Fast and Exact (Poisson) Solvers on Symmetric Geometries
In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a natural question to consider is whether and how the symmetry can be leveraged to derive an efficient solver for the underlying system of linear e...
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| Published in | Computer graphics forum Vol. 34; no. 5; pp. 153 - 165 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Blackwell Publishing Ltd
01.08.2015
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| Abstract | In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a natural question to consider is whether and how the symmetry can be leveraged to derive an efficient solver for the underlying system of linear equations. In this work we provide a simple representation‐theoretic analysis that demonstrates how symmetries of the geometry translate into block diagonalization of the linear operators and we show how this results in efficient linear solvers for surfaces of revolution with and without angular boundaries. |
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| AbstractList | In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a natural question to consider is whether and how the symmetry can be leveraged to derive an efficient solver for the underlying system of linear equations. In this work we provide a simple representation‐theoretic analysis that demonstrates how symmetries of the geometry translate into block diagonalization of the linear operators and we show how this results in efficient linear solvers for surfaces of revolution with and without angular boundaries. |
| Author | Kazhdan, M. |
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| Cites_doi | 10.1145/882262.882269 10.1145/1391989.1391995 10.1145/1141911.1142001 10.1137/S1064827598339402 10.1145/1400885.1400961 10.1137/1.9781611971057.ch4 10.1137/1.9780898717532 10.1137/1.9780898718003 10.1145/280814.280831 10.1137/1.9780898719505 10.1090/conm/180/2003 10.1080/10867651.2001.10487540 10.1137/S1064827503430138 10.1007/978-3-540-88688-4_9 10.1007/s00041-003-0018-9 10.1145/1365490.1365500 10.1145/321250.321259 10.1016/0021-9991(88)90102-7 10.1145/1189762.1189766 10.1145/1882261.1866175 10.1145/1015706.1015718 10.1007/978-1-4684-9458-7 10.1109/99.660313 10.1145/882262.882338 10.1090/S0025-5718-1965-0178586-1 10.1111/j.1467-8659.2008.01122.x 10.1109/JPROC.2004.840301 10.1111/j.1467-8659.2009.01524.x 10.1145/311535.311548 10.1007/978-3-540-24673-2_31 |
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| Copyright | 2015 The Author(s) Computer Graphics Forum © 2015 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2015 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Stam J.: Flows on surfaces of arbitrary topology. ACM Transactions on Graphics (SIGGRAPH '03) 22 (2003), 724-731. 7 – reference: Healy D., Rockmore D., Kostelec P., Moore S.: FFTs on the 2-sphere - improvements and variations. Journal of Fourier Analysis and Applications 9 (2003), 341-385. 6, 7 – reference: Cooley J., Tukey J.: An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19 (1965), 297-301. 2 – reference: Nickolls J., Buck I., Garland M., Skadron K.: Scalable parallel programming with CUDA. Queue 6, 2 (2008), 40-53. 9 – reference: Schumann U., Sweet R.: Fast Fourier transforms for direct solution of Poisson's equation with staggered boundary conditions. Journal of Computational Physics 75, 1 (1988), 123-137. 2 – reference: Chen Y., Davis T., Hager W., Rajamanickam S.: Algorithm 887: Cholmod, supernodal sparse Cholesky factorization and update/downdate. ACM Transactions on Mathematical Software 35, 3 (2008), 22:1-22:14. 7 – reference: Dagum L., Menon R.: OpenMP: An industry-standard API for shared-memory programming. IEEE Computational Science and Engineering 5, 1 (1998), 46-55. 6 – reference: Briggs W., Henson V., McCormick S.: A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 2000. 2 – reference: Aksoylu B., Khodakovsky A., Schröder P.: Multilevel solvers for unstructured surface meshes. SIAM Journal of Scientific Computing 26, 4 (2005), 1146-1165. 2 – reference: Frigo M., Johnson S.: The design and implementation of FFTW3. Proceedings of the IEEE 93, 2 (2005), 216-231. Special issue on "Program Generation, Optimization, and Platform Adaptation". 6 – reference: Serre J.: Linear representations of finite groups. Springer-Verlag, New York, 1977. 3 – reference: Stam J.: A simple fluid solver based on the FFT. Journal of Graphics Tools 6 (2001), 43-52. 7 – reference: Bindel D., Bruyns C.: Shape-changing symmetric objects for sound synthesis. In Audio Engineering Society 121 (2006), pp. 875-881. 2, 3, 4 – reference: Vallet B., LÃl'vy B.: Spectral geometry processing with manifold harmonics. Computer Graphics Forum (Proceedings Eurographics) 27, 2 (2008), 251-260. 2 – reference: Höllig K.: Finite Element Methods with B-Splines. Society for Industrial and Applied Mathematics, 2003. 11 – reference: Shewchuk J.: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Tech. rep., Carnegie Mellon University, Pittsburgh, PA, USA, 1994. 2 – reference: Maslen D., Rockmore D.: Generalized FFTS - A Survey of Some Recent Results. Tech. rep., Dartmouth College, Hanover, NH, USA, 1996. 9 – reference: Golub G., Loan C.V.: Matrix Computations (3rd Ed.). Johns Hopkins University Press, Baltimore, MD, USA, 1996. 2 – reference: Hockney R.: A fast direct solution of Poisson's equation using Fourier analysis. Journal of the ACM 12, 1 (1965), 95-113. 2, 5, 11 – reference: Shi L., Yu Y., Bell N., Feng W.: A fast multigrid algorithm for mesh deformation. ACM Transactions on Graph 25, 3 (2006), 1108-1117. 2 – reference: Fulton W., Harris J.: Representation Theory: A First Course. Springer-Verlag, New York, 1991. 3 – reference: Saad Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, 2003. 2, 6 – reference: Elcott S., Tong T., Kanso E., Schröder P., Desbrun M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1 (2007). 7, 8 – reference: Cleary A., Falgout R., Henson V., Jones J., Manteuffel T., McCormick S., Miranda G., Ruge J.: Robustness and scalability of algebraic multigrid. 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| Snippet | In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a... |
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| SubjectTerms | Analysis and systems-Fluid Simulation Blocking Boundaries Categories and Subject Descriptors (according to ACM CCS) Computer graphics I.3.5 [Computer Graphics]: Geometric algorithms I.3.5 [Computer Graphics]: Geometric algorithms, languages, and systems—Fluid Simulation Image processing systems languages Linear operators Mathematical models Poisson distribution Poisson equation Solvers Studies Symmetry |
| Title | Fast and Exact (Poisson) Solvers on Symmetric Geometries |
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