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Register a new userRandomized maximum likelihood based posterior sampling
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Added by
Dean Oliver
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Minimization of a stochastic cost function is commonly used for approximate sampling in high-dimensional Bayesian inverse problems with Gaussian prior distributions and multimodal posterior distributions. The density of the samples generated by minimization is not the desired target density, unless the observation operator is linear, but the distribution of samples is useful as a proposal density for importance sampling or for Markov chain Monte Carlo methods. In this paper, we focus on applications to sampling from multimodal posterior distributions in high dimensions. We first show that sampling from multimodal distributions is improved by computing all critical points instead of only minimizers of the objective function. For applications to high-dimensional geoscience problems, we demonstrate an efficient approximate weighting that uses a low-rank Gauss-Newton approximation of the determinant of the Jacobian. The method is applied to two toy problems with known posterior distributions and a Darcy flow problem with multiple modes in the posterior.
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This work considers the problem of computing the CANDECOMP/PARAFAC (CP) decomposition of large tensors. One popular way is to translate the problem into a sequence of overdetermined least squares subproblems with Khatri-Rao product (KRP) structure. In this work, for tensor with different levels of importance in each fiber, combining stochastic optimization with randomized sampling, we present a mini-batch stochastic gradient descent algorithm with importance sampling for those special least squares subproblems. Four different sampling strategies are provided. They can avoid forming the full KRP or corresponding probabilities and sample the desired fibers from the original tensor directly. Moreover, a more practical algorithm with adaptive step size is also given. For the proposed algorithms, we present their convergence properties and numerical performance. The results on synthetic data show that our algorithms outperform the existing algorithms in terms of accuracy or the number of iterations.
A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constant must grow in a controlled manner. In this paper we generalize the notion of a WAM to a hierarchical subspaces of not necessarily polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAMs. Our main results show that WAMs and their stronger variant, admissible meshes, can be generated by random sampling, and our analysis provides concrete estimates for growth of both the meshes and the discrete-continuum comparability constants.
The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matern covariances with smoothness index $ u$ and correlation length $lambda$, we analyse the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $ u$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $ u$, $lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.
The unscented Kalman inversion (UKI) method presented in [1] is a general derivative-free approach for the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be differentiable. The regularization strategies, convergence property, and speed-up strategies [1,2] of the UKI are thoroughly studied, and the method is capable of handling noisy observation data and solving chaotic inverse problems. In this paper, we study the uncertainty quantification capability of the UKI. We propose a modified UKI, which allows to well approximate the mean and covariance of the posterior distribution for well-posed inverse problems with large observation data. Theoretical guarantees for both linear and nonlinear inverse problems are presented. Numerical results, including learning of permeability parameters in subsurface flow and of the Navier-Stokes initial condition from solution data at positive times are presented. The results obtained by the UKI require only $O(10)$ iterations, and match well with the expected results obtained by the Markov Chain Monte Carlo method.
In this paper, we analyze the convergence behavior of the randomized extended Kaczmarz (REK) method for all types of linear systems (consistent or inconsistent, overdetermined or underdetermined, full-rank or rank-deficient). The analysis shows that the larger the singular value of $A$ is, the faster the error decays in the corresponding right singular vector space, and as $krightarrowinfty$, $x_{k}-x_{star}$ tends to the right singular vector corresponding to the smallest singular value of $A$, where $x_{k}$ is the $k$th approximation of the REK method and $x_{star}$ is the minimum $ell_2 $-norm least squares solution. These results explain the phenomenon found in the extensive numerical experiments appearing in the literature that the REK method seems to converge faster in the beginning. A simple numerical example is provided to confirm the above findings.
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