A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions

Pflaum, Christoph; Hartmann, R.

doi:10.1137/15m101508x

Cited by 5 publications

(30 citation statements)

References 13 publications

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“…Together with | max(l, l ′ )| 1 > N , we claim that there are two different dimension directions m 1 = m 2 so that l m1 = l ′ m1 and l m2 = l ′ m2 . Due to the orthonormal property of the multiwavelet bases, we follow the argument for the case of the prewavelet bases in [22] and prove that (3.1) holds.…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 92%

“…For the completeness of the paper, we choose to provide the proof. First, in [22], it was shown that if |l| 1 ≤ N , |l ′ | 1 ≤ N , and |l max | 1 > N , then…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%

“…We bound the right-hand side of above equation as follows. First, we need the inequality proved in [22] that, for any l with |l| 1 ≤ N ,…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%

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A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Guo¹,

Cheng²

2016

SIAM J. Sci. Comput.

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Abstract. In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for hyperbolic problems and is proven to be L 2 stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is verified by numerical tests in up to four dimensions.Key words. discontinuous Galerkin methods; sparse grid; high-dimensional transport equations; Vlasov equation; Boltzmann equation.1. Introduction. In this paper, we develop a sparse grid DG method for high-dimensional transport equations. High-dimensional transport problems are ubiquitous in science and engineering, and most evidently in kinetic simulations where it is necessary to track the evolution of probability density functions of particles. Deterministic kinetic simulations are very demanding due to the large computational and storage cost. To make the schemes more attractive comparing with the alternative probabilistic methods, an appealing approach is to explore the sparse grid techniques [6,15] with the aim of breaking the curse of dimensionality [4]. In the context of wavelets or sparse grid methods for kinetic transport equations, we mention the work of using wavelet-MRA methods for Vlasov equations [5], the combination technique for linear gyrokinetics [19], sparse adaptive finite element method [25], sparse discrete ordinates method [16] and sparse tensor spherical harmonics [17] for radiative transfer, among many others. This paper focuses on the DG method [13], which is a class of finite element methods using discontinuous approximation space for the numerical solution and the test functions. The RKDG scheme [14] developed in a series of papers for hyperbolic equations became very popular due to its provable convergence, excellent conservation properties and accommodation for adaptivity and parallel implementations. Recent years have seen great growth in the interest of applying DG methods to kinetic systems (see for example [3,18,20,9, 10]) because of the conservation properties and long time performance of the resulting simulations. However, the DG method is still deemed too costly in a realistic setting, often requiring more degrees of freedom than other high order numerical calculations.Recently, we developed a sparse grid DG method for high-dimensional elliptic problems [24]. A sparse DG finite element space has been constructed, reducing the degrees of freedom from the standardwhere h is the uniform mesh size in each dimension. The resulting scheme retains main properties of standard DG methods while making the computational cost tangebile for high-dimensional simulations. This motivates the current work for the transport equations, and we use kinetic problems as a...

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Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 92%

“…For the completeness of the paper, we choose to provide the proof. First, in [22], it was shown that if |l| 1 ≤ N , |l ′ | 1 ≤ N , and |l max | 1 > N , then…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%

See 1 more Smart Citation

A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Guo¹,

Cheng²

2016

SIAM J. Sci. Comput.

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“…O(N D ). So, they become computationally very expensive for D > 2 [5,9,10]. However, for highly sparse matrices, better scaling can be achieved by further exploitation of the sparsity and by imposing locality to reduce the number of required operations.…”

Section: Introductionmentioning

confidence: 99%

“…However, for highly sparse matrices, better scaling can be achieved by further exploitation of the sparsity and by imposing locality to reduce the number of required operations. For example, in the sparse grid method, a computational cost of O(N(log N) D−1 ) can be achieved [9,11,12]. The second class is meshfree methods, where the unknown quantities are approximated by high order basis sets without the need for the mesh [1,7].…”

Section: Introductionmentioning

confidence: 99%

Efficient Mapping of High Order Basis Sets for Unbounded Domains

Mumtaz¹,

Alharbi²

2018

CiCP

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Many physical problems involve unbounded domains where the physical quantities vanish at infinities. Numerically, this has been handled using different techniques such as domain truncation, approximations using infinitely extended and vanishing basis sets, and mapping bounded basis sets using some coordinate transformations. Each technique has its own advantages and disadvantages. Yet, approximating simultaneously and efficiently a wide range of decaying rates has persisted as major challenge. Also, coordinate transformation, if not carefully implemented, can result in non-orthogonal mapped basis sets. In this work, we revisited this issue with an emphasize on designing appropriate transformations using sine series as basis set. The transformations maintain both the orthogonality and the efficiency. Furthermore, using simple basis set (sine function) help avoid the expensive numerical integrations. In the calculations, four types of physically recurring decaying behaviors are considered, which are: non-oscillating and oscillating exponential decays, and non-oscillating and oscillating algebraic decays. The results and the analyses show that properly designed high-order mapped basis sets can be efficient tools to handle challenging physical problems on unbounded domains. Decay rate ranges as large of 6 orders of magnitudes can be approximated efficiently and concurrently.

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Discretization of PDEs with variable coefficients using locally adaptive sparse grids

Scherner‐Grießhammer,

Pflaum

2023

Proc Appl Math and Mech

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Elliptic partial differential equations with variable coefficients can be discretized on sparse grids. With prewavelets being L2‐orthogonal, one can apply the Ritz‐Galerkin discretization to obtain a linear equation system with unknowns. However, for several applications like partial differential equations with corner singularities or the high‐dimensional Schrödinger equation, locally adaptive grids are needed to obtain optimal convergence. Therefore, we introduce a new kind of locally adaptive sparse grid and a corresponding algorithm that allows solving the resulting finite element discretization equation with optimal complexity. These grids are constructed by local tensor product grids to generate adaptivity but still maintain a local unidirectional approach. First simulation results of a two‐dimensional Helmholtz problem are presented.

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A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions

Cited by 5 publications

References 13 publications

A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Efficient Mapping of High Order Basis Sets for Unbounded Domains

Discretization of PDEs with variable coefficients using locally adaptive sparse grids

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