Adaptive Solution of Partial Differential Equations in Multiwavelet Bases

Alpert, Bradley K.; Beylkin, Gregory; Gines, D.; Vozovoi, L.

doi:10.1006/jcph.2002.7160

Cited by 223 publications

(269 citation statements)

References 20 publications

(52 reference statements)

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“…Right: The parallel speedup of one iteration of madness-hfb, for solving the DFT problem for 1,640 3D quasiparticle wavefunctions with over 4.4 billion equations and unknowns; this simulation was performed within a box with a spatial dimension of 120 fermis, using 8 multiwavelets, up to level 8+ of refinement, and with a relative precision of 10 −6 . the multiwavelets consist of smooth, singular, and discontinuous functions with spatial locality (compact support), they are well suited for localized approximation of weak singularities and discontinuities or regions of high curvature [90,91,92]. Gibbs effects are also reduced.…”

Section: Multiresolution 3d Dft Frameworksupporting

confidence: 38%

Computational nuclear quantum many-body problem: The UNEDF project

Bogner

Bulgac

Carlson

et al. 2013

Computer Physics Communications

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The UNEDF project was a large-scale collaborative effort that applied high-performance computing to the nuclear quantum many-body problem. UNEDF demonstrated that close associations among nuclear physicists, mathematicians, and computer scientists can lead to novel physics outcomes built on algorithmic innovations and computational developments. This review showcases a wide range of UNEDF science results to illustrate this interplay.

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Section: Multiresolution 3d Dft Frameworksupporting

confidence: 38%

Computational nuclear quantum many-body problem: The UNEDF project

Bogner

Bulgac

Carlson

et al. 2013

Computer Physics Communications

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“…In constructing derivative operators we incorporate boundary conditions into the derivative matrix. In the case of discontinuous interface conditions, these conditions are also incorporated into the derivative matrix in a way similar to [11]. We also use the spectral projector to remove spurious large eigenvalues and corresponding eigenspaces from the derivative operators, thus further reducing their norm.…”

Section: Introductionsupporting

confidence: 39%

“…We follow the method in [11] (essentially the tau method, see e.g. [25,26]) and extend the technique to a nonorthogonal basis since the approximate PSWFs {Ψ i (x)} M i=1 are not orthonormal.…”

Section: Derivative Matrices With Boundary and Interface Conditionsmentioning

confidence: 48%

Wave propagation using bases for bandlimited functions

Beylkin

Sandberg

2005

Wave Motion

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We develop a two-dimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a first-order system, we evolve the equation in time using the matrix exponential. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of one-dimensional operators. We also use a partitioned low-rank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and computational time when compared with a fourth-order finite difference scheme in space and an explicit fourth-order Runge-Kutta solver in time.

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“…In practice, these coefficients are efficiently computed using the quadrature mirror filter coefficients [2,3]. The multiwavelets ψ have been developed by Alpert [1] and are also explained in [17].…”

supporting

confidence: 43%

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

Vuik

Ryan

2016

SIAM J. Sci. Comput.

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Abstract. In Vuik and Ryan [J. Comput. Phys., 270 (2014), pp. 138-160] we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has an impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in the detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond the others [J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, 1977]. We provide an algorithm that can be applied to various troubled-cell indication variables. With this technique, the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically.

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Adaptive Solution of Partial Differential Equations in Multiwavelet Bases

Cited by 223 publications

References 20 publications

Computational nuclear quantum many-body problem: The UNEDF project

Computational nuclear quantum many-body problem: The UNEDF project

Wave propagation using bases for bandlimited functions

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

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