Numerical Methods for Stochastic Computations: A Spectral Method Approach

By Dongbin Xiu. 2010

1,454 citations recorded by the source corpus.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/c338045b467615acab7c85d4b875eedf23d01db6

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Citations in the corpus

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks (2019, background): “Y i(t;!); !2 : (1) This approach can evolve the time-dependent basis of modes u i(x;t) and stochastic coecients Y i(t;!) simultaneously, and it is dierent than the standard polynomial chaos methods [11, 12, 13, 14]. To remove the redundancy in this representation, we need some constraints. For examp”
Least squares polynomial chaos expansion: A review of sampling strategies (2017, background): “Among the probabilistic UQ approaches, stochastic spectral methods based on polynomial chaos (PC) expansions [122, 15, 40, 125] have received special attention due to their advantages over traditional UQ techniques; see, e.g., [40, 117, 86, 27, 42].”
B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data (2020, background): “n the KL expansion are solved analytically, and the prior for the unknown parameters is the product of independent standard Gaussian distributions. We refer the readers to example 4.1 in Chapter 4 in [26] for details. The predicted uand f are illustrated in Figs. 10-11. The results from the KLHMC an”
Chaospy: An open source tool for designing methods of uncertainty quantification (2015, methodology): “For a review of the former, there is very useful book by Rubinstein, Reuven and Kroese [1], while for he latter, we refer to the excellent book by Xiu [2].”
Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives] (2013, methodology): “The rate of convergence depends on the smoothness of the function f and the type of orthogonal polynomial basis functions { } i z used for approximation (for example, [16] and [20]).”
Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices (2014, methodology): “It should be noted that sparse grid collocation method [23] may converge much faster than Monte Carlo when the number of uncertain loads is moderate.”
On Prediction Properties of Kriging: Uniform Error Bounds and Robustness (2017, background): “Such a rate is known as a spectral convergence order (Xiu, 2010; Wendland, 2004).”
The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions (2019, neutral): “For insightful reviews see [30–34].”
Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos (2013, methodology): “To reduce the rapid increase of full tensor product quadrature nodes in high dimensions d ≫ 1, sparse grids can be applied [27,46,48–50].”
Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, background): “Theoretical results suggest that with increasing degree N the truncated GN (Z) converges to G(Z) in the mean-square sense under mild conditions [41].”
Length scale and manufacturability in density-based topology optimization (2016, background): “Even though MC possesses dimension independent convergence, its convergence speed is relatively slow [157].”
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients (2015, neutral): “This task has been intensively studied since the 1990s, see in particular [14, 15, 20, 22] for general treatments.”
Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems (2018, positive): “at importance to construct a relatively cheaper “surrogate” of the forward model in the oﬄine [11]. In the next section, we will review the most commonly used polynomial chaos (PC) surrogate approach [13, 19, 40]. 2.2. Polynomial chaos expansions. In order to simplify the discussion and without loss”
Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression (2014, background): “Uncertainty Quantification (UQ) is an emerging field that aims at addressing these issues in an integral and rigorous manner: it intends for a meaningful characterization of uncertainties from the available information and efficient propagation of these uncertainties for a quantitative validation of”
Approximation of high-dimensional parametric PDEs * (2015, result): “We refer in particular to [40, 46, 55, 56, 90] for general introductions to these approaches, and to [1, 41, 37, 53, 54, 83, 84] for related work prior to the results exposed in our paper.”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, neutral): “, high-dimensional random inputs) [75, 74].”
Machine Learning in Aerodynamic Shape Optimization (2022, methodology): “In particular, PC-Kriging combines the advantages of PCE as a promising trend function to capture the general shape and kriging which leads the approximation to interpolate through training data (Fig.”