die einfachsten Grundbegriffe der linearen Funktionalanalysis vorausgesetzt werden. Herrn Professor Dr. K. Ritter mochten wir fur die Ermutigung danken, uberhaupt mit der Arbeit zu beginnen. Fraulein R. -M. Wedekind gilt unser besonderer Dank fur das Schreiben des Manuskripts. Gottingen, November 1977 Andreas Kirsch Wolfgang Warth Jochen Werner Inhaltsverzeichnis Einleitung 8 I Funktionalanalytische Hilfsmittel Konvexe Mengen in linearen Raumen 8 1 2 Konvexe Mengen in linearen normierten Raumen 23 28 II Notwendige Optimalitatsbedingungen Problemstellung, Definitionen, Hilfssatze 28 1 Ein Alternativsatz und Maximumprinzipien 48 2 Konvexe Optimierungsaufgaben 61 3 4 Das Maximumprinzip fur differenzierbare Funk tionen 68 5 Das Maximumprinzip bei Optimierungsaufgaben mit affin linearen Ungleichungsrestriktionen 76 84 III Anwendungen 1 Notwendige Optimalitatsbedingungen bei opti malen Steuerungsproblemen 84 2 Notwendi e Optimalitatsbedingungen bei dis kreten optimal en Steuerungsproblemen 111 3 Notwendige Optimalitatsbedingungen in der Approximationstheorie 118 4 Einige spezielle Beispiele 127 Literaturverzeichnis 149 Symbolverzeichnis 155 Sachverzeichnis 156 Einleitung Eines der wichtigsten Teilgebiete der Optimierung ist die Theorie notwendiger Bedingungen. Untersucht wird hierbei die Frage, wel chen Bedingungen eine Lasung einer gegebenen Optimierungsaufgabe notwendig zu genugen hat. Bei konkreten Fragestellungen hofft man, mit Hilfe dieser notwendigen Optimalitatsbedingungen Aussagen zu gewinnen, die zu einer Berechnung maglicher Lasungen ausgenutzt werden kannen.
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Mathematics and Computers in Simulation, 2004
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Proceedings of the IEEE, 1991
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Oberwolfach Reports, 2007
The workshop treated inverse problems for partial differential equations, especially inverse scat... more The workshop treated inverse problems for partial differential equations, especially inverse scattering problems, and their applications in technology. While special attention was paid to sampling methods, decom- position methods, Newton methods and questions of unique determination were also investigated.
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Inverse Problems, 2016
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Applied Mathematical Sciences, 2021
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An Introduction to the Mathematical Theory of Inverse Problems, 2021
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SIAM J. Appl. Math., 2021
With this erratum we correct some errors in [A. Kirsch and A. Rieder, Inverse Problems for abstra... more With this erratum we correct some errors in [A. Kirsch and A. Rieder, Inverse Problems for abstract evolution equations II: higher order differentiability for viscoelasticity, SIAM J. Appl. Math. 79-6 (2019), pp. 2639-2662].
Bookmarks Related papers MentionsView impact
Applied Mathematical Sciences, 2014
In Sect. 2.6 of the previous chapter we have studied the scattering of plane waves by balls. In t... more In Sect. 2.6 of the previous chapter we have studied the scattering of plane waves by balls. In this chapter we investigate the same problem for arbitrary shapes. To treat this boundary value problem we introduce the boundary integral equation method which reformulates the boundary value problem in terms of an integral equation on the boundary of the region. For showing existence of a solution of this integral equation we will apply the Riesz-Fredholm theory.
Bookmarks Related papers MentionsView impact
Applied Mathematical Sciences, 2014
In this chapter we want to introduce the reader to a second powerful approach for solving boundar... more In this chapter we want to introduce the reader to a second powerful approach for solving boundary value problems for the Maxwell system (or more general partial differential equations) which is the basis of, e.g., the Finite Element technique. We introduce this idea for the cavity problem as our reference problem which has been formulated in the introduction, see (1.21a)–(1.21c).
Bookmarks Related papers MentionsView impact
Applied Mathematical Sciences, 1996
Bookmarks Related papers MentionsView impact
Applied Mathematical Sciences, 1996
Bookmarks Related papers MentionsView impact
Inverse Problems, 2012
Bookmarks Related papers MentionsView impact
Mathematical Methods in the Applied Sciences, 2017
Bookmarks Related papers MentionsView impact
Inverse Problems, 2016
Bookmarks Related papers MentionsView impact
GAMM-Mitteilungen
Bookmarks Related papers MentionsView impact
The Factorization Method for Inverse Problems, 2007
Bookmarks Related papers MentionsView impact
The Factorization Method for Inverse Problems, 2007
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Springer Monographs in Mathematics, 2004
Contents Preface 1 Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna ... more Contents Preface 1 Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna Optimization 1.2 Arrays of Point Sources 1.2.1 The Linear Array 1.2.2 Circular Arrays 1.3 Maximization of Directivity and Super-gain 1.3.1 Directivity and Other Measures of Performance 1.3.2 Maximization of Directivity 1.4 Dolph-Tschebyshe. Arrays 1.4.1 Tschebyshe. Polynomials 1.4.2 The Dolph Problem 1.5 Line Sources 1.5.1 The Linear Line Source 1.5.2 The Circular Line Source 1.5.3 Numerical Quadrature 1.6 Conclusion 2 Discussion of Maxwell's Equations 2.1 Introduction 2.2 Geometry of the Radiating Structure 2.3 Maxwell's Equations in Integral Form 2.4 The Constitutive Relations 2.5 Maxwell's Equations in Differential Form 2.6 Energy Flow and the Poynting Vector 2.7 Time Harmonic Fields 2.8 Vector Potentials 2.9 Radiation Condition, Far Field Pattern 2.10 Radiating Dipoles and Line Sources 2.11 Boundary Conditions on Interfaces 2.12 Hertz Potentials and Classes of Solutions 2.13 Radiation Problems in Two Dimensions 3 Optimization Theory for Antennas 3.1 Introductory Remarks 3.2 The General Optimization Problem 3.2.1 Existence and Uniqueness 3.2.2 The Modeling of Constraints 3.2.3 Extreme Points and Optimal Solutions 3.2.4 The Lagrange Multiplier Rule 3.2.5 Methods of Finite Dimensional Approximation 3.3 Far Field Patterns and Far Field Operators 3.4 Measures of Antenna Performance 4 The Synthesis Problem 4.1 Introductory Remarks 4.2 Remarks on Ill-Posed Problems 4.3 Regularization by Constraints 4.4 The Tikhonov Regularization 4.5 The Synthesis Problem for the Finite Linear Line Source 4.5.1 Basic Equations 4.5.2 The Nystrom Method 4.5.3 Numerical Solution of the Normal Equations 4.5.4 Applications of the Regularization Techniques 5Boundary Value Problems for the Two-Dimensional Helmholtz Equation 5.1 Introduction and Formulation of the Problems 5.2 Rellich's Lemma and Uniqueness 5.3 Existence by the Boundary Integral Equation Method 5.4 L2-Boundary Data 5.5 Numerical Methods 5.5.1 Nystrom's Method for Periodic Weakly Singular Kernels 5.5.2 Complete Families of Solutions 5.5.3 Finite Element Methods for Absorbing Boundary Conditions 5.5.4 Hybrid Methods 6 Boundary Value Problems for Maxwell's Equations 6.1 Introduction and Formulation of the Problem 6.2 Uniqueness and Existence 6.3 L2-Boundary Data 7 Some Particular Optimization Problems 7.1 General Assumptions 7.2 Maximization of Power 7.2.1 Input Power Constraints 7.2.2 Pointwise Constraints on Inputs 7.2.3 Numerical Simulations 7.3 The Null-Placement Problem 7.3.1 Maximization of Power with Prescribed Nulls 7.3.2 A Particular Example - The Line Source 7.3.3 Pointwise Constraints 7.3.4 Minimization of Pattern Perturbation 7.4 The Optimization of Signal-to-Noise Ratio and Directivity 7.4.1 The Existence of Optimal Solutions 7.4.2 Necessary Conditions 7.4.3 The Finite Dimensional Problems 8 Conflicting Objectives: The Vector Optimization Problem . 8.1 Introduction 8.2 General Multi-criteria Optimization Problems 8.2.1 Minimal Elements and Pareto Points 8.2.2 The Lagrange Multiplier Rule 8.2.3 Scalarization 8.3 The Multi-criteria Dolph Problem for Arrays 8.3.1 The Weak Dolph Problem 8.3.2 Two Multi-criteria Versions 8.4 Null Placement Problems and Super-gain 8.4.1 Minimal Pattern Deviation 8.4.2 Power and Super-gain 8.5 The Signal-to-noise Ratio Problem 8.5.1 Formulation of the Problem and Existence of Pareto Points 8.5.2 The Lagrange Multiplier Rule 8.5.3 An Example A Appendix A.1 Introduction A.2 Basic Notions and Examples A.3
Bookmarks Related papers MentionsView impact
Lecture Notes in Economics and Mathematical Systems, 1978
1m Wintersemester 1974/75 hielt der letztgenannte Autor an der Universitat Gottingen eine Vorlesu... more 1m Wintersemester 1974/75 hielt der letztgenannte Autor an der Universitat Gottingen eine Vorlesung uber Optimierung. Diese horte der erstgenannte Autor als Student, der zweitgenannte betreute als Assistent die Ubungen. 1m AnschluB an die nachfolgenden Diskussio nen untereinander entstand der Plan, die vorliegende Arbeit zu schreiben. An Vorkenntnissen sollten dabei nur die einfachsten Grundbegriffe der linearen Funktionalanalysis vorausgesetzt werden. Herrn Professor Dr. K. Ritter mochten wir fur die Ermutigung danken, uberhaupt mit der Arbeit zu beginnen. Fraulein R. -M. Wedekind gilt unser besonderer Dank fur das Schreiben des Manuskripts. Gottingen, November 1977 Andreas Kirsch Wolfgang Warth Jochen Werner Inhaltsverzeichnis Einleitung 8 I Funktionalanalytische Hilfsmittel Konvexe Mengen in linearen Raumen 8 1 2 Konvexe Mengen in linearen normierten Raumen 23 28 II Notwendige Optimalitatsbedingungen Problemstellung, Definitionen, Hilfssatze 28 1 Ein Alternativsatz und Maximumprinzipien 48 2 Konvexe Optimierungsaufgaben 61 3 4 Das Maximumprinzip fur differenzierbare Funk tionen 68 5 Das Maximumprinzip bei Optimierungsaufgaben mit affin linearen Ungleichungsrestriktionen 76 84 III Anwendungen 1 Notwendige Optimalitatsbedingungen bei opti malen Steuerungsproblemen 84 2 Notwendi e Optimalitatsbedingungen bei dis kreten optimal en Steuerungsproblemen 111 3 Notwendige Optimalitatsbedingungen in der Approximationstheorie 118 4 Einige spezielle Beispiele 127 Literaturverzeichnis 149 Symbolverzeichnis 155 Sachverzeichnis 156 Einleitung Eines der wichtigsten Teilgebiete der Optimierung ist die Theorie notwendiger Bedingungen. Untersucht wird hierbei die Frage, wel chen Bedingungen eine Lasung einer gegebenen Optimierungsaufgabe notwendig zu genugen hat. Bei konkreten Fragestellungen hofft man, mit Hilfe dieser notwendigen Optimalitatsbedingungen Aussagen zu gewinnen, die zu einer Berechnung maglicher Lasungen ausgenutzt werden kannen.
Bookmarks Related papers MentionsView impact
Mathematics and Computers in Simulation, 2004
Bookmarks Related papers MentionsView impact
Proceedings of the IEEE, 1991
Bookmarks Related papers MentionsView impact
Oberwolfach Reports, 2007
The workshop treated inverse problems for partial differential equations, especially inverse scat... more The workshop treated inverse problems for partial differential equations, especially inverse scattering problems, and their applications in technology. While special attention was paid to sampling methods, decom- position methods, Newton methods and questions of unique determination were also investigated.
Bookmarks Related papers MentionsView impact
Inverse Problems, 2016
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