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Showing papers on "Sparse grid published in 2022"


Journal ArticleDOI
TL;DR: In this paper , the authors proposed a new extreme value moment method (EVMM) combining adaptive Kriging model and weighted approach based on sparse grid numerical integration (WA-SGNI) to estimate the statistical moments of the approximated EVF responses.

9 citations


Journal ArticleDOI
TL;DR: In this article , Tamellini et al. compare the efficiency and accuracy of different unbounded sparse grids (Gauss-Hermite, Gauss-Leja and Kronrod-Patterson) with Monte Carlo simulations.

5 citations


Journal ArticleDOI
TL;DR: In this article , a multi-fidelity Monte Carlo approach was proposed to address the high computational costs of the required gyrokinetic simulations and the large number of parameters render standard Monte Carlo techniques intractable.

5 citations


Journal ArticleDOI
TL;DR: In this paper , a sparse grid stochastic collocation method was developed to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity.
Abstract: <p style='text-indent:20px;'>In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface.</p>

4 citations


Journal ArticleDOI
TL;DR: In this paper , the authors proposed an iterated Crank-Nicolson (CN) based domain decomposition method for the termination of unbounded uniform finite-difference time-domain domains.
Abstract: In multi-dimension problems, huge sparse matrices must be calculated according to Crank-Nicolson (CN) procedure which results in degeneration of efficiency and accuracy. Based on the iterated CN procedure, domain decomposition method, an alternative scheme is proposed for the termination of unbounded uniform finite-difference time-domain domains. Meanwhile, absorbing boundary condition is proposed in the iterated CN procedure which is incorporated with the higher order concept. The proposed scheme employs the explicit scheme during the calculation rather than the implicit one. Thus, the calculation of matrices can be avoided. It shows the advantages especially in efficiency and accuracy compared with implicit schemes. Such conclusion can be further demonstrated through the numerical example. From results, it can be concluded that the proposed scheme shows efficiency improvement and accurate maintenance. Compared with other procedures, it also holds its considerable effectiveness in nonuniform domains. For comparison, it can maintain considerable performance compared with the others.

3 citations


Journal ArticleDOI
TL;DR: In this article , a parallel formulation of a WENO multiresolution scheme based on OpenMP has been proposed, which has the capability of accurately capturing the formation of shocks and the evolution of rarefaction waves.
Abstract: In the current work a seven-equation model of two-dimensional two-phase flow problems is analyzed by a parallel formulation of a WENO multiresolution scheme. The scheme adaptivity is obtained by a third order interpolating wavelet transform associated to a threshold operator. In this way, a sparse representation of the vector solution is obtained at each time step. The evolution in time is performed by a third order TVD Runge-Kutta scheme. For the spatial integration on the sparse grid, the Lax-Friedrich flux splitting scheme is considered in which the flux derivatives are approximated by the standard fifth-order WENO scheme. The parallel formulation of the code is based on OpenMP, which is crucial for the computation of long term simulations with shorter computational times. The considered adaptive parallel WENO scheme has the capability of accurately capturing the formation of shocks and the evolution of rarefaction waves, as evinced by the presented numerical simulations.

2 citations


Posted ContentDOI
17 Mar 2022
TL;DR: The Sparse Grids Matlab Kit as discussed by the authors is a collection of Matlab functions for high-dimensional interpolation and quadrature, based on the combination technique form of sparse grids.
Abstract: The Sparse Grids Matlab Kit is a collection of Matlab functions for high-dimensional interpolation and quadrature, based on the combination technique form of sparse grids. It is lightweight, high-level and easy to use, good for quick prototyping and teaching. However, it has some features that allow its use also in realistic applications: in particular, the Sparse Grids Matlab Kit is somehow geared towards Uncertainty Quantification, but it is flexible enough for other purposes. The goal of this paper is to provide an overview of the data structure and of mathematical aspects forming the backbone of the software, as well as to compare it with similar software available in literature.

2 citations


Journal ArticleDOI
TL;DR: In this article , the authors explore numerical techniques for improving the simulation of exposures, aiming to decimate the number of portfolio evaluations, particularly for large portfolios involving multiple, correlated risk factors.

2 citations



Journal ArticleDOI
TL;DR: The uniform sparse Fast Fourier Transform (usFFT) as discussed by the authors is an adaptive algorithm for the solution of elliptic partial differential equations with random coefficients, which tries to detect the most important frequencies in a given search domain and therefore adaptively generate a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function.
Abstract: Abstract We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously for multiple fixed spatial nodes, e.g., nodes of a finite element mesh. The key idea of joining the detected frequency sets in each dimension increment results in a Fourier approximation space, which fits uniformly for all these spatial nodes. This strategy allows for a faster and more efficient computation due to a significantly smaller amount of samples needed, than just using other algorithms, e.g., the sFFT for each spatial node separately. We test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems.

1 citations



Journal ArticleDOI
TL;DR: In this paper , a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem is presented, where the authors consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties.
Abstract: Abstract This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, and kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF). The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier–Stokes (RANS) multi-grid solver $$\chi$$ χ navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementation of both methods, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBF seems to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.


Journal ArticleDOI
TL;DR: In this paper , a numerical method for computing the solutions of high-dimensional Markovian backward stochastic difference equations (BS$\Delta$Es), in which they apply a sparse grid quadrature/interpolation formula, was proposed.
Abstract: We propose a numerical method for computing the solutions of high-dimensional Markovian backward stochastic difference equations (BS$\Delta$Es), in which we apply a sparse grid quadrature/interpolation formula. This allows us to calculate the approximated solution accurately and efficiently, which we justify through a rigorous error analysis. Finally, we use a simple numerical experiment to demonstrate the performance of our method.

Posted ContentDOI
09 Nov 2022
TL;DR: In this article , the sparse grids stochastic collocation method is applied to perform uncertainty quantification for structural engineering in the scenario described above, where the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects.
Abstract: When dealing with timber structures, the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects. In this work we apply the sparse grids stochastic collocation method to perform uncertainty quantification for structural engineering in the scenario described above. Sparse grids have been developed by the mathematical community in the last decades and their theoretical background has been rigorously and extensively studied. The document proposes a brief practice-oriented introduction with minimal theoretical background, provides detailed instructions for the use of the already implemented Sparse Grid Matlab kit (freely available on-line) and discusses two numerical examples inspired from timber engineering problems that highlight how sparse grids exhibit superior performances compared to the plain Monte Carlo method. The Sparse Grid Matlab kit requires only a few lines of code to be interfaced with any numerical solver for mechanical problems (in this work we used an isogeometric collocation method) and provides outputs that can be easily interpreted and used in the engineering practice.

Journal ArticleDOI
TL;DR: In this paper , a new sparse grid combination technique was developed to reduce the computational cost of functionals. But it is hard to obtain a concrete error splitting model for complicated approximations.
Abstract: Functionals related to a solution of a problem, usually modelled by partial differential equations, can be important quantities used to capture features of the problem. For high dimensional problems the computational cost of the functionals can be large since the numerical solution of a high dimensional partial differential equation is usually expensive to compute. We develop a new sparse grid combination technique to reduce the computational cost of such functionals. Our method is based on error splitting models of the functionals. However, it is hard to obtain a concrete error splitting model for complicated approximations. We show the connection between the decay of the surpluses and the error splitting models. By using the connection, we can also apply our combination technique to functionals when we only know their computed surpluses. Numerical experiments are provided to illustrate our idea and test the performance of our method. References A. J. Brizard and T. S. Hahm. Foundations of nonlinear gyrokinetic theory. In: Rev. Mod. Phys. 79.2 (2007), pp. 421–468. doi: 10.1103/RevModPhys.79.421 H.-J. Bungartz and M. Griebel. Sparse grids. In: Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182 T. Gerstner and M. Griebel. Numerical integration using sparse grids. In: Numer. Algor. 18 (1998), pp. 209–232. doi: 10.1023/A:1019129717644. M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In: Iterative methods in linear algebra: Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, 1991. Ed. by P. de Groen and R. Beauwens. North-Holland, Amsterdam, 1992, pp. 263–281. url: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.3530 B. Harding. Fault tolerant computation of hyperbolic partial differential equations with the sparse grid combination technique. PhD thesis. The Australian National University, 2016. url: https://openresearch-repository.anu.edu.au/bitstream/1885/101226/1/Harding%20Thesis%202016.pdf M. Hegland. Adaptive sparse grids. In: Proceedings of the 10th Computational Techniques and Applications Conference CTAC-2001. Ed. by K. Burrage and R. B. Sidje. Vol. 44. 2003, pp. C335–C353. doi: 10.21914/anziamj.v44i0.685 Gene Development Team; F. Jenko et al. The Gyrokinetic Plasma Turbulence Code Gene: User Manual. 2013. url: http://genecode.org/

Posted ContentDOI
22 Aug 2022
TL;DR: In this paper , a probabilistic wirebasket-based coarse grid for a two-level domain decomposition solver is devised in three dimensions, which provides an efficient mechanism for global error propagation and thus improves the convergence.
Abstract: Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for instance, arising in linear elasticity, the existing two-level domain decomposition solvers with the vertex-based coarse grid show poor numerical and parallel scalabilities. Therefore, new algorithms with a better resolved coarse grid are needed. The probabilistic wirebasket-based coarse grid for a two-level solver is devised in three dimensions. This enriched coarse grid provides an efficient mechanism for global error propagation and thus improves the convergence. This development enhances the scalability of the two-level solver in handling stochastic PDEs in three dimensions. Numerical and parallel scalabilities of this algorithm are studied using MPI and PETSc libraries on high-performance computing (HPC) systems. Implementational challenges of the intrusive spectral stochastic finite element methods (SSFEM) are addressed by coupling domain decomposition solvers with FEniCS general purpose finite element package. This work generalizes the applications of intrusive SSFEM to tackle a variety of stochastic PDEs and emphasize the usefulness of the domain decomposition-based solvers and HPC for uncertainty quantification.



Book ChapterDOI
01 Jan 2022
TL;DR: In this paper , a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations is used to model the short-term transient dynamics of the gas flow.
Abstract: In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.

Journal ArticleDOI
TL;DR: In this article, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation.

Journal ArticleDOI
TL;DR: In this article , a generalized sparse grid is proposed for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems, which reduces the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy.
Abstract: Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.

Posted ContentDOI
12 Apr 2022
TL;DR: In this article , an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo-ve expansion is presented.
Abstract: We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo\`eve expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Lo\`eve coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the rate of the spatial discretization as asymptotic convergence rate with respect to the computational cost.

Posted ContentDOI
21 Feb 2022
TL;DR: The dimension-adaptive sparse grid quadrature (DASG) as discussed by the authors is a generalization of the classical sparse grid method, which refines different dimensions according to their importance.
Abstract: Stochastic optimisation problems minimise expectations of random cost functions. We use 'optimise then discretise' method to solve stochastic optimisation. In our approach, accurate quadrature methods are required to calculate the objective, gradient or Hessian which are in fact integrals. We apply the dimension-adaptive sparse grid quadrature to approximate these integrals when the problem is high dimensional. Dimension-adaptive sparse grid quadrature shows high accuracy and efficiency in computing an integral with a smooth integrand. It is a kind of generalisation of the classical sparse grid method, which refines different dimensions according to their importance. We show that the dimension-adaptive sparse grid quadrature has better performance in the optimise then discretise' method than the 'discretise then optimise' method.

Posted ContentDOI
24 Nov 2022
TL;DR: In this paper , the Galerkin method was combined with cubic spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping for European-style options on several underlying assets.
Abstract: The paper focuses on pricing European-style options on several underlying assets under the Black-Scholes model represented by a nonstationary partial differential equation. The proposed method combines the Galerkin method with $L^2$-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping. To this end, we construct an orthogonal cubic spline wavelet basis on the interval satisfying homogeneous Dirichlet boundary conditions and design a wavelet basis on the unit cube using the sparse tensor product. The method brings the following advantages. First, the number of basis functions is significantly smaller than for the full grid, which makes it possible to overcome the so-called curse of dimensionality. Second, some matrices involved in the computation are identity matrices, which significantly simplifies and streamlines the algorithm, especially in higher dimensions. Further, we prove that discretization matrices have uniformly bounded condition numbers, even without preconditioning, and that the condition numbers do not depend on the dimension of the problem. Due to the use of cubic spline wavelets, the method is higher-order convergent. Numerical experiments are presented for options on the geometric average.

Posted ContentDOI
05 Jan 2022
TL;DR: In this paper , the authors established sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs.
Abstract: We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphy-based argument allows a unified, ``differentiation-free'' proof of sparsity (expressed in terms of $\ell^p$-summability or weighted $\ell^2$-summability) of sequences of Wiener-Hermite coefficients in polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding analyticity and sparsity results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various \emph{constructive} high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of Hermite-Smolyak anisotropic sparse-grid interpolation and quadrature in both forward and inverse computational uncertainty quantification.

Posted ContentDOI
02 Jun 2022
TL;DR: In this paper , a compressive Fourier collocation (CFC) method was proposed to approximate the Fourier coefficients of a high-dimensional partial differential equation (PDE) with periodic boundary conditions.
Abstract: High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

Posted ContentDOI
02 Nov 2022
TL;DR: Huang et al. as discussed by the authors proposed an adaptive sparse grid discontinuous Galerkin (aSG-DG) method for computing high dimensional partial differential equations (PDEs) and its software implementation.
Abstract: This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG) method for computing high dimensional partial differential equations (PDEs) and its software implementation. The C\texttt{++} software package called AdaM-DG, implementing the aSG-DG method, is available on Github at \url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package is capable of treating a large class of high dimensional linear and nonlinear PDEs. We review the essential components of the algorithm and the functionality of the software, including the multiwavelets used, assembling of bilinear operators, fast matrix-vector product for data with hierarchical structures. We further demonstrate the performance of the package by reporting numerical error and CPU cost for several benchmark test, including linear transport equations, wave equations and Hamilton-Jacobi equations.

Book ChapterDOI
01 Jan 2022

Posted ContentDOI
25 Feb 2022
TL;DR: In this article , a sparse grid version of Gaussian convolution was introduced, where substantially fewer grid points are required, and it was shown that the sparse grid convolution can achieve a saturation rate of O(n−d-1}2−2n−1−n−n) where n−d is the scale of the Gaussian kernel.
Abstract: We consider the problem of approximating $[0,1]^{d}$-periodic functions by convolution with a scaled Gaussian kernel. We start by establishing convergence rates to functions from periodic Sobolev spaces and we show that the saturation rate is $O(h^{2}),$ where $h$ is the scale of the Gaussian kernel. Taken from a discrete point of view, this result can be interpreted as the accuracy that can be achieved on the uniform grid with spacing $h.$ In the discrete setting, the curse of dimensionality would place severe restrictions on the computation of the approximation. For instance, a spacing of $2^{-n}$ would provide an approximation converging at a rate of $O(2^{-2n})$ but would require $(2^{n}+1)^{d}$ grid points. To overcome this we introduce a sparse grid version of Gaussian convolution approximation, where substantially fewer grid points are required, and show that the sparse grid version delivers a saturation rate of $O(n^{d-1}2^{-2n}).$ This rate is in line with what one would expect in the sparse grid setting (where the full grid error only deteriorates by a factor of order $n^{d-1}$) however the analysis that leads to the result is novel in that it draws on results from the theory of special functions and key observations regarding the form of certain weighted geometric sums.