Showing papers on "Sparse grid published in 2021"

Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes

Abstract: We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1] , [2] , [3] , we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4] . This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.

Application of sparse grid combination techniques to low temperature plasmas particle-in-cell simulations. I. Capacitively coupled radio frequency discharges

Abstract: The use of a particle-in-cell (PIC) algorithm with an explicit scheme to model low temperature plasmas is challenging due to computational time constrains related to resolving both the electron Debye length in space and the inverse of a fraction of the plasma frequency in time. One recent publication [Ricketson and Cerfon, Plasma Phys. Control. Fusion 59, 024002 (2017)] has demonstrated the interest of using a sparse grid combination technique to accelerate the explicit PIC model. Simplest plasma conditions were considered. This paper is the demonstration of the capability and the effectiveness of the sparse grid combination technique embedded in the PIC algorithm (hereafter called “sparse PIC”) to self-consistently model capacitively coupled radio frequency discharges. For two-dimensional calculations, the sparse PIC approach is shown to accurately reproduce the plasma profiles and the energy distribution functions compared to the standard PIC model. The plasma parameters obtained by these two numerical methods differ by less than 5%, while a speed up in the executable time between 2 and 5 is obtained depending on the setup.

Turbulence suppression by energetic particles: a sensitivity-driven dimension-adaptive sparse grid framework for discharge optimization

Abstract: A newly developed sensitivity-driven approach is employed to study the role of energetic particles in suppressing turbulence-inducing micro-instabilities for a set of realistic JET-like cases with NBI deuterium and ICRH $^3$He fast ions. First, the efficiency of the sensitivity-driven approach is showcased for scans in a $21$-dimensional parameter space, for which only $250$ simulations are necessary. The same scan performed with traditional Cartesian grids with only two points in each of the $21$ dimensions would require $2^{21} = 2,097,152$ simulations. Then, a $14$-dimensional parameter subspace is considered, using the sensitivity-driven approach to find an approximation of the parameter-to-growth rate map averaged over nine bi-normal wave-numbers, indicating pathways towards turbulence suppression. The respective turbulent fluxes, obtained via nonlinear simulations for the optimized set of parameters, are reduced by more than two order of magnitude compared to the reference results.

An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

Abstract: In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loeve decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.

A Graph Learning Algorithm Based On Gaussian Markov Random Fields And Minimax Concave Penalty

Abstract: This paper presents a graph learning framework to produce sparse and accurate graphs from network data. While our formulation is inspired by the graphical lasso, a key difference is the use of a nonconvex alternative of the l 1 norm as well as a quadratic term to ensure overall convexity. Specifically, the weakly-convex minimax concave penalty (MCP) is used, which is given by subtracting the Huber function from the l 1 norm, inducing a less-biased sparse solution than l 1 . In our framework, the graph Laplacian is represented by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on the Moreau decomposition, the problem can be solved by the primal-dual splitting method. An admissible choice of parameters for provable convergence is presented. Numerical examples show that the proposed method significantly outperforms its l 1 -based counterpart for sparse grid graphs.

Uncertainty Quantification Using Parameter Space Partitioning

Abstract: A new method is presented for high-dimensional variability analysis based on two main concepts, namely, node tearing for parameter partitioning and sparse grid interpolation. Node tearing is used to localize the parameters and, thus, reducing the number of stochastic parameters within the subcircuits and sparse grids reduce the required number of samples for a targeted accuracy. MC analysis of the overall circuit is carried out using interface equations of a much smaller dimension than the original circuit equations. Pertinent computational results are presented to validate the efficiency and accuracy of the proposed method.

Application of sparse grid combination techniques to low temperature plasmas Particle-In-Cell simulations. II. Electron drift instability in a Hall thruster

Abstract: Three-dimensional simulations of partially magnetized plasma are real challenges that actually limit the understanding of the discharge operations such as the role of kinetic instabilities using explicit Particle-In-Cell (PIC) schemes. The transition to high performance computing cannot overcome all the limits inherent to very high plasma densities and thin mesh sizes employed to avoid numerical heating. We have applied a recent method proposed in the literature [L. F. Ricketson and A. J. Cerfon, Plasma Phys. Controlled Fusion 59, 024002 (2017)] to model low temperature plasmas. This new approach, namely, the sparse grid combination technique, offers a gain in computational time by solving the problem on a reduced number of grid cells, hence allowing also the reduction of the total number of macroparticles in the system. We have modeled the example of the two-dimensional electron drift instability, which was extensively studied in the literature to explain the anomalous electron transport in a Hall thruster. Comparisons between standard and sparse grid PIC methods show an encouraging gain in the computational time with an acceptable level of error. This method offers a unique opportunity for future three-dimensional simulations of instabilities in partially magnetized low temperature plasmas.

A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

Abstract: 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-Loeve 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.

A Generalized Spatially Adaptive Sparse Grid Combination Technique with Dimension-wise Refinement

Abstract: Today, high-dimensional calculations can be found in almost all scientific disciplines. The application of machine learning and uncertainty quantification methods are common examples where high-dim...

Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters

Abstract: The paper is concerned with the issues of modeling dynamic systems with interval parameters In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that interpolates the solution of the problem with a given accuracy The main problem of applying the algorithm is related to the curse of dimension, ie, exponential complexity relative to the number of interval uncertainties in parameters The main objective of this work is to apply the previously proposed adaptive interpolation algorithm to dynamic systems with a large number of interval parameters In order to reduce the computational complexity of the algorithm, the authors propose using adaptive sparse grids This article introduces a novelty approach of applying sparse grids to problems with interval uncertainties The efficiency of the proposed approach has been demonstrated on representative interval problems of nonlinear dynamics and computational materials science

Pricing American Options under High-Dimensional Models with Recursive Adaptive Sparse Expectations

Abstract: We introduce a novel numerical framework for pricing American options in high dimensions. Our scheme manages to alleviate the problem of dimension scaling through the use of adaptive sparse grids. We approximate the value function with a low number of points and recursively apply fast approximations of the expectation operator from an exercise period to the previous period. Given that available option databases gather several thousands of prices, there is a clear need for fast approaches in empirical work. Our method processes an entire cross section of options in a single execution and offers an immediate solution to the estimation of hedging coefficients through finite differences. It thereby brings valuable advantages over Monte Carlo simulations, which are usually considered to be the tool of choice in high dimensions, and satisfies the need for fast computation in empirical work with current databases containing thousands of prices. We benchmark our algorithm under the canonical model of Black and Scholes and the stochastic volatility model of Heston, the latter in the presence of discrete dividends. We illustrate the massive improvement of complexity scaling over dense grids with a basket option study including up to eight underlying assets. We show how the high degree of parallelism of our scheme makes it suitable for deployment on massively parallel computing units to scale to higher dimensions or further speed up the solution process.

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

Abstract: By combining a certain approximation property in the spatial domain, and weighted 𝓁2 -summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L 1 (ℝ ∞ , V , γ ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

A Novel Method for Intrusive Stochastic Estimation of Geometric Tolerance Effects in Finite Element Electromagnetic Analysis

Abstract: The robust design of microwave, millimeter-wave, or Terahertz components and structures incorporating manufacturing process tolerances would enhance the fabrication yield and thereby reduce the overall production cost. Based on the spectral stochastic finite element method (SSFEM) for material variations, a novel geometrical SSFEM (GSSFEM) is proposed to analyze dimensional uncertainty in 3-D microwave models. The Jacobian for Piola mapping is represented as a function of stochastic variables to capture mesh level uncertainties. Polynomial chaos expansion is used to approximate the electric field as a random process. The technique is validated by applying it to waveguide problems with uncertainty in geometric dimensions. GSSFEM results are compared with analytical formulations of the transmission coefficient for a physical insight and further with the Monte Carlo simulations for quantitative evaluation. The results using the proposed approach are in excellent agreement with conventional methods and are found to be faster than sparse grid stochastic collocation, while the scaling of the computational requirements with the number of degrees of freedom and stochastic dimension is as good as this efficient scheme. Furthermore, this approach has been employed to analyze uncertainty in multiple geometric parameters to compare their sensitivity. The proposed GSSFEM can be employed for analyzing the impact of fabrication tolerance of passive waveguide components at microwave frequencies and beyond.

A stochastic collocation approach for parabolic PDEs with random domain deformations.

Abstract: In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. The stochastic moments of the QoI are computed by employing a collocation method in conjunction with an isotropic Smolyak sparse grid. Theoretical sub-exponential convergence rates as a function to the number of collocation interpolation knots are derived. Numerical experiments are performed and they confirm the theoretical error estimates.

Nested Sparse Successive Galerkin Approximation for Nonlinear Optimal Control Problems

Abstract: Solving the Hamilton-Jacobi-Bellman (HJB) equation for nonlinear optimal control problems usually suffers from the so-called curse of dimensionality. In this letter, a nested sparse successive Galerkin method is presented for HJB equations, and the computational cost only grows polynomially with the dimension. Based on successive approximation techniques, the nonlinear HJB partial differential equation (PDE) is transformed into a sequence of linear PDEs. Then the nested sparse grid methods are employed to solve the resultant linear PDEs. The designed method is sparse in two aspects. Firstly, the solution of the linear PDE is constructed based on sparse combinations of nested basis functions. Secondly, the multi-dimensional integrals in the Galerkin method are efficiently calculated using nested sparse grid quadrature rules. Once the successive approximation process is finished, the optimal controller can be analytically given based on the sparse basis functions and coefficients. Numerical results also demonstrate the accuracy and efficiency of the designed nested sparse successive Galerkin method.

Data-driven low-fidelity models for multi-fidelity Monte Carlo sampling in plasma micro-turbulence analysis

Abstract: The linear micro-instabilities driving turbulent transport in magnetized fusion plasmas (as well as the respective nonlinear saturation mechanisms) are known to be sensitive with respect to various physical parameters characterizing the background plasma and the magnetic equilibrium. Therefore, uncertainty quantification is essential for achieving predictive numerical simulations of plasma turbulence. However, the high computational costs of the required gyrokinetic simulations and the large number of parameters render standard Monte Carlo techniques intractable. To address this problem, we propose a multi-fidelity Monte Carlo approach in which we employ data-driven low-fidelity models that exploit the structure of the underlying problem such as low intrinsic dimension and anisotropic coupling of the stochastic inputs. The low-fidelity models are efficiently constructed via sensitivity-driven dimension-adaptive sparse grid interpolation using both the full set of uncertain inputs and subsets comprising only selected, important parameters. We illustrate the power of this method by applying it to two plasma turbulence problems with up to $14$ stochastic parameters, demonstrating that it is up to four orders of magnitude more efficient than standard Monte Carlo methods measured in single-core performance, which translates into a runtime reduction from around eight days to one hour on 240 cores on parallel machines.

Error Analysis of Sound Source Directivity Interpolation Based on Spherical Harmonics

Abstract: Precise measurement of the sound source directivity not only requires special equipment, but also is time-consuming. Alternatively, one can reduce the number of measurement points and apply spatial interpolation to retrieve a high-resolution approximation of directivity function. This paper discusses the interpolation error for different algorithms with emphasis on the one based on spherical harmonics. The analysis is performed on raw directivity data for two loudspeaker systems. The directivity was measured using sampling schemes of different densities and point distributions (equiangular and equiareal). Then, the results were interpolated and compared with these obtained on the standard 5° regular grid. The application of the spherical harmonic approximation to sparse measurement data yields a mean error of less than 1 dB with the number of measurement points being reduced by 89%. The impact of the sparse grid type on the retrieval error is also discussed. The presented results facilitate optimal sampling grid choice for low-resolution directivity measurements.

A hybrid collocation-perturbation approach for PDEs with random domains.

Abstract: Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of N random variables. In many application, quantify the uncertainty propagated to a quantity of interest (QoI) is an important problem. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method and added as a correction term to the large variation sparse grid component. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem making it suitable for large dimensional problems. The computational cost of the correction term increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.

Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties

Abstract: We consider the theoretical aspects of generalization of the adaptive interpolation algorithm to the case of a large number of interval uncertainties with the use of sparse grids. The classical adaptive interpolation algorithm essentially constructs an adaptive hierarchical partition of the uncertainty domain into subdomains each of which corresponds to a polynomial function of some degree interpolating the dependence of the solution of the problem on the point values of interval parameters with a given accuracy. The main disadvantage of the algorithm is its exponential dependence on the number of interval parameters. Already in the presence of five to six interval uncertainties, the algorithm becomes practically inapplicable. In this regard, it is proposed to use interpolation on adaptive sparse meshes instead of interpolation on a regular mesh; in some cases, this permits considerably expanding the scope of the algorithm. An estimate is produced for the global error of the algorithm on sparse meshes. The linear dependence of the global algorithm error on the mesh level is shown. The efficiency of our approach is demonstrated on a representative series of problems.

Metro Maps on Flexible Base Grids

Abstract: We present new generic methods to efficiently draw schematized metro maps for a wide variety of layouts, including octilinear, hexalinear, and orthoradial maps. The maps are drawn by mapping the input graph to a suitable grid graph. Previous work was restricted to regular octilinear grids. In this work, we investigate a variety of grids, including triangular grids and orthoradial grids. In particular, we also construct sparse grids where the local node density adapts to the input graph (e.g. octilinear Hanan grids, which we introduce in this work). For octilinear maps, this reduces the grid size by a factor of up to 5 compared to previous work, while still achieving close-to-optimal layouts. For many maps, this reduction also leads to up to 5 times faster solution times of the underlying optimization problem. We evaluate our approach on five maps. All octilinear maps can be computed in under 0.5 seconds, all hexalinear and orthoradial maps can be computed in under 2.5 seconds.

Algorithm of Gaussian Sum Filter Based on SGQF for Nonlinear Non-Gaussian Models

Abstract: To improve the filtering effect of the sparse grid quadrature filter (SGQF) under non-Gaussian conditions, the Gaussian sum technique is introduced, and the Gaussian sum sparse grid quadrature filter (GSSGQF) is developed. We present a systematic formulation of the SGQF and extend it to the discrete-time nonlinear system with the non-Gaussian noise. The proposed algorithm approximates the non-Gaussian probability densities by a finite number of weighted sums of Gaussian densities, and takes the SGQF as the Gaussian sub-filter to conduct the time and measurement update for each Gaussian component. An application in the discrete-time nonlinear system with the non-Gaussian noise has been shown to demonstrate the accuracy of the GSSGQF. It outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the GSSGQF provides significant performance improvement in the calculation accuracy for nonlinear non-Gaussian filtering problems.

Approximation of Multivariate Functions on Sparse Grids By Kernel-Based Quasi-Interpolation

Abstract: In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constru...

M-estimation based sparse grid quadrature filter and stochastic stability analysis

Abstract: In this study, a novel M-estimation based sparse grid quadrature filter (MSGQF) is proposed to improve the robust performance of the nonlinear system. We present a systematic formulation of the sparse grid quadrature filter (SGQF), and extend it to the discrete-time nonlinear system with abnormal measurement values. The M-estimation method is introduced in the SGQF, which uses the Huber’s cost function to update the measurement covariance. Convergence on the modified robust SGQF is established and proved. The sufficient conditions are shown to ensure stochastic stability of the MSGQF. A target tracking problem has been conducted to demonstrate the accuracy of the MSGQF. When measurement abnormal values appear, it outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the MSGQF provides significant performance improvement in the robustness of the nonlinear system.