429 research outputs found
A local hybrid surrogateâbased finite element tearing interconnecting dualâprimal method for nonsmooth random partial differential equations
A domain decomposition approach for highâdimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dualâprimal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problemâdependent hpârefinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the KarhunenâLoĂšve expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions
A local hybrid surrogate-based finite element tearing interconnecting dual-primal method for nonsmooth random partial differential equations
A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the KarhunenâLoĂšve expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. © 2020 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd
A hybrid FETI-DP method for non-smooth random partial differential equations
A domain decomposition approach exploiting the localization of random parameters in high-dimensional random PDEs is presented. For high efficiency, surrogate models in multi-element representations are computed locally when possible. This makes use of a stochastic Galerkin FETI-DP formulation of the underlying problem with localized representations of involved input random fields. The local parameter space associated to a subdomain is explored by a subdivision into regions where the parametric surrogate accuracy can be trusted and where instead Monte Carlo sampling has to be employed. A heuristic adaptive algorithm carries out a problem-dependent hp refinement in a stochastic multi-element sense, enlarging the trusted surrogate region in local parametric space as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration in the involved surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on sub-domains, e.g. in a multi-physics setting, or when the Karhunen-Loeve expansion of a random field can be localized. The efficiency of this hybrid technique is demonstrated with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and non-trusted sampling regions
Around the circular law
These expository notes are centered around the circular law theorem, which
states that the empirical spectral distribution of a nxn random matrix with
i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the
complex plane as the dimension tends to infinity. This phenomenon is the
non-Hermitian counterpart of the semi circular limit for Wigner random
Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random
covariance matrices. We present a proof in a Gaussian case, due to Silverstein,
based on a formula by Ginibre, and a proof of the universal case by revisiting
the approach of Tao and Vu, based on the Hermitization of Girko, the
logarithmic potential, and the control of the small singular values. Beyond the
finite variance model, we also consider the case where the entries have heavy
tails, by using the objective method of Aldous and Steele borrowed from
randomized combinatorial optimization. The limiting law is then no longer the
circular law and is related to the Poisson weighted infinite tree. We provide a
weak control of the smallest singular value under weak assumptions, using
asymptotic geometric analysis tools. We also develop a quaternionic
Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment
Perturbations and projections of KalmanâBucy semigroups
© 2017 Elsevier B.V. We analyse various perturbations and projections of KalmanâBucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and KalmanâBucy filtering distributions. New projection-type models are also discussed; e.g. BoseâMesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control
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