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An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations

G. Horton, S. Vandewalle, P. Worley
1995 SIAM Journal on Scientific Computing  
This paper describes an algorithm for the time-accurate solution of certain classes of parabolic partial differential equations that can be parallelized in both time and space.  ...  The standard numerical algorithms for solving parabolic partial differential equations are inherently sequential in the time direction.  ...  , Sankt-Augustin, Germany, for letting them use the CM/2 machine at the GMD.  ... 
doi:10.1137/0916034 fatcat:66o4yawudfabzhxsx65pwdk6dm

Page 7630 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
(B-KUL-C; Leuven (Heverlee)) ; Worley, P. (1-ORNL; Oak Ridge, TN) An algorithm with polylog parallel complexity for solving parabolic partial differential equations. (English summary) SIAM J. Sci.  ...  This paper describes an algorithm for the time- accurate solution of certain classes of parabolic partial differential equations that can be parallelized in both time and space.  ... 

Waveform Relaxation with Fast Direct Methods as Preconditioner

Jo Simoens, Stefan Vandewalle
2000 SIAM Journal on Scientific Computing  
For a restricted class of parabolic PDEs one can devise a practical numerical solver with a parallel complexity that is theoretically optimal.  ...  Waveform relaxation was developed as an iterative method for solving large systems of ODEs. It is the continuous-in-time analogue of stationary iterative methods for linear algebraic equations.  ...  We consider the numerical solution of parabolic partial differential equations (PPDEs) of initial-boundary value type.  ... 
doi:10.1137/s1064827598338986 fatcat:kgk5wagrf5f45bxltq42bgprla

A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations [article]

Shuonan Wu, Zhi Zhou
2021 arXiv   pre-print
In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations.  ...  In particular, we apply the k-step backward differentiation formula, and then develop an iterative solver by using the waveform relaxation technique.  ...  Zhou is partially supported by a Hong Kong RGC grant (project No. 15304420).  ... 
arXiv:2007.13125v2 fatcat:chry5yt2rfdczhtob5yl7xljeq

Scalable algorithms for the solution of Navier's equations of elasticity

Owe Axelsson, Maya Neytcheva
1995 Journal of Computational and Applied Mathematics  
For a class of multilevel methods for elliptic partial differential equations it is shown how to balance the coarsest mesh size to the finest and the number of processors to the size of the problem to  ...  time of the parallel algorithm on p processors.  ...  When we make the comparison with respect to an optimal order algorithm to solve elliptic type difference equations any scalable parallel algorithm must also have an optimal order of computational complexity  ... 
doi:10.1016/0377-0427(95)00054-2 fatcat:4b32jmbgzraz5ec75vbyjmsu64

Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning [article]

Julius Berner, Markus Dablander, Philipp Grohs
2020 arXiv   pre-print
We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs).  ...  Successful numerical experiments are presented, which empirically confirm the functionality and efficiency of our proposed algorithm in the case of heat equations and Black-Scholes option pricing models  ...  Introduction Linear parabolic partial differential equations (PDEs) of the form ∂uγ ∂t = 1 2 Trace σ γ [σ γ ] * ∇ 2 x u γ + µ γ , ∇ x u γ , u γ (x, 0) = ϕ γ (x), (1) are referred to as Kolmogorov PDEs,  ... 
arXiv:2011.04602v1 fatcat:qjohbcbkvvc3din4mchcvue3vy

Chapter 14 Numerical dynamic programming in economics [chapter]

John Rust
1996 Handbook of Computational Economics  
with an approximate Bellman operator r, that is everywhere differentiable (unlike r which has kinks at certain points V E B). 4 In some cases parametric methods allow us to exploit certain types of prior  ...  Note: this is a preliminary and incomplete: do not quote without permission of authors. 3 In this vein section 5 discusses the desirability of solving a slightly perturbed version of Bellman's equation  ...  frequently reduces to a rather difficult parabolic partial differential equation (see Kushner, 1990 and Kushner and Dupuis, 1992) .' 5 For this reason, the complexity of solving continuous time MDP's  ... 
doi:10.1016/s1574-0021(96)01016-7 fatcat:qjxlxm44wvdb3dmnz4ems7bqou

Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions [article]

Yu Tong, Dong An, Nathan Wiebe, Lin Lin
2020 arXiv   pre-print
Besides solving linear systems, fast inversion also allows us to develop fast algorithms for computing matrix functions, such as the efficient preparation of Gibbs states.  ...  We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems.  ...  We provide a concrete example of using fast inversion to solve a translational invariant elliptic partial differential equation in Section 2.2.  ... 
arXiv:2008.13295v1 fatcat:y2cv3gomafahbngmvrsgsev6xm

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Linas Vepštas
2008 Numerical Algorithms  
The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4.  ...  As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function", to more general series.  ...  invert the formulas 6.1 or 6.2, solving for the value of n which is to be used in equation 5.1.  ... 
doi:10.1007/s11075-007-9153-8 fatcat:gdr5roie6je3botugztbhhidyi

Page 1323 of Mathematical Reviews Vol. 27, Issue Index [page]

Mathematical Reviews  
(English summary) 95a:65158 Horton, Graham Arthur (with Vandewalle, Stefan; Worley, P.) An algorithm with polylog parallel complexity for solving parabolic partial differential equations.  ...  (with Scherzer, Otmar; Yamamoto, Masahiro) Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data.  ... 

Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis [article]

Houman Owhadi, Clint Scovel
2017 arXiv   pre-print
We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information  ...  When the solution space is a Banach space B endowed with a quadratic norm ·, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of u∈ B, given partial measurements [ϕ_i,  ...  A (k),locπ(k,k−1) ) on Line 11 of Algorithm 6 are solved to accuracy |y − y app | X ≤ C −1 a H 3−k+kd/2 /k 2 (4)The equation w (1),loc = (A (1),loc ) −1 g (1),loc on Line 17 of Algorithm 6 is solved to  ... 
arXiv:1703.10761v2 fatcat:2qkxmnmwdrg35jcpaybamfg7eq

Page 1406 of Mathematical Reviews Vol. 27, Issue Index [page]

Mathematical Reviews  
(see 95e:65005) 65M55 — (with Horton, Graham Arthur; Worley, P.) An algorithm with polylog parallel complexity for solving parabolic partial differential equations. (English summary) SIAM J. Sci.  ...  (Summary) 95e:94017 94A55 (94A60) Vandewalle, Stefan (with Horton, Graham Arthur) Multicomputer-multigrid solution of parabolic partial differential equations.  ... 

Optimally sparse approximations of 3D functions by compactly supported shearlet frames [article]

Gitta Kutyniok, Jakob Lemvig, Wang-Q Lim
2012 arXiv   pre-print
We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L^2(R^3) with this particular shearlet structure.  ...  As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images.  ...  As it is only a matter of scaling we replace the right hand side of the last equation with 1 for simplicity.  ... 
arXiv:1109.5993v2 fatcat:n6fbnz4i3ngqdngal66edhpfky

Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames

Gitta Kutyniok, Jakob Lemvig, Wang-Q Lim
2012 SIAM Journal on Mathematical Analysis  
Clearly, this model provides us with two new smoothness parameters: β being a classical smoothness parameter and α being an anisotropic smoothness parameter; see Figure 1 .1 for an illustration.  ...  We introduce a pyramid-adapted, hybrid shearlet system for the 3D setting and construct frames for L 2 (R 3 ) with this particular shearlet structure.  ...  As it is only a matter of scaling, we replace the right-hand side of the last equation with 1 for simplicity.  ... 
doi:10.1137/110844726 fatcat:f4n3bktlsrfx3pp4u7asrcphqi

Solving parametric PDEs with neural networks: unfavorable structure vs. expressive power [article]

Mones Konstantin Raslan, Technische Universität Berlin
2021
The second part of this thesis deals with the concrete application of solving parametric partial differential equations (PDEs) by NNs.  ...  The focus of this part of the thesis lies on an examination of the expressive power of NNs for solutions of parametric PDEs.  ...  P.P was supported by a DFG Research Fellowship "Shearlet-based energy functionals for anisotropic phasefield methods". M.R. is supported by the Berlin Mathematical School.  ... 
doi:10.14279/depositonce-11332 fatcat:2hlsf5j5fnepzk4xyd6azjpg2i
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