Search Results - Smolyak approximation. Point Collocation

Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients by Nobile, F., Tempone, Raul

“…We consider the problem of numerically approximating statistical moments of the solution of a time‐dependent linear parabolic partial differential equation…”
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A SPARSE GRID STOCHASTIC COLLOCATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA by NOBILE, F., TEMPONE, R., WEBSTER, C. G.

“…This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations…”
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A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data by Babuška, Ivo, Nobile, Fabio, Tempone, Raúl

“… These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points…”
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AN ANISOTROPIC SPARSE GRID STOCHASTIC COLLOCATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA by NOBILE, F., TEMPONE, R., WEBSTER, C. G.

“… (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw…”
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A sparse grid stochastic collocation method for structural reliability analysis by He, Jun, Gao, Shengbin, Gong, Jinghai

“…•A sparse-grid collocation method is developed for structural reliability analysis…”
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A SPARSE COMPOSITE COLLOCATION FINITE ELEMENT METHOD FOR ELLIPTIC SPDEs by BIERI, MARCEL

“… The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable…”
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SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos by Ahlfeld, R., Belkouchi, B., Montomoli, F.

“…A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The…”
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Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example by Shen, Danfeng, Wu, Hao, Xia, Bingqing, Gan, Deqiang

“… Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e…”
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Efficient uncertainty quantification with the polynomial chaos method for stiff systems by Cheng, Haiyan, Sandu, Adrian

“…). 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…”
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A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data by Lang, J., Scheichl, R., Silvester, D.

“… A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost…”
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Reducing the cost of using collocation to compute vibrational energy levels: Results for CH2NH by Avila, Gustavo, Carrington, Tucker

“…)]. Known quadrature and collocation methods using a Smolyak grid require storing intermediate vectors with more elements than points on the Smolyak grid…”
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Sparse collocation method for global sensitivity analysis and calculation of statistics of solutions in SPDEs by Zaicenco, Anton G.

“…Summary This paper explores advantages offered by the stochastic collocation method based on the Smolyak grids for the solution of differential equations with random inputs in the parameter space…”
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A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty by Kouri, D. P., Heinkenschloss, M., Ridzal, D., van Bloemen Waanders, B. G.

“… The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution…”
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Koopman Operator Based Numerical Methods and Neural Network Based Models for Differential Equations by Li, Bian

“…The study of differential equations is of paramount importance as it facilitates algorithms that are applied to a number of fields. However, most differential…”
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A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data by Lang, Jens, Scheichl, Robert, Silvester, David

“… The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually…”
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Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients: Uncertainty Quantification in Computational and Prediction Science by NOBILE, F, TEMPONE, Raul

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Computing Dynamic Heterogeneous-Agent Economies: Tracking the Distribution by Gordon, Grey

“… The contribution of this paper is to introduce a method for computing equilibrium in these models by including an entire distribution if finite-dimensional-or a fine approximation of it if…”
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Analysis and Design of Robust Control for Linear Parameter-Varying Systems by Hsu, Shao-Chen

“…, which can be formulated from a nonlinear system via either approximation or function substitution…”
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