Inverse and Ill-Posed Problems

Heinz W. Engl, C. W. Groetsch

• Publisher: Academic Press
• Year: 2014
• ISBN:
  9781483272658
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Inverse and Ill-Posed Problems is a collection of papers presented at a seminar of the same title held in Austria in June 1986. The papers discuss inverse problems in various disciplines; mathematical solutions of integral equations of the first kind; general considerations for ill-posed problems; and the various regularization methods for integral and operator equations of the first kind. Other papers deal with applications in tomography, inverse scattering, detection of radiation sources, optics, partial differential equations, and parameter estimation problems. One paper discusses three topics on ill-posed problems, namely, the imposition of specified types of discontinuities on solutions of ill-posed problems, the use of generalized cross validation as a data based termination rule for iterative methods, and also a parameter estimation problem in reservoir modeling. Another paper investigates a statistical method to determine the truncation level in Eigen function expansions and for Fredholm equations of the first kind where the data contains some errors. Another paper examines the use of singular function expansions in the inversion of severely ill-posed problems arising in confocal scanning microscopy, particle sizing, and velocimetry. The collection can benefit many mathematicians, students, and professor of calculus, statistics, and advanced mathematics.

Table of contents

• Front Cover
• Inverse and Ill-Posed Problems
• Copyright Page
• Table of Contents
• Contributors
• Preface
• CHAPTER 1. A FEW GEOMETRICAL FEATURES OF INVERSE AND ILL-POSED PROBLEMS
• CHAPTER 2. THE INVERSE PROBLEM OF AQUIFER TRANSMISSIVITY IDENTIFICATION
• CHAPTER 3. RELIABILITY OF INFORMATION OBTAINED FROM APPROXIMATELY-SOLVED PROBLEMS
• CHAPTER 4. THREE TOPICS IN ILL-POSED PROBLEMS
• CHAPTER 5. A NEW APPROACH TO CLASSIFICATION AND REGULARIZATION OF ILL-POSED OPERATOR EQUATIONS
• CHAPTER 6. ON THE OPTIMALITY OF REGULARIZATION METHODS
• CHAPTER 7. OPTIMAL PARAMETER CHOICE FOR ORDINARY AND ITERATED TIKHONOV REGULAR!ZATION
• CHAPTER 8. PARAMETER CHOICE FOR TIKHONOV REGULARIZATION OF ILL-POSED PROBLEMS
• CHAPTER 9. FREDHOLM INTEGRAL EQUATIONS OF FIRST KIND AND THE METHOD OF CORRELOGRAM
• CHAPTER 10. ON ILL-POSED PROBLEMS AND THE METHOD OF CONJUGATE GRADIENTS
• CHAPTER 11. CONVERGENCE OF THE CONJUGATE GRADIENT METHOD FOR COMPACT OPERATORS
• CHAPTER 12. COMPARISON PRINCIPLES FOR ITERATIVE METHODS
• CHAPTER 13. COMPUTATION OF ROUGH SOLUTIONS OF ABEL INTEGRAL EQUATIONS
• CHAPTER 14. ITERATIVE METHODS FOR THE APPROXIMATE SOLUTION OF ILL-POSED PROBLEMS WITH A PRIORI INFORMATION AND THEIR APPLICATIONS
• CHAPTER 15. An Overview of Numerical Methods for Nonlinear Ill-Posed Problems
• CHAPTER 16. SEVERELY ILL-POSED RADON PROBLEMS
• CHAPTER 17. PROJECTION THEOREMS FOR FAR FIELD PATTERNS AND THE INVERSE SCATTERING PROBLEM
• CHAPTER 18. A NUMERICAL METHOD FOR AN INVERSE SCATTERING PROBLEM
• CHAPTER 19. APPLIED INVERSE PROBLEMS IN OPTICS
• CHAPTER 20. SOME REMARKS ON LOCATING RADIATION SOURCES
• CHAPTER 21. ON THE APPROXIMATE SOLUTION OF A TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM
• CHAPTER 22. MODIFIED EQUATIONS FOR APPROXIMATING THE SOLUTION OF A CAUCHY PROBLEM FOR THE HEAT EQUATION
• CHAPTER 23. STABILITY ESTIMATES FOR ILL-POSED CAUCHY PROBLEMS FOR PARABOLIC EQUATIONS
• CHAPTER 24. A BOUNDARY ELEMENT COLLOCATION METHOD FOR THE NEUMANN PROBLEM OF THE HEAT EQUATION
• CHAPTER 25. SUFFICIENT CONDITIONS FOR THE SOLUTION OF THE INVERSE VIBRATING BEAM PROBLEM
• CHAPTER 26. ON STABILIZING ILL-POSED PROBLEMS AGAINST ERRORS IN GEOMETRY AND MODELING
• CHAPTER 27. ON AN ILL-POSED PROBLEM FOR CONSTANT ALPHA FORCE-FREE FIELDS
• CHAPTER 28. SOME INVERSE AND ILL-POSED PROBLEMS IN COMPUTATIONAL FLUID DYNAMICS
• CHAPTER 29. IMPROVED CONTINUOUS DEPENDENCE RESULTS FOR A CLASS OF EVOLUTIONARY EQUATIONS
• CHAPTER 30. SOME BOUNDARY VALUE PROBLEMS FOR THE WAVE EQUATION
• CHAPTER 31. ON THE LOW FREQUENCY ASYMPTOTICS OF THE EXTERIOR 2-D DIRICHLET PROBLEM IN DYNAMIC ELASTICITY
• CHAPTER 32. INVERSE AND ILL-POSED PROBLEMS IN RESERVOIR SIMULATION
• CHAPTER 33. RATE OF CONVERGENCE FOR THE ESTIMATION OF A COEFFICIENT IN A TWO POINT BOUNDARY VALUE PROBLEM
• CHAPTER 34. IDENTIFIABILITY OF DISTRIBUTED PARAMETERS
• CHAPTER 35. ON THE REGULARIZATION OF LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS
• CHAPTER 36. LIMITS OF ABSTRACT SPLINES
• LIST OF PARTICIPANTS
• NOTES AND REPORTS IN MATHEMATICS IN SCIENCE AND ENGINEERING