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Combination technique based k-th moment analysis of elliptic problems with random diffusion
2013
Journal of Computational Physics
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We consider the efficient deterministic solution of elliptic boundary value problems with random diffusion matrix. ...
Assuming random perturbations with known k moments, we derive, to leading order in the random perturbation's amplitude, deterministic equations for k moments of the random solution. ...
Section 2 is concerned with the modelling of elliptic boundary value problems with random diffusion. ...
doi:10.1016/j.jcp.2013.06.013
fatcat:hwjy25xocjhr5h5dp32n4gkzsa
A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach
2008
Journal of Computational Physics
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Various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework. Fig. 15. ...
A novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems. ...
The computing was conducted using the resources of the Cornell Center for Advanced Computing. ...
doi:10.1016/j.jcp.2008.01.019
fatcat:gxzt3w267re3hgr3gd3yhlskui
A least-squares approximation of partial differential equations with high-dimensional random inputs
2009
Journal of Computational Physics
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For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. ...
Stochastic collocation schemes, [27, 1, 20, 19 ] based on isotropic and anisotropic sparse grids reduce the problem of curse of dimensionality normally associated with conventional tensor-product schemes ...
Acknowledgment This material is based upon work supported by the Department of Energy, National Nuclear Security Administration under Award Number NA28614. ...
doi:10.1016/j.jcp.2009.03.006
fatcat:tqkmookv3jbcfgjizwb7w47siy
A priori error analysis of a numerical stochastic homogenization method
[article]
2020
arXiv
pre-print
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This paper provides an a priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. ...
If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in form of the spectral gap inequality, then the ...
, random fields with moment bounds on E[|A| p ] + E[|A −1 | p ] for p ≫ 1 by using an adaptation of our strategy. ...
arXiv:1912.11646v2
fatcat:pyhraqpowndszjjhtp6xdarkoq
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos
2002
Computer Methods in Applied Mechanics and Engineering
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In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. ...
The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. ...
Computations were performed at Brown's TCASCV and NCSA's (University of Illinois) facilities. ...
doi:10.1016/s0045-7825(02)00421-8
fatcat:zi3fo5fejbb2to67vqdehycg3q
Algorithms for Propagating Uncertainty Across Heterogeneous Domains
2015
SIAM Journal on Scientific Computing
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The effectiveness of these new algorithms is demonstrated in numerical examples involving elliptic problems with random diffusion coefficients, stochastically advected scalar fields, and nonlinear advection-reaction ...
problems with random reaction rates. ...
Jared Knap of ARL for stimulating discussions and useful suggestions. This work was supported by ARO grant W911NF-14-1-0425 and ARL grant W911NF-12-2-0023. ...
doi:10.1137/140992060
fatcat:2lowqeho7rezxkojx2r6ls3yae
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
2006
Journal of Computational Physics
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Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. ...
The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. ...
To address this problem, we have implemented a WCE based technique for computing moments of all orders. ...
doi:10.1016/j.jcp.2006.01.008
fatcat:iu7j54e7gbgkvdaiihpz5xgu5m
Multi-fidelity stochastic collocation method for computation of statistical moments
2017
Journal of Computational Physics
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We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. ...
By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. ...
One dimensional stochastic elliptic equation. We now consider an elliptic equation with random diffusivity, a standard benchmark problem in stochastic computing. ...
doi:10.1016/j.jcp.2017.04.022
fatcat:l645p6q7wvc6feigxyxadf726e
Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains
[chapter]
2016
Lecture Notes in Computational Science and Engineering
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In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. ...
We make the ansatz Combination technique based second moment analysis In complete analogy to the non-tensor product case, we obtain the solution scheme (1) Solve ( This lemma tells us that, given Cov[δ ...
This linearization technique has already been applied to random diffusion coefficients or even to elliptic equations on random domains in [18, 22, 23] . ...
doi:10.1007/978-3-319-28262-6_3
fatcat:qg4zrktrmfchxfqnzbm4efnm3y
RBF-based meshless methods for 2D elastostatic problems
2004
Engineering analysis with boundary elements
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The work presented here concerns the use of radial basis functions (RBFs) for the analysis of two dimensional elastostatic problems. ...
The basic characteristic of the formulation is the definition of a global approximation for the variables of interest in each problem (the deflection for the plate bending problem and the stress function ...
Acknowledgements The author is a member of ICIST (Instituto de Engenharia de Estruturas, Território e Construçáo do Instituto Superior Técnico) and acknowledges the financial support of FCT (Financiamento ...
doi:10.1016/j.enganabound.2003.06.003
fatcat:isgynzorirfuxeupo5j6ur5u4e
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
[article]
2021
arXiv
pre-print
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In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loève decomposition, then a probabilistic ...
On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. ...
The results are then statistically analyzed based on all K samples to calculate leading moments of variables of interest. ...
arXiv:2107.01386v2
fatcat:rosrafxdmbf45g44aciibq6g2m
Transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface
2014
International Journal of Solids and Structures
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The paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface. ...
The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. ...
Our analysis employs the technique of complex potentials (e.g., Muskhelishvili, 1953) . ...
doi:10.1016/j.ijsolstr.2014.03.024
fatcat:wwmq4uuzljfafdzgbf5zqc5szq
Surface processing methods for point sets using finite elements
2004
Computers & graphics
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The definition of tangent planes relies on moment-based computation with proven scaling and stability properties. ...
Our framework efficiently and effectively brings well-known PDE-based processing techniques to the field of point-based surfaces. ...
When combined with advanced rendering techniques such as splatting [1, 2] , point based surfaces can be superior to triangle meshes in terms of rendering quality and data storage flexibility. ...
doi:10.1016/j.cag.2004.08.013
fatcat:sys7rsjxi5bnjh5lynehbn2x3m
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
2010
SIAM Review
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Anal. 42(2004)] and allows one to treat easily a wider range of situations, such as: input data that depend non-linearly on the random variables, diffusivity coefficients with unbounded second moments ...
In this paper we propose and analyze a Stochastic-Collocation method to solve elliptic Partial Differential Equations with random coefficients and forcing terms (input data of the model). ...
or exponential ones. • deals with no difficulty with a diffusivity coefficient a with unbounded second moment. ...
doi:10.1137/100786356
fatcat:fdxdhxwppvaxdkvmh7jei5e64m
Aperiodic Photonics of Elliptic Curves
2019
Crystals
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Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green's matrix solution of the multiple wave scattering problem for electric and magnetic ...
environments beyond the limitations of traditional random media. ...
Abbreviations The following abbreviations are used in this manuscript:
EC Elliptic curve structure UR Uniform random structure UR e Ensemble averaged uniform random structures EC DL Elliptic curve discrete ...
doi:10.3390/cryst9090482
fatcat:kftf5zrmlnaqrjo3m47bbr2aji
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