Reconstruction of algebraic sets from dynamic moments
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, p. 647-664

We discuss an exact reconstruction algorithm for time expanding semi-algebraic sets given by a single polynomial inequality. The theoretical motivation comes from the classical L-problem of moments, while some possible applications to 2D fluid moving boundaries are sketched. The proofs rely on an adapted co-area theorem and a Hankel form minimization.

Nous présentons un algorithme de reconstruction exacte pour des domaines sémi-algèbriques croissants en temps, qui sont donnés par une seule inegalité polyno ^miale. La motivation théoretique vient du L-problème classique des moments, et nous esquissons une application possible aux fluides 2D avec des frontières mobiles. Les démonstrations sont basées sur le théorème de la co-aire et utilisent aussi la minimization d’une forme de Hankel.

@article{AFST_2007_6_16_3_647_0,
     author = {Putinar, Gabriela and Putinar, Mihai},
     title = {Reconstruction of algebraic sets from dynamic moments},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {3},
     year = {2007},
     pages = {647-664},
     doi = {10.5802/afst.1163},
     mrnumber = {2379056},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_3_647_0}
}
Putinar, Gabriela; Putinar, Mihai. Reconstruction of algebraic sets from dynamic moments. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, pp. 647-664. doi : 10.5802/afst.1163. http://www.numdam.org/item/AFST_2007_6_16_3_647_0/

[1] Akhiezer (N. I.), Krein (M.).— Some questions in the theory of moments. Translations of Mathematical Monographs, Vol. 2 American Mathematical Society, Providence, R.I. (1962). | MR 167806 | Zbl 0117.32702

[2] Berg (C.).— The multidimensional moment problem and semigroups. Moments in mathematics, 110-124, Proc. Sympos. Appl. Math., 37, Amer. Math. Soc., Providence, RI (1987). | MR 921086 | Zbl 0636.44007

[3] Bochnak (J.), Coste (M.), Roy (M.-F.).— Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3)36. Springer-Verlag, Berlin (1998). | MR 1659509 | Zbl 0912.14023

[4] Elad (M.), Milanfar (P.), Golub (G. H.).— Shape from moments- an estimation theory perspective. IEEE Trans. Signal Process. 52, no. 7, p. 1814-1829 (2004). | MR 2087688

[5] Fuglede (B.).— The multidimensional moment problem. Expositiones Math. 1, no. 1, p. 47-65 (1983). | MR 693807 | Zbl 0514.44006

[6] Golub (G. H.), Milanfar (P.), Varah (J.).— A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21, no. 4, p. 1222-1243 (1999/00). | MR 1740393 | Zbl 0956.65030

[7] Golub (G. H.), Gustafsson (B.), Milanfar (P.), Putinar (M.), Varah (J.).— Shape reconstruction from moments: theory, algorithms, and applications, SPIE Proceedings vol. 4116 (2000), Advanced Signal Processing, Algorithms, Architecture, and Implementations X (Franklin T.Luk, ed.), p. 406-416.

[8] Gustafsson (B.), He (C.), Milanfar (P.), Putinar (M.).— Reconstructing planar domains from their moments. Inverse Problems 16, no. 4, p. 1053-1070 (2000). | MR 1776483 | Zbl 0959.44010

[9] Gustafsson (B.), Putinar (M.).— Linear analysis of quadrature domains. II. Israel J. Math. 119, p. 187-216 (2000). | MR 1802654 | Zbl 0968.30016

[10] Gustafsson (B.), Vasiliev (A.).— Conformal and Potential Analysis in Hele-Shaw Cells, Birkhauser, Basel (2006). | MR 2245542 | Zbl 05065065

[11] Karlin (S.), Studden (W. J.).— Tchebycheff systems: With applications in analysis and statistics. Pure and Applied Mathematics, Vol. XV Interscience Publishers John Wiley & Sons (1966). | MR 204922 | Zbl 0153.38902

[12] Krein (M. G.), Nudelman (A. A.).— The Markov moment problem and extremal problems. Translations of Mathematical Monographs, Vol. 50. A. M. S., Providence, R.I. (1977). | Zbl 0361.42014

[13] Putinar (G.).— Asymptotics for extremal moments and monodromy of complex singularities. preprint UCSB, no. 2006-48.

[14] Putinar (G.).— Semi-local micro-differential theory and computations of moments for semi-algebraic domains. preprint UCSB, no. 2006-46.

[15] Putinar (M.).— Extremal solutions of the two-dimensional L-problem of moments. II. J. Approx. Theory 92, no. 1, p. 38-58 (1998). | MR 1492857 | Zbl 0910.47011