This was a course for beginning graduate students in mathematics and related subjects. It started on March 3 and ended on June 3.
The notion of a well-posed problem, un problème bien posé, goes back to a famous paper by Jacques Hadamard published in 1902. In an earlier paper in 1901 he mentioned questions mal posées. He argued that the problems that are physically important are both possible and déterminé, i.e., solvable and uniquely solvable. He gave examples of problems that are not well posed; they are also, he claimed, dépourvu de signification physique. However, I shall show that important problems in technology, medicine, and the natural sciences that are ill-posed abound. In fact, any measurement, except for the most trivial ones, gives rise to an inverse problem that is ill-posed.
Today one defines a well-posed problem as a problem that is uniquely solvable and is such that the solution depends in a continuous way on the data. If the solution depends in a discontinuous way on the data, then small errors, whether rounding off errors, measurement errors, or perturbations caused by noise, can create large deviations. Therefore the numerical treatment of ill-posed problems is a challenge.
In this course I presented examples of well-posed and ill-posed problems. I discussed mathematical methods to treat them. I took the material from the book Lavrent'ev & Savel'ev (1999), and started with its third part, that on ill-posed problems. I explained results from functional analysis and partial differential equations as needed when we went along.
Prerequisites: A general acquaintance with analysis. A basic course on Fourier analysis and differential equations sufficed to follow the course successfully.
Exercises. Seven excercises related to ill-posed problems were handed out. The last two are about the Mumford--Shah energy and a discrete variant of it.
1. March 3: Definitions of well-posed and ill-posed problems. Hadamard's 1902 paper with his definition of an ill-posed problem and his two examples: the Cauchy problem for the Lapacian and the Cauchy problem for the wave equation with data on a timelike manifold. Examples of ill-posed problems: the backward heat equation; the Cauchy problem for the heat equation with data on a timelike manifold. Example from medicine and the geosciences.
2. March 10: From the categories well-posed, ill-posed, interesting, not interesting we can form four kinds of problems. Are the problems Hadamard mentioned really not interesting? Should certain problems be promoted from not interesting to interesting? Why? Some physical problems giving rise to well-posed problems. Ill-posed problems in complex analysis. Strongly and weakly ill-posed problems. Integral equations arising from some ill-posed problems.
3. March 11: More problems in physics that lead to ill-posed problems. Looking for minerals, or gravimetrical measurements showing that the Cauchy problem for the Laplace equation is interesting. Temperature measurements. Analysis of the sideways heat equation. (Eldén, Berntsson & Reginska 2000.)
4. March 24: More on the sideways heat equation. More precise notions of wellposedness.
5. March 25: Well-posedness in the sense of Tihonov relative to a subset of the space of solutions. Regularizing operators. Variational principle for regularizations. Stabilizing functionals. Minimizing sequences. Construction of a stabilizing functional on the infinite-dimensional Hilbert space l2, where an ellipsoid is compact if and only if the axes tend to zero.
6. April 1: How the stabilizing functional steers the approximate solutions towards the desired solution. Regularizing families for linear operators in Hilbert spaces. The adjoint of an operator in Hilbert space. Construction of a regularizing family for an injective, self-adjoint, positive, compact linear operator: Rc = (cI+A)-1, c > 0. Estimates for Rc in case A has a discrete spectrum. Regularizing families for injective compact linear operators: Rc = (cI+A*A)-1A*.
7. April 7. Other constructions of regularizing families. Iterative solutions. Fredholm integral equations of the first and second kind. Reduction of some special equations of the first kind to the second kind. Volterra integral equations of the first and second kind. Reduction of some special equations of the first kind to the second kind. Construction of approximate solutions to Volterra equations.
8. April 8. Presentation of the Greenland problem (Exercise 5). Operator Volterra equations. Evolution equations (beginning).
9. April 15. Fredholm integral equations of the first kind with convex constraints: an equation Au = g, where A is compact, injective and linear. We are looking for a solution u in a given, boundedly compact subset of a Hilbert space. (Brynielsson 1974.)
10. April 28. Evolution equations (cont'd). Inverse problems in image processing. The Mumford--Shah energy (definition).
11. April 29. The Mumford--Shah energy and the weak Mumford--Shah energy. Functions of bounded variation. Existence of an L2 function with minimal total variation in two dimensions (the Rudin--Osher approach).
12. May 12. More on functions of bounded variations. Special functions of bounded variation. Existence of a minimizer for the Mumford--Shah energy in one variable; in several variables.
13. June 2. The Radon transformation.
14. June 3. Discussion of the exercises.
Christer
Lennart Brynielsson (1974): On Fredholm integral equations of the first kind with convex constraints. SIAM J. Math. Anal. 5, No. 6, 955--962.
A. Chambolle (2000): Inverse problems in image processing and image separation: some mathematical and numerical aspects. Lecture given at the School on Mathematical Problems in Image Processing, Trieste 2000-09-4--22. 65 pp.
D. Dahl-Jensen; K. Mosegaard; N. Gundestrup; G. D. Clow; S. J. Johnsen; A. W. Hansen; N. Balling (1998): Past temperatures directly from the Greeenland ice sheet. Science 282, 268--271.
Lars Eldén; Fredrik Berntsson; Teresa Reginska (2000): Wavelet and Fourier mathods for solving the sideways heat equation. SIAM J. Sci. Computing 21 , No. 6, 2187--2205.
Jaques Hadamard (1902): Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 49--52.
M. M. Lavrent'ev; L. Ja. Savel'ev (1999): Teorija operatorov i nekorrektnye zada^ci. Novosibirsk: Izdatel'stvo Rossijskoj Akademii Nauk, Sibirskoe Otdelenie, Institut Matematiki im. S. L. Soboleva. 702 pp. ISBN 5-86134-077-3.
M. M. Lavrent'ev; S. M. Zerkal; O. E. Trofimov (2001): Computer modelling in tomography and ill-posed problems. Utrecht, Boston, Köln, Tokyo: VSP. ISBN 90-6764-350-5.
Valdimir Maz'ya; Tatyana Shaposhnikova (1998): Jacques Hadamard, a universal mathematician. American Mathematical Society; London Mathematical Society. ISBN 0-8218-0841-9.
David Mumford; Jayant Shah (1989): Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 42, 577--685.
A. N. Tikhonov; A. V. Goncharsky (Eds.) (1987): Ill-posed problems in the natural sciences. Moscow: MIR Publishers.
A. N. Tikhonov; A. S. Leonov; A. G. Yagola (1998): Non-linear ill-posed problems. Vol 1 & 2. London, Weinheim, New York, Tokyo, Melbourne, Madras: Chapman & Hall. ISBN 0-412-78660-5.
Other books and articles of interest:
V. P. Goluyatnikov (2000): Uniqueness questions in reconstruction of multidimensional objects from tomography-type projection data. Utrecht, Boston, Köln, Tokyo: VSP. ISBN 90-6764-332-7.
Gabor T. Herman (1980): Image reconstruction from projections: the fundamentals of computerized tomography. New York, London, Toronto, Sydney, San Francisco: Academic Press. ISBN 0-12-342050-4.
Henrik Kalisch; Jerry L. Bona (2000): Models for internal waves in deep water. Discrete and Continuous Dynamical Systems 6, No. 1, 1--20.
Charles Crosby Lyon (1984): The solution of ill-posed systems of linear equations in the presence of noise, with applications in geotomography. Master Thesis, The University of Arizona.