On bounds for diffusion, discrepancy and fill distance metrics

Lecture Notes in Computational Science and Engineering

Volumn 58, Issue , 2008, Pages 261-270

  Damelin, Steven B ^a  

^a GEORGIA SOUTHERN UNIVERSITY   (United States)

Author keywords

Affinity matrix; Diffusion distance; Dimension reduction; Discrepancy; Discretization; Distribution; Eigenfunction expansion; Energy; Equilibrium measure; Euclidean space; Fill distance; Graph; Kernels; Laplacian; Mesh norm; Numerical integration; Orthonormal basis; Positive definite

Indexed keywords

AFFINITY MATRIX; DIFFUSION DISTANCE; DIMENSION REDUCTION; DISCREPANCY; DISCRETIZATION; DISTRIBUTION; EIGENFUNCTION EXPANSION; ENERGY; EQUILIBRIUM MEASURE; EUCLIDEAN SPACE; FILL DISTANCE; GRAPH; KERNELS; LAPLACIAN; MESH NORM; NUMERICAL INTEGRATION; ORTHONORMAL BASIS; POSITIVE DEFINITE;

COMPUTER NETWORKS; COMPUTER PROGRAMMING; COMPUTER VISION; DATA VISUALIZATION; DIFFUSION; EDUCATION; EIGENVALUES AND EIGENFUNCTIONS; LAPLACE TRANSFORMS; SYSTEMS ENGINEERING; WIRELESS SENSOR NETWORKS; WIRELESS TELECOMMUNICATION SYSTEMS;

SET THEORY;

EID: 77955282452     PISSN: 14397358     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-73750-6_11     Document Type: Conference Paper

References (25)

1. Bajnok, B., Damelin, S.B., Li, J., and Mullen, G.: A constructive finite field method for scattering points on the surface of a d-dimensional sphere. Computing, 68, 97-109 (2002)
2. Chung F.: Spectral Graph Theory. In: CBNS-AMS, 92. AMS Publications, Providence, RI (1997)
3. Damelin, S.B. and Grabner, P.: Numerical integration, energy and asymptotic equidistributon on the sphere. Journal of Complexity, 19, 231-246 (2003)
4. Damelin, S.B.: Marcinkiewicz-Zygmund inequalities and the Numerical approximation of singular integrals for exponential weights: Methods, Results and Open Problems, some new, some old. Journal of Complexity, 19, 406-415 (2003)
5. Damelin, S.B., Hero, A., and Wang, S.J.: Kernels, hitting time metrics and diffusion in clustering and data analysis, in preparation.
6. Damelin, S.B., Hickernell F., and Zeng, X.: On the equivalence of discrepancy and energy on closed subsets of Euclidean space, in preparation.
7. Damelin, S.B. and Kuijlaars, A.: The support of the extremal measure for monomial external fields on [-1, 1]. Trans. Amer. Math. Soc. 351, 4561-4584 (1999)
8. Damelin, S.B., Levesley, J., and Sun, X.: Energies, Group Invariant Kernels and Numerical Integration on Compact Manifolds. Journal of Complexity, submitted.
9. Damelin, S.B., Levesley, J., and Sun, X.: Energy estimates and the Weyl criterion on compact homogeneous manifolds. In: Algorithms for Approximation V. Springer, to appear.
10. n. Journal of Complexity, 21 (6), 845-863 (2005)
11. Damelin S.B. and Maymeskul, V.: Minimal discrete energy problems and numerical integration on compact sets in Euclidean spaces. In: Algorithms for Approximation V. Springer, to appear.
12. Damelin S.B. and Maymeskul, V.: On regularity and dislocation properties of minmial energy configurations on compact sets, submitted.
13. Du, Q., Faber, V., and Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev, 41 (4), 637-676 (1999)
14. Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. (Graduate Studies in Mathematics.) American Mathematical Society (2001)
15. Hickernell, F.J.: Goodness of fit statistics, discrepancies and robust designs. Statist. Probab. Lett. 44 (1), 73-78 (1999)
16. Hickernell, F.J.: What affects the accuracy of quasi-Monte Carlo quadrature? In: Niederreiter, H. and Spanier, J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods. Springer-Verlag, Berlin, 16-55 (2000)
17. Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67 (221), 299-322 (1998)
18. Hickernell, F.J.: An algorithm driven approach to error analysis for multidimensional integration, submitted.
19. Hickernell, F.J.: My dream quadrature rule. J. Complexity, 19, 420-427 (2003)
20. Huffman, W.C. and Pless, V.: Fundamentals of error-correcting codes, Cambridge University Press, Cambridge (2003)
21. Johnson, M.E., Moore, L.M., and Ylvisaker, D.: Minimax and maxmin distance designs. J. Statist. Plann. Inference 26, 131-148 (1990)
22. Kuipers, L. and Niederreiter, H.: Uniform Distribution of Sequences. Wiley-Interscience, New York (1974)
23. Lafon, S. and Coifman, R.R.: Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5-30 (2006)
24. Niederreiter, H.: Random Number Generation and Quasi- Monte Carlo Methods. Volume 63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992)
25. Lubotzky, A., Phillips, R., and Sarnak, P.: Hecke operators and distributing points on the sphere (I-II). Comm. Pure App. Math. 39-40, 148-186, 401-420 (1986, 1987)