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An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations
1995
SIAM Journal on Scientific Computing
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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
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(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
2000
SIAM Journal on Scientific Computing
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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]
2021
arXiv
pre-print
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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
1995
Journal of Computational and Applied Mathematics
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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]
2020
arXiv
pre-print
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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]
1996
Handbook of Computational Economics
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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]
2020
arXiv
pre-print
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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
2008
Numerical Algorithms
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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
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(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]
2017
arXiv
pre-print
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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
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(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]
2012
arXiv
pre-print
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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
2012
SIAM Journal on Mathematical Analysis
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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]
2021
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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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