Efficient Rigorous Numerics for Higher-Dimensional PDEs via One-Dimensional Estimates
Abstract
We present an efficient rigorous computational method which is an extension of the work Analytic Estimates and Rigorous Continuation for Equilibria of Higher-Dimensional PDEs (M. Gameiro and J.-P. Lessard, J. Differential Equations, 249 (2010), pp. 2237--2268). The idea is to generate sharp one-dimensional estimates using interval arithmetic which are then used to produce high-dimensional estimates. These estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable. Computing the equilibria using a finite-dimensional projection, the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the PDE. A new construction of the polynomials is presented where the nonlinearities are bounded by products of one-dimensional estimates as opposed to using FFT with large inputs. It is demonstrated that with this approach it is much cheaper to prove that the numerical output is correct than to recompute at a finer resolution. We apply this method to PDEs defined on three- and four-dimensional spatial domains.
MSC codes
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12 (2000), pp. 273--288.
2.
H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer., 13 (2004), pp. 147--269.
3.
S. Day, Y. Hiraoka, K. Mischaikow, and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dynam. Syst., 4 (2005), pp. 1--31.
4.
S. Day, J.-P. Lessard, and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal., 45 (2007), pp. 1398--1424.
5.
G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer Ser. Oper. Res., Springer, New York, 1996.
6.
M. Gameiro, J.-P. Lessard, and K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Math. Comput. Simulation, 79 (2008), pp. 1368--1382.
7.
M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations, 249 (2010), pp. 2237--2268.
8.
M. Gameiro and J.-P. Lessard, Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation, Numer. Math., 117 (2011), pp. 753--778.
9.
P. Glasserman, Monte Carlo methods in Financial Engineering, Stoch. Model. Appl. Probab. 53, Springer, New York, 2004.
10.
M. Griebel and S. Knapek, Optimized tensor-product approximation spaces, Constr. Approx., 16 (2000), pp. 525--540.
11.
Y. Hiraoka and T. Ogawa, An efficient estimate based on FFT in topological verification method, J. Comput. Appl. Math., 199 (2007), pp. 238--244.
12.
S. Maier-Paape, U. Miller, K. Mischaikow, and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Rev. Mat. Complut., 21 (2008), pp. 351--426.
13.
R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
14.
F. Nobile, R. Tempone, and C. G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), pp. 2411--2442.
15.
F. Nobile, R. Tempone, and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), pp. 2309--2345.
16.
C. Schwab and R. Stevenson, Adaptive wavelet algorithms for elliptic PDE's on product domains, Math. Comp., 77 (2008), pp. 71--92.
17.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci. 68, Springer, New York, 1997.
18.
J. B. van den Berg, J.-P. Lessard, and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), pp. 1565--1584.
19.
C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Frontiers in Appl. Math. 10, SIAM, Philadelphia, PA, 1992.
20.
T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 93--127.
Information & Authors
Information
Published In
Copyright
© 2013, Society for Industrial and Applied Mathematics.
History
Submitted: 7 June 2011
Accepted: 23 April 2013
Published online: 16 July 2013
MSC codes
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.