Minimization of energy functionals applied to some inverse problems

Applied Mathematics and Optimization

Volumn 44, Issue 3, 2001, Pages 273-297

  Leonard C ^a,b  

^a UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Convex integral functionals; Fenchel duality; Fredholm integral equations; Marginal problems; Maximum entropy method

Indexed keywords

CONSTRAINT THEORY; FUNCTIONS; INTEGRAL EQUATIONS; INVERSE PROBLEMS; PROBABILITY; QUANTUM THEORY; RANDOM PROCESSES; VARIATIONAL TECHNIQUES;

CONVEX INTEGRAL FUNCTIONALS; ENERGY FUNCTIONALS; FENCHEL DUALITY; FREDHOLM INTEGRAL EQUATIONS; LINEAR CONSTRAINTS; MARGINAL PROBLEMS; MAXIMUM ENTROPY METHOD;

MAXIMUM LIKELIHOOD ESTIMATION;

EID: 0041389085     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-001-0019-5     Document Type: Article

References (37)

1. R. Aebi. Schrödinger Diffusion Processes. Probability and Its Applications. Birkhäuser, Boston, MA, 1996.
2. S. Bernstein. Sur les liaisons entre les grandeurs aléatoires. Vehr. des intern. Mathematikerkongr., Zürich, 1932, Band 1.
3. A. Beurling. An automorphism of product measures. Ann. Math. 72(1) (1960), 189-200.
4. J. M. Borwein. On the failure of maximum entropy reconstruction for Fredholm equations and other finite systems. Math. Programming 61 (1993), 251-261.
5. 1 and the existence of maximum entropy estimates. SIAM J. Optim. 3(2) (1993), 248-267.
6. J. M. Borwein, A. S. Lewis and R. D. Nussbaum. Entropy minimization, DAD problems and doubly stochastic kernels. J. Funct. Anal. 123 (1994), 264-307.
7. E. Carlen. Conservative diffusions. Comm. Math. Phys. 94 (1984), 293-315.
8. P. Cattiaux and F. Gamboa. Large deviations and variational theorems for marginal problems, Bernoulli 5 (1999).
9. P. Cattiaux and C. Léonard. Large deviations and Nelson processes. Forum Math., 7 (1995), 95-115.
10. A. B. Cruzeiro, Liming Wu and J. C. Zambrini. Bernstein processes associated with a Markov process. Proceedings of the Third International Workshop on Stochastic Analysis and Mathematical Physics: ANESTOC '98, Ed. R. Rebolledo, Trends in Mathematics. Birkhaüser, Boston, MA.
11. I. Csiszár. I-Divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975), 146-158.
12. I. Csiszár. Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12 (1984), 768-793.
13. I. Csiszár. Generalized projections for non-negative functions. Acta Math. Hungar. 68(1-2) (1995), 161-185.
14. D. Dacunha-Castelle and F. Gamboa. Maximum d'entropie et problèmes de moments. Ann. Inst. H. Poincaré 26 (1990), 567-596.
15. P. Dai Pra. A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23, (1991), 313-329.
16. D. A. Dawson and J. Gärthner. Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics Stochastics Rep., 20 (1987), 247-308.
17. A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, second edition. Springer-Verlag, New York, 1998.
18. H. Föllmer. Random Fields and Diffusion Processes. Cours à 'Ecole d'Été de Probabiités de Saint-Flour. Lecture Notes in Mathematics 1362 Springer-Verlag, Berlin, 1988.
19. H. Föllmer and N. Gantert. On Schrödinger processes in infinite dimensions. Ann. Probab., 25 (1997), 901-926.
20. R. Fortet. Résolution d'un système d'équations de M. Schrödinger. J. Math. Pures Appl. IX (1940), 83-105.
21. F. Gamboa and E. Gassiat. Sets of superresolution and the maximum entropy method on the mean. SIAM J. Math. Anal., 27(4) (1996), 1129-1152.
22. F. Gamboa and E. Gassiat. Bayesian methods and maximum entropy for ill posed problems. Ann. Statist. 25(1) (1997), 328-350.
23. B. Jamison. Reciprocal processes. Z. Wahrsch. Verw. Gebiete 30 (1974), 65-86.
24. A. Kozek. Convex integral functionals on Orlicz spaces. Ann. Soc. Math. Polon. Ser. I (1979), 109-135.
25. C. Léonard. Large deviations for particle systems associated with spatially homogeneous Boltzmann type equations. Probab. Theory Rel. Fields, 101(1) (1995), 1-44.
26. C. Léonard. Convex conjugates of integral functionals, Acta Math. Hungar., to appear.
27. C. Léonard. Minimizers of energy functionals, Acta Math. Hungar., to appear.
28. C. Léonard. Minimizers of energy functionals under not very integrable constraints. Preprint, 2000.
29. A. S. Lewis. Consistency of moment systems. Canad. J. Math., 47 (1995), 995-1006.
30. M. Nagasawa, Schrödinger Equations and Diffusion Theory. Monographs in Mathematics. Birkhaüser, Boston, MA, 1993.
31. E. Nelson. Stochastic mechanics and random fields. Cours à l'Ecole d'été dde Probabilités de Saint-Flour. Lecture Notes in Mathematics 1362. Springer-Verlag, Berlin, 1988.
32. M. M. Rao and Z. D. Ren. Theory of Orlicz Spaces. Marcel Dekker, New York, 1991.
33. R. T. Rockafellar. Integrals which are convex functionals. Pacific J. Math. 24(3) (1968), 525-539.
34. R. T. Rockafellar. Integrals which are convex functionals, II. Pacific J. Math. 39(2) (1971), 439-469.
35. R. T. Rockafallar. Conjugate Duality and Optimization. SIAM, Philadelphia, PA, 1974.
36. E. Schrödinger. Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2 (1932), 269-310.
37. J. C. Zambrini. Variational processes and stochastic versions of mechanics. J. Math. Phys. 27(9) (1986), 2307.