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Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  03 May 2017

Zhen Li ,
Zuoqiang Shi  and
Jian Sun
Zhen Li*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Zuoqiang Shi*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Jian Sun*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
*
*Corresponding author. Email addresses:zlin12@mails.tsinghua.edu.cn (Z. Li), zqshi@mail.tsinghua.edu.cn (Z. Shi), jsun@math.tsinghua.edu.cn (J. Sun)
*Corresponding author. Email addresses:zlin12@mails.tsinghua.edu.cn (Z. Li), zqshi@mail.tsinghua.edu.cn (Z. Shi), jsun@math.tsinghua.edu.cn (J. Sun)
*Corresponding author. Email addresses:zlin12@mails.tsinghua.edu.cn (Z. Li), zqshi@mail.tsinghua.edu.cn (Z. Shi), jsun@math.tsinghua.edu.cn (J. Sun)
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Abstract

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

Keywords

Point integral methodpoint cloudLaplace-Beltrami operatorconvergence

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N25: Eigenvalue problems 65N75: Probabilistic methods, particle methods, etc.
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 228 - 258
Copyright
Copyright © Global-Science Press 2017 

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References

[1] MeshLab. http://meshlab.sourceforge.net/.Google Scholar
[2] Belkin, M. and Niyogi, P.. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003.Google Scholar
[3] Belkin, M. and Niyogi, P.. Towards a theoretical foundation for laplacian-based manifold methods. In COLT, pages 486–500, 2005.Google Scholar
[4] Belkin, M. and Niyogi, P.. Convergence of laplacian eigenmaps. preprint, short version NIPS 2008, 2008.Google Scholar
[5] Belkin, M., Sun, J., and Wang, Y.. Constructing laplace operator from point clouds in rd. In SODA ′09: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms, pages 1031–1040, Philadelphia, PA, USA, 2009. Society for Industrial and Applied Mathematics.Google Scholar
[6] Bertalmio, M., Cheng, L.-T., Osher, S., and Sapiro, G.. Variational problems and partial differential equations on implicit surfaces. Journal of Computational Physics, 174(2):759–780, 2001.Google Scholar
[7] Bertalmio, M., Memoli, F., Cheng, L.-T., Sapiro, G., and Osher, S.. Variational problems and partial differential equations on implicit surfaces: Bye bye triangulated surfaces? In Geometric Level Set Methods in Imaging, Vision, and Graphics, pages 381–397. Springer New York, 2003.Google Scholar
[8] Cao, H.-D. and Yau, S.-T.. Geometric flows, volume 12 of Surveys in Differential Geometry. International Press of Boston, Inc., 2007.Google Scholar
[9] da Costa, R. C. T.. Quantum mechanics of a constrained particle. PHYSICAL REVIEW A, 25(6), April 1981.Google Scholar
[10] Defay, R. and Priogine, I.. Surface Tension and Adsorption. John Wiley & Sons, New York, 1966.Google Scholar
[11] Dey, T. K.. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, New York, NY, USA, 2006.Google Scholar
[12] Dey, T. K., Ranjan, P., and Wang, Y.. Convergence, stability, and discrete approximation of laplace spectra. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ′10, pages 650–663, Philadelphia, PA, USA, 2010. Society for Industrial and Applied Mathematics.Google Scholar
[13] Dodziuk, J. and Patodi, V. K.. Riemannian structures and triangulations of manifolds. Journal of Indian Math. Soc., 40:152, 1976.Google Scholar
[14] Dziuk, G.. Finite elements for the beltrami operator on arbitrary surfaces. In Hildebrandt, S. and Leis, R., editors, Partial differential equations and calculus of variations, volume 1357 of Lecture Notes in Mathematics, pages 142–155. Springer, 1988.Google Scholar
[15] Hein, M., Audibert, J.-Y., and von Luxburg, U.. From graphs to manifolds - weak and strong pointwise consistency of graph laplacians. In Proceedings of the 18th Annual Conference on Learning Theory, COLT’05, pages 470–485, Berlin, Heidelberg, 2005. Springer-Verlag.Google Scholar
[16] Jean-Daniel Boissonnat, A. G., Dyer, Ramsay. Stability of delaunay-type structures for manifolds: [extended abstract]. In Symposium on Computational Geometry, pages 229–238, 2012.Google Scholar
[17] Lafon, S.. Diffusion Maps and Geometric Harmonics. PhD thesis, 2004.Google Scholar
[18] Lai, R., Liang, J., and Zhao, H.. A local mesh method for solving pdes on point clouds. Inverse Problem and Imaging, to appear.Google Scholar
[19] Levy, B.. Laplace-beltrami eigenfunctions towards an algorithm that “understands” geometry. In Shape Modeling and Applications, 2006. SMI 2006. IEEE International Conference on, pages 13–13, June 2006.Google Scholar
[20] Liang, J. and Zhao, H.. Solving partial differential equations on point clouds. SIAM Journal of Scientific Computing, 35:1461–1486, 2013.Google Scholar
[21] Lindeberg, T.. Scale selection properties of generalized scale-space interest point detectors. Journal of Mathematical Imaging and Vision, 46(2):177–210, 2013.Google Scholar
[22] Luo, C., Sun, J., and Wang, Y.. Integral estimation from point cloud in d-dimensional space: a geometric view. In Symposium on Computational Geometry, pages 116–124, 2009.Google Scholar
[23] Macdonald, C. B. and Ruuth, S. J.. The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput., 31(6):4330–4350, 2009.Google Scholar
[24] März, T. and Macdonald, C. B.. Calculus on surfacewith feneral closest point functions. SIAM J. Numer. Anal., 50(6):3303–3328, 2012.Google Scholar
[25] Ovsjanikov, M., Sun, J., and Guibas, L. J.. Global intrinsic symmetries of shapes. Comput. Graph. Forum, 27(5):1341–1348, 2008.Google Scholar
[26] Reuter, M., Wolter, F.-E., and Peinecke, N.. Laplace-beltrami spectra as “shape-dna” of surfaces and solids. Computer-Aided Design, 38(4):342–366, 2006.Google Scholar
[27] Ruuth, S. J. and Merriman, B.. A simple embedding method for solving partial differential equations on surfaces. Journal of Computational Physics, 227(3):1943–1961, 2008.Google Scholar
[28] Saito, N.. Data analysis and representation on a general domain using eigenfunctions of laplacian. Applied and Computational Harmonic Analysis, 25(1):68–97, 2008.Google Scholar
[29] Schuster, P. and Jaffe, R.. Quantum mechanics on manifolds embedded in euclidean space. Annals of Physics, 307(1):132–143, 2003.Google Scholar
[30] Shewchuk, J. R.. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In Lin, M. C. and Manocha, D., editors, Applied Computational Geometry: Towards Geometric Engineering, volume 1148 of Lecture Notes in Computer Science, pages 203–222. Springer-Verlag, May 1996. From the First ACM Workshop on Applied Computational Geometry.Google Scholar
[31] Shewchuk, J. R.. What is a good linear finite element? - interpolation, conditioning, anisotropy, and quality measures. Technical report, In Proc. of the 11th International Meshing Roundtable, 2002.Google Scholar
[32] Shi, Z. and Sun, J.. Convergence of laplacian spectra from point clouds. arXiv:1506.01788.Google Scholar
[33] Shi, Z. and Sun, J.. Convergence of the point integral method for the poisson equation on manifolds i: the neumann boundary. arXiv:1403.2141.Google Scholar
[34] Shi, Z. and Sun, J.. Convergence of the point integral method for the poisson equation on manifolds ii: the dirichlet boundary. arXiv:1312.4424.Google Scholar
[35] Singer, A. and tieng Wu, H.. Spectral convergence of the connection laplacian from random samples. arXiv:1306.1587.Google Scholar
[36] Strang, G. and Fix, G. J.. An analysis of the finite element method. Prentice-Hall, 1973.Google Scholar
[37] The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 4.4 edition, 2014.Google Scholar
[38] Wardetzky, M.. Discrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation. PhD thesis, 2006.Google Scholar
[39] Yau, S.-T.. The role of partial differential equations in differential geometry. In Proceedings of the International Congress of Mathematicians (Helsinki 1978), pages 237–250. Acad. Sci. Fennica, Helsinki, 1980.Google Scholar