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

1995
*
SIAM Journal on Scientific Computing
*

This paper describes

doi:10.1137/0916034
fatcat:66o4yawudfabzhxsx65pwdk6dm
*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. ...##
###
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

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. ...

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

2021
*
arXiv
*
pre-print

In this paper, we propose a

arXiv:2007.13125v2
fatcat:chry5yt2rfdczhtob5yl7xljeq
*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). ...##
###
Scalable algorithms for the solution of Navier's equations of elasticity

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*...

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

2020
*
arXiv
*
pre-print

We present a deep learning

arXiv:2011.04602v1
fatcat:qjohbcbkvvc3din4mchcvue3vy
*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, ...##
###
Chapter 14 Numerical dynamic programming in economics
[chapter]

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 ...

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

2020
*
arXiv
*
pre-print

Besides

arXiv:2008.13295v1
fatcat:y2cv3gomafahbngmvrsgsev6xm
*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. ...##
###
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

2008
*
Numerical Algorithms
*

The

doi:10.1007/s11075-007-9153-8
fatcat:gdr5roie6je3botugztbhhidyi
*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. ...##
###
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]

2017
*
arXiv
*
pre-print

We show how the discovery of robust scalable numerical solvers

arXiv:1703.10761v2
fatcat:2qkxmnmwdrg35jcpaybamfg7eq
*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 ...##
###
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]

2012
*
arXiv
*
pre-print

We introduce a pyramid-adapted, hybrid shearlet system

arXiv:1109.5993v2
fatcat:n6fbnz4i3ngqdngal66edhpfky
*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. ...##
###
Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames

2012
*
SIAM Journal on Mathematical Analysis
*

Clearly, this model provides us

doi:10.1137/110844726
fatcat:f4n3bktlsrfx3pp4u7asrcphqi
*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. ...##
###
Solving parametric PDEs with neural networks: unfavorable structure vs. expressive power
[article]

2021

The second part of this thesis deals

doi:10.14279/depositonce-11332
fatcat:2hlsf5j5fnepzk4xyd6azjpg2i
*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. ...
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