70 research outputs found
An adaptive ANOVA stochastic Galerkin method for partial differential equations with random inputs
It is known that standard stochastic Galerkin methods encounter challenges
when solving partial differential equations with high dimensional random
inputs, which are typically caused by the large number of stochastic basis
functions required. It becomes crucial to properly choose effective basis
functions, such that the dimension of the stochastic approximation space can be
reduced. In this work, we focus on the stochastic Galerkin approximation
associated with generalized polynomial chaos (gPC), and explore the gPC
expansion based on the analysis of variance (ANOVA) decomposition. A concise
form of the gPC expansion is presented for each component function of the ANOVA
expansion, and an adaptive ANOVA procedure is proposed to construct the overall
stochastic Galerkin system. Numerical results demonstrate the efficiency of our
proposed adaptive ANOVA stochastic Galerkin method
Domain-decomposed Bayesian inversion based on local Karhunen-Loève expansions
In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method
Streaming data recovery via Bayesian tensor train decomposition
In this paper, we study a Bayesian tensor train (TT) decomposition method to
recover streaming data by approximating the latent structure in high-order
streaming data. Drawing on the streaming variational Bayes method, we introduce
the TT format into Bayesian tensor decomposition methods for streaming data,
and formulate posteriors of TT cores. Thanks to the Bayesian framework of the
TT format, the proposed algorithm (SPTT) excels in recovering streaming data
with high-order, incomplete, and noisy properties. The experiments in synthetic
and real-world datasets show the accuracy of our method compared to
state-of-the-art Bayesian tensor decomposition methods for streaming data
Domain-decomposed Bayesian inversion based on local Karhunen-Lo\`{e}ve expansions
In many Bayesian inverse problems the goal is to recover a spatially varying
random field. Such problems are often computationally challenging especially
when the forward model is governed by complex partial differential equations
(PDEs). The challenge is particularly severe when the spatial domain is large
and the unknown random field needs to be represented by a high-dimensional
parameter. In this paper, we present a domain-decomposed method to attack the
dimensionality issue and the method decomposes the spatial domain and the
parameter domain simultaneously. On each subdomain, a local Karhunen-Lo`eve
(KL) expansion is constructed, and a local inversion problem is solved
independently in a parallel manner, and more importantly, in a
lower-dimensional space. After local posterior samples are generated through
conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel
projection procedure is developed to effectively reconstruct the global field.
In addition, the domain decomposition interface conditions are dealt with an
adaptive Gaussian process-based fitting strategy. Numerical examples are
provided to demonstrate the performance of the proposed method
An adaptive reduced basis ANOVA method forhigh-dimensional Bayesian inverse problems
In Bayesian inverse problems sampling the posterior distribution is often a
challenging task when the underlying models are computationally intensive. To
this end, surrogates or reduced models are often used to accelerate the
computation. However, in many practical problems, the parameter of interest can
be of high dimensionality, which renders standard model reduction techniques
infeasible. In this paper, we present an approach that employs the ANOVA
decomposition method to reduce the model with respect to the unknown
parameters, and the reduced basis method to reduce the model with respect to
the physical parameters. Moreover, we provide an adaptive scheme within the
MCMC iterations, to perform the ANOVA decomposition with respect to the
posterior distribution. With numerical examples, we demonstrate that the
proposed model reduction method can significantly reduce the computational cost
of Bayesian inverse problems, without sacrificing much accuracy
- …