Deriving the continuity of maximum-entropy basis functions via variational analysis

SIAM Journal on Optimization

Volumn 18, Issue 3, 2007, Pages 914-925

  Sukumart, N ^a   Wets, R J B ^b  

^a University of California Davis   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Convex approximation; Epiconvergence; Maximum entropy; Meshfree methods; Relative entropy

Indexed keywords

BASIS FUNCTIONS; CONVEX APPROXIMATION; DATA APPROXIMATION; DISCRETE PROBABILITY DISTRIBUTION; EPICONVERGENCE; INDUCTIVE INFERENCES; LINEAR CONSTRAINTS; MAXIMUM ENTROPY; MESHFREE METHODS; RECENT DEVELOPMENT; RELATIVE ENTROPY; VARIATIONAL ANALYSIS; VARIATIONAL PRINCIPLES;

INFORMATION THEORY; POLYNOMIAL APPROXIMATION; PROBABILITY DISTRIBUTIONS; RISK ASSESSMENT; VARIATIONAL TECHNIQUES;

DISTRIBUTION FUNCTIONS;

EID: 49449118360     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/06066480X     Document Type: Conference Paper

References (30)

1. M. ARROYO AND M. ORTIZ, Local maximum-entropy approximation schemes: A seamless bridge between finite elements and meshfree methods, Int. J. Numer. Methods Engrg., 65 (2006), pp. 2167-2202.
2. H. ATTOUCH, Variational Convergence for Functions and Operators, Appl. Math. Ser., Pitman, Boston, 1984.
3. J.-P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Birkhäuser Boston, Boston, MA, 1990.
4. T. BELYTSCHKO, Y. KRONGAUZ, D. ORGAN, M. FLEMING, AND P. KRYSL, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139 (1996), pp. 3-47.
5. A. L. BERGER, S. A. DELLA PIETRA, AND V. J. DELLA PIETRA, A maximum entropy approach to natural language processing, Comput. Linguist., 22 (1996), pp. 39-71.
6. S. BOYD AND L. VANDENBERGHE, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
7. T. M. COVER AND J. A. THOMAS. Elements of Information Theory, Wiley, New York, 1991.
8. R. T. FAROUKI AND T. N. T. GOODMAN, On the optimal stability of the Bernstein basis, Math. Comp., 65 (1996), pp. 1553-1566.
9. M. R. GUPTA, An Information Theory Approach to Supervised Learning, Ph.D. thesis, Department of Electrical Engineering, Stanford University, Palo Alto, CA, 2003.
10. J. SKILLING AND R. K. BRYAN, Maximum entropy image reconstruction: General algorithm, Monthly Notices Roy. Astronom. Soc., 211 (1984), pp. 111-118.
11. E. T. JAYNES, Information theory and statistical mechanics, Phys. Rev., 106 (1957), pp. 620-630.
12. E. T. JAYNES, Information Theory and Statistical Mechanics, in Statistical Physics: The 1962 Brandeis Lectures, K. Ford, ed., W. A. Benjamin, New York, 1963, pp. 181-218.
13. E. T. JAYNES, Probability Theory: The Logic of Science, Cambridge University Press, Cambridge, UK, 2003.
14. A. KHINCHIN, Mathematical Foundations of Information Theory, Dover, New York, 1957.
15. S. KULLBACK, Information Theory and Statistics, Wiley, New York, 1959.
16. S. KULLBACK AND R. A. LEIBLER, On information and sufficiency, Ann. Math. Statist., 22 (1951), pp. 79-86.
17. P. LANCASTER AND K. SALKAUSKAS, Surfaces generated by moving least squares methods, Math. Comp., 37 (1981), pp. 141-158.
18. R. W. MINICH, C. A. SCHUH, AND M. KUMAR, Role of topological constraints on the statistical properties of grain boundary networks, Phys. Rev. B, 66 (2003), 052101.
19. J. M. PEÑA, B-splines and optimal stability, Math. Comp., 66 (1997), pp. 1555-1560.
20. d, Discrete Comput. Geom., 12 (1994), pp. 189-202.
21. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer-Verlag, Heidelberg, 2004.
22. C. E. SHANNON, A mathematical theory of communication, Bell Systems Tech. J., 27 (1948), pp. 379-423.
23. J. E. SHORE AND R. W. JOHNSON, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inform. Theory, 26 (1980), pp. 26-36.
24. D. S. SIVIA, Data Analysis: A Bayesian Tutorial, Oxford University Press, Oxford, UK, 1996.
25. G. STRANG AND G. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
26. N. SUKUMAR, Construction of polygonal interpolants: A maximum entropy approach, Int. J. Numer. Methods Engrg., 61 (2004), pp. 2159-2181.
27. N. SUKUMAR, Maximum entropy approximation, AIP Conf. Proc., 803 (2005), pp. 337-344.
28. N. SUKUMAR AND R. W. WRIGHT, Overview and construction of meshfree basis functions: From moving least squares to entropy approximants, Int. J. Numer. Methods Engrg., 70 (2007), pp. 181-205.
29. D. WALKUP AND R. J.-B. WETS, A Lipschitzian characterization of convex polyhedra, Proc. Amer. Math. Soc., 23 (1969), pp. 167-173.
30. H. WENDLAND, Scattered Data Approximation, Cambridge University Press, Cambridge, UK, 2005.