Numerical solution of parabolic equations in high dimensions

Petersdorff, Tobias von; Schwab, Christoph

doi:10.1051/m2an:2004005

Cited by 105 publications

(91 citation statements)

References 14 publications

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“…Remark 1.2. Although we consider in this paper only elliptic PDEs, our results on s * -compressibility and s * -computability of the stiffness matrix also have immediate applications to the efficient solution of parabolic problems by implicit timestepping procedures as discussed, e.g., in [vPS04]. Remark 1.3.…”

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 87%

Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

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Abstract. With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D ⊂ R d , the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in L 2 . It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best N -term approximation from tensor product wavelet bases. When the function is the solution of some well-posed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, non-separable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.

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Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 87%

Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

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“…Specifically, given the numerical solution on the entire interval (0, T ), we use the local error estimators η cG,m and η dG,m in (18) and (29), respectively, to identify those time-steps on which large errors occur. More precisely, an element I m is marked for refinement if η cG,m > θ max i η cG,i , respectively η dG,m > θ max i η dG,i .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For detailed implementational techniques for hp-version dG time-stepping methods we refer to [24,29]. We discretize our model problems using a high-order finite element method in space so that the spatial errors are dominated by the errors resulting from the cG and dG time discretizations.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Furthermore, it has been proved in [20,21] that the combination of hp-version time-stepping with suitable wavelet spatial discretizations results in log-linear complexity solution algorithms for nonlocal evolution processes involving pseudo-differential operators. In addition, the discretization of high-dimensional parabolic problems, using sparse grids in space, has been analyzed in [29].…”

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confidence: 99%

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A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

Schötzau

Wihler

2010

Numer. Math.

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The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.

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“…This argument can be made rigorous for linear parabolic equations upon using Theorem 4.5 of [26] and carefully approximating the solution on the first time interval. To test this heuristic argument for parabolic variational inequalities, we take β = 1.5 and report the results in Table 7.…”

Section: D American Optionmentioning

confidence: 99%

A posteriorierror analysis for parabolic variational inequalities

et al. 2007

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Abstract.Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ R d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L 2 (0, T ; H 1 (Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d = 1, 2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.Mathematics Subject Classification. 58E35, 65N15, 65N30.

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Numerical solution of parabolic equations in high dimensions

Cited by 105 publications

References 14 publications

Adaptive wavelet algorithms for elliptic PDE's on product domains

Adaptive wavelet algorithms for elliptic PDE's on product domains

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

A posteriorierror analysis for parabolic variational inequalities

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