The Inverse Moment Problem for Convex Polytopes

Discrete and Computational Geometry

Volumn 48, Issue 3, 2012, Pages 596-621

  Gravin, Nick ^a,c   Lasserre, Jean ^b   Pasechnik, Dmitrii V ^a   Robins, Sinai ^a  

^a NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

^b LAAS CNRS   (France)

^c STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

Axial moments; Brion Lawrence Khovanskii Pukhlikov Barvinok formula; Inverse moment problem; Polytopes

Indexed keywords

EID: 84865687754     PISSN: 01795376     EISSN: 14320444     Source Type: Journal    
DOI: 10.1007/s00454-012-9426-4     Document Type: Article

References (20)

1. Barvinok, A. I.: Calculation of exponential integrals. Zap. Nauč. Semin. POMI 192(5), 149-162, 175-176 (1991).
2. Barvinok, A. I.: Exponential integrals and sums over convex polyhedra. Funkc. Anal. Prilozh. 26(2), 64-66 (1992).
3. Barvinok, A.: Integer Points in Polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008).
4. Baldoni, V., Berline, N., De Loera, J. A., Köppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comput. 80(273), 297-325 (2011).
5. Boley, D. L., Luk, F. T., Vandevoorde, D.: Vandermonde factorization of a Hankel matrix. In: Scientific Computing, Hong Kong, 1997, pp. 27-39. Springer, Singapore (1997).
6. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003). Revised version of the first edition online at http://perso. univ-rennes1. fr/marie-francoise. roy/.
7. Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York (2007).
8. Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. Éc. Norm. Super. 21(4), 653-663 (1988).
9. Cuyt, A., Golub, G., Milanfar, P., Verdonk, B.: Multidimensional integral inversion, with applications in shape reconstruction. SIAM J. Sci. Comput. 27(3), 1058-1070 (2005) (electronic).
10. Davis, P. J.: Triangle formulas in the complex plane. Math. Comput. 18, 569-577 (1964).
11. Demillo, R. A., Lipton, R. T.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7(4), 192-195 (1978).
12. Elad, M., Milanfar, P., Golub, G. H.: Shape from moments-an estimation theory perspective. IEEE Trans. Signal Process. 52(7), 1814-1829 (2004).
13. Golub, G. H., Milanfar, P., Varah, J.: A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21(4), 1222-1243 (1999) (electronic).
14. Grigoriev, D., Pasechnik, D. V.: Polynomial-time computing over quadratic maps. I. Sampling in real algebraic sets. Comput. Complex. 14(1), 20-52 (2005).
15. Lawrence, J.: Polytope volume computation. Math. Comput. 57(195), 259-271 (1991).
16. Lenstra, A. K., Lenstra, H. W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515-534 (1982).
17. Pukhlikov, A. V., Khovanskiǐ, A. G.: The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes. Algebra Anal. 4(4), 188-216 (1992).
18. Schwartz, J. T.: Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach. 27(4), 701-717 (1980).
19. Stanley, R. P.: Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997). With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original.
20. Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Symbolic and Algebraic Computation, EUROSAM'79, Internat. Sympos., Marseille, 1979. Lecture Notes in Comput. Sci., vol. 72, pp. 216-226. Springer, Berlin (1979).