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Weighted smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2011年第11期85卷 1365-1389页

作者： Agarwal, Nitin Aluru, N. R. Univ Illinois Dept Mech Sci & Engn Beckman Inst Adv Sci & Technol Urbana IL 61801 USA

This paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non-uniform probability measures. Recently, there has been a growing interest in the stochastic collocation approach, which seeks to approximate the unknown stochastic solution using polynomial interpolation in the multi-dimensional random domain. Existing approaches employ sparse grid interpolation based on the smolyak algorithm, which leads to orders of magnitude reduction in the number of support nodes as compared with usual tensor product. However, such sparse grid interpolation approaches based on piecewise linear interpolation employ uniformly sampled nodes from the random domain and do not take into account the probability measures during the construction of the sparse grids. Such a construction based on uniform sparse grids may not be ideal, especially for highly skewed or localized probability measures. To this end, this work proposes a weighted smolyak algorithm based on piecewise linear basis functions, which incorporates information regarding non-uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： smolyak algorithm sparse grid non-uniform distributions stochastic collocation method stochastic differential equations uncertainty quantification

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On weak tractability of the smolyak algorithm for approximation problems

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JOURNAL OF APPROXIMATION THEORY 2015年 192卷 347-361页

作者： Xu, Guiqiao Tianjin Normal Univ Dept Math Tianjin 300387 Peoples R China

We consider the problems of L-P-approximation of d-variate analytic functions defined on the cube with directional derivatives of all orders bounded by 1. For 1 <= p < infinity, it is shown that the smolyak algorithm based on polynomial interpolation at the extrema of the Chebyshev polynomials leads to weak tractability of these problems. This gives an affirmative answer to one of the open problems raised recently by Hinrichs et al. (2014). Our proof uses the polynomial exactness of the algorithm and an explicit bound on the operator norm of the algorithm. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Weak tractability smolyak algorithm Infinitely differentiable function class Standard information

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Approximation by quasi-interpolation operators and smolyak's algorithm

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JOURNAL OF COMPLEXITY 2022年 69卷

作者： Kolomoitsev, Yurii Univ Lubeck Inst Math Ratzeburger Allee 160 D-23562 Lubeck Germany NAS Ukraine Inst Appl Math & Mech Gen Batyuk Str 19 UA-84116 Slovyansk Donetsk Region Ukraine

We study approximation of multivariate periodic functions from Besov and Triebel-Lizorkin spaces of dominating mixed smoothness by the smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding smolyak algorithm in the Lq-norm for functions from the Besov spaces Bsp,theta(Td) and the Triebel-Lizorkin spaces Fsp,theta(Td) for all s > 0 and admissible 1 < p, theta < infinity as well as provide analogues of the Littlewood- Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators. (c) 2021 Elsevier Inc. All rights reserved.

关键词： smolyak algorithm Quasi-interpolation operators Kantorovich operators Besov-Triebel-Lizorkin spaces of mixed smoothness Error estimates Littlewood-Paley-type characterizations

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Randomized smolyak algorithms based on digital sequences for multivariate integration

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IMA JOURNAL OF NUMERICAL ANALYSIS 2007年第4期27卷 655-674页

作者： Dick, Josef Leobacher, Gunther Pillichshammer, Friedrich Univ New S Wales Div Engn Sci & Technol Singapore 248922 Singapore Univ Linz Inst Finanzmathemat A-4040 Linz Austria

In this paper, we consider smolyak algorithms based on quasi-Monte Carlo rules for high-dimensional numerical integration. The quasi-Monte Carlo rules employed here use digital (t, alpha, beta, sigma, d)-sequences as quadrature points. We consider the worst-case error for multivariate integration in certain Sobolev spaces and show that our quadrature rules achieve the optimal rate of convergence. By randomizing the underlying digital sequences, we can also obtain a randomized smolyak algorithm. The bound on the worst-case error holds also for the randomized algorithm in a statistical sense. Further, we also show that the randomized algorithm is unbiased and that the integration error can be approximated as well.

关键词： numerical integration quasi-Monte Carlo algorithm smolyak algorithm

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On the smolyak cubature error for analytic functions

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2000年第1期12卷 71-93页

作者： Petras, K Tech Univ Carolo Wilhelmina Braunschweig Inst Angew Math D-38106 Braunschweig Germany

We consider smolyak's construction for the numerical integration over the d-dimensional unit cube. The underlying class of integrands is a tensor product space consisting of functions that are analytic in the Cartesian product of ellipses. The Kronrod-Patterson quadrature formulae are proposed as the corresponding basic sequence and this choice is compared with Clenshaw-Curtis quadrature formulae. First, error bounds are derived for the one-dimensional case, which lead by a recursion formula to error bounds for higher dimensional integration. The applicability of these bounds is shown by examples from frequently used test packages. Finally, numerical experiments are reported.

关键词： cubature smolyak algorithm error bounds analytic functions Clenshaw-Curtis quadrature Kronrod-Patterson quadrature

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A data-driven stochastic collocation approach for uncertainty quantification in MEMS

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2010年第5期83卷 575-597页

作者： Agarwal, Nitin Aluru, N. R. Univ Illinois Dept Mech Sci & Engn Beckman Inst Adv Sci & Technol Urbana IL 61801 USA

This work presents a data-driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi-physics simulation of Micro-ElectroMechanical Systems (MEMS). The proposed framework comprises of two key steps: first, probabilistic characterization of the input uncertain parameters based on available experimental information, and second, propagation of these uncertainties through the predictive model to relevant quantities of interest. The uncertain input parameters are modeled as independent random variables, for which the distributions are estimated based on available experimental observations, using a nonparametric diffusion-mixing-based estimator, Botev (Nonparametric density estimation via diffusion mixing. Technical Report, 2007). The diffusion-based estimator derives from the analogy between the kernel density estimation (KDE) procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent. The diffusion model allows for the incorporation of the prior density and leads to an improved density estimate, in comparison with the standard KDE approach, as demonstrated through several numerical examples. Following the characterization step, the uncertainties are propagated to the output variables using the stochastic collocation approach, based on sparse grid interpolation, smolyak (Soviet Math. Dokl. 1963;4:240-243). The developed framework is used to study the effect of variations in Young's modulus, induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a MEMS switch. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： uncertainty quantification nonparametric density estimation diffusion estimator stochastic collocation smolyak algorithm sparse grids polysilicon Young's modulus

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Stochastic Analysis of Electrostatic MEMS Subjected to Parameter Variations

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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS 2009年第6期18卷 1454-1468页

作者： Agarwal, Nitin Aluru, Narayana R. Univ Illinois Dept Mech Sci & Engn Urbana IL 61801 USA Univ Illinois Beckman Inst Adv Sci & Technol Urbana IL 61801 USA

This paper presents an efficient stochastic framework for quantifying the effect of stochastic variations in various design parameters such as material properties, geometrical features, and/or operating conditions on the performance of electrostatic microelectromechanical systems (MEMS) devices. The stochastic framework treats uncertainty as a separate dimension, in addition to space and time, and seeks to approximate the stochastic dependent variables using sparse grid interpolation in the multidimensional random space. This approach can be effectively used to compute important information, such as moments (mean and variance), failure probabilities, and sensitivities with respect to design variables, regarding relevant quantities of interest. The approach is straightforward to implement and, depending on the accuracy required, can be orders of magnitude faster than the traditional Monte Carlo method. We consider two examples-MEMS switch and resonator-and employ the proposed approach to study the effect of uncertain Young's modulus and various geometrical parameters, such as dimensions of electrodes and gap between microstructures, on relevant quantities of interest such as actuation behavior, resonant frequency, and quality factor. It is demonstrated that, in addition to computing the required statistics and probability density function, the proposed approach effectively identifies critical design parameters, which can then be controlled during fabrication, in order to improve device performance and reliability. [2009-0065]

关键词： Microelectromechanical systems (MEMS) resonator MEMS switch parameter variation reliability smolyak algorithm sparse grid interpolation uncertainty propagation

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Parametric uncertainty and global sensitivity analysis in a model of the carotid bifurcation: Identification and ranking of most sensitive model parameters

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MATHEMATICAL BIOSCIENCES 2015年 269卷 104-116页

作者： Gul, R. Bernhard, S. Free Univ Berlin Fachbereich Math Berlin Germany Pforzheim Univ Appl Sci Dept Elect Engn & Informat Technol Pforzheim Germany

In computational cardiovascular models, parameters are one of major sources of uncertainty, which make the models unreliable and less predictive. In order to achieve predictive models that allow the investigation of the cardiovascular diseases, sensitivity analysis (SA) can be used to quantify and reduce the uncertainty in outputs (pressure and flow) caused by input (electrical and structural) model parameters. In the current study, three variance based global sensitivity analysis (GSA) methods;Sobol, FAST and a sparse grid stochastic collocation technique based on the smolyak algorithm were applied on a lumped parameter model of carotid bifurcation. Sensitivity analysis was carried out to identify and rank most sensitive parameters as well as to fix less sensitive parameters at their nominal values (factor fixing). In this context, network location and temporal dependent sensitivities were also discussed to identify optimal measurement locations in carotid bifurcation and optimal temporal regions for each parameter in the pressure and flow waves, respectively. Results show that, for both pressure and flow, flow resistance (R), diameter (d) and length of the vessel (I) are sensitive within right common carotid (RCC), right internal carotid (RIG) and right external carotid (REC) arteries, while compliance of the vessels (C) and blood inertia (L) are sensitive only at RCC. Moreover, Young's modulus (E) and wall thickness (h) exhibit less sensitivities on pressure and flow at all locations of carotid bifurcation. Results of network location and temporal variabilities revealed that most of sensitivity was found in common time regions i.e. early systole, peak systole and end systole. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Global sensitivity analysis Lumped parameter model Sobol method FAST Sparse grid stochastic collocation technique smolyak algorithm

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Efficient uncertainty quantification with the polynomial chaos method for stiff systems

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MATHEMATICS AND COMPUTERS IN SIMULATION 2009年第11期79卷 3278-3295页

作者： Cheng, Haiyan Sandu, Adrian Virginia Polytech Inst & State Univ Dept Comp Sci Blacksburg VA 24061 USA

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear system of equations at every time step. Using the Galerkin approach the size of the system state increases from n to S x n, where S is the number of PC basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n(3)) to O(S(3)n(3)). The S-3-fold increase can make the computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with no negative impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Uncertainty quantification Polynomial chaos Least-squares collocation smolyak algorithm Low-discrepancy data sets

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Stochastic optimization using a sparse grid collocation scheme

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PROBABILISTIC ENGINEERING MECHANICS 2009年第3期24卷 382-396页

作者： Sankaran, Sethuraman Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

In computational sciences, optimization problems are frequently encountered in solving inverse problems for computing system parameters based on data measurements at specific sensor locations, or to perform design of system parameters. This task becomes increasingly complicated in the presence of uncertainties in boundary conditions or material properties. The task of computing the optimal probability density function (PDF) of parameters based on measurements of physical fields of interest in the form of a PDF, is posed as a stochastic optimization problem. This stochastic optimization problem is solved by dividing it into two problems-an auxiliary optimization problem to construct stochastic space representations from the PDF of measurement data, and a stochastic optimization problem to compute the PDF of problem parameters. The auxiliary optimization problem is solved using a downhill simplex method, whilst a gradient based approach is employed for solving the stochastic optimization problem. The gradients required for stochastic optimization are defined, using appropriate stochastic sensitivity problems. A computationally efficient sparse grid collocation scheme is utilized to compute the solution of these stochastic sensitivity problems. The implementation discussed, requires minimum intrusion into existing deterministic solvers, and it is thus applicable to a variety of problems. Numerical examples involving stochastic inverse heat conduction problems, contamination source identification problems and large deformation robust design problems are discussed. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Stochastic optimization Sparse grid collocation smolyak algorithm Inverse problems Stochastic sensitivities Robust design

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