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We propose a general method for distributed Bayesian model choice, using the marginal likelihood, where a data set is split in non-overlapping subsets. These subsets are only accessed locally by individual workers and no data is shared between the workers. We approximate the model evidence for the full data set through Monte Carlo sampling from the posterior on every subset generating a model evidence per subset. The results are combined using a novel approach which corrects for the splitting using summary statistics of the generated samples. Our divide-and-conquer approach enables Bayesian model choice in the large data setting, exploiting all available information but limiting communication between workers. We derive theoretical error bounds that quantify the resulting trade-off between computational gain and loss in precision. The embarrassingly parallel nature yields important speed-ups when used on massive data sets as illustrated by our real world experiments. In addition, we show how the suggested approach can be extended to model choice within a reversible jump setting that explores multiple feature combinations within one run.

In this paper, we propose a deep learning based reduced order modeling method for stochastic underground flow problems in highly heterogeneous media. We aim to utilize supervised learning to build a reduced surrogate model from the stochastic parameter space that characterizes the possible highly heterogeneous media to the solution space of a stochastic flow problem to have fast online simulations. Dominant POD modes obtained from a well-designed spectral problem in a global snapshot space are used to represent the solution of the flow problem. Due to the small dimension of the solution, the complexity of the neural network is significantly reduced. We adopt the generalized multiscale finite element method (GMsFEM), in which a set of local multiscale basis functions that can capture the heterogeneity of the media and source information are constructed to efficiently generate globally defined snapshot space. Rigorous theoretical analyses are provided and extensive numerical experiments for linear and nonlinear stochastic flows are provided to verify the superior performance of the proposed method.

We consider a biochemical model that consists of a system of partial differential equations based on reaction terms and subject to non--homogeneous Dirichlet boundary conditions. The model is discretised using the gradient discretisation method (GDM) which is a framework covering a large class of conforming and non conforming schemes. Under classical regularity assumptions on the exact solutions, the GDM enables us to establish the existence of the model solutions in a weak sense, and strong convergence for the approximate solution and its approximate gradient. Numerical test employing a finite volume method is presented to demonstrate the behaviour of the solutions to the model.

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

Just-in-time adaptive interventions (JITAIs) are time-varying adaptive interventions that use frequent opportunities for the intervention to be adapted--weekly, daily, or even many times a day. The micro-randomized trial (MRT) has emerged for use in informing the construction of JITAIs. MRTs can be used to address research questions about whether and under what circumstances JITAI components are effective, with the ultimate objective of developing effective and efficient JITAI. The purpose of this article is to clarify why, when, and how to use MRTs; to highlight elements that must be considered when designing and implementing an MRT; and to review primary and secondary analyses methods for MRTs. We briefly review key elements of JITAIs and discuss a variety of considerations that go into planning and designing an MRT. We provide a definition of causal excursion effects suitable for use in primary and secondary analyses of MRT data to inform JITAI development. We review the weighted and centered least-squares (WCLS) estimator which provides consistent causal excursion effect estimators from MRT data. We describe how the WCLS estimator along with associated test statistics can be obtained using standard statistical software such as R (R Core Team, 2019). Throughout we illustrate the MRT design and analyses using the HeartSteps MRT, for developing a JITAI to increase physical activity among sedentary individuals. We supplement the HeartSteps MRT with two other MRTs, SARA and BariFit, each of which highlights different research questions that can be addressed using the MRT and experimental design considerations that might arise.

We develop a rapid and accurate contour method for the solution of time-fractional PDEs. The method inverts the Laplace transform via an optimised stable quadrature rule, suitable for infinite-dimensional operators, whose error decreases like $\exp(-cN/\log(N))$ for $N$ quadrature points. The method is parallisable, avoids having to resolve singularities of the solution as $t\downarrow 0$, and avoids the large memory consumption that can be a challenge for time-stepping methods applied to time-fractional PDEs. The ODEs resulting from quadrature are solved using adaptive sparse spectral methods that converge exponentially with optimal linear complexity. These solutions of ODEs are reused for different times. We provide a complete analysis of our approach for fractional beam equations used to model small-amplitude vibration of viscoelastic materials with a fractional Kelvin-Voigt stress-strain relationship. We calculate the system's energy evolution over time and the surface deformation in cases of both constant and non-constant viscoelastic parameters. An infinite-dimensional ``solve-then-discretise'' approach considerably simplifies the analysis, which studies the generalisation of the numerical range of a quasi-linearisation of a suitable operator pencil. This allows us to build an efficient algorithm with explicit error control. The approach can be readily adapted to other time-fractional PDEs and is not constrained to fractional parameters in the range $0<\nu<1$.

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner. The decomposition creates an hierarchical expansion for an output random variable of interest with respect to measure-consistent orthonormalized basis splines (B-splines) in independent input random variables. A dimensionwise decomposition of a spline space into orthogonal subspaces, each spanned by a reduced set of such orthonormal splines, results in SDD. Exploiting the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. The computational complexity of the SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to the extent possible. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation of nonsmooth functions calculates the probabilistic characteristics of an output variable with an accuracy matching or surpassing those obtained by high-order approximations from several existing methods. Finally, a 34-dimensional random eigenvalue analysis demonstrates the utility of SDD in solving practical problems.

We aim to reconstruct the latent space dynamics of high dimensional systems using model order reduction via the spectral proper orthogonal decomposition (SPOD). The proposed method is based on three fundamental steps: in the first, we compress the data from a high-dimensional representation to a lower dimensional one by constructing the SPOD latent space; in the second, we build the time-dependent coefficients by projecting the realizations (also referred to as snapshots) onto the reduced SPOD basis and we learn their evolution in time with the aid of recurrent neural networks; in the third, we reconstruct the high-dimensional data from the learnt lower-dimensional representation. The proposed method is demonstrated on two different test cases, namely, a compressible jet flow, and a geophysical problem known as the Madden-Julian Oscillation. An extensive comparison between SPOD and the equivalent POD-based counterpart is provided and differences between the two approaches are highlighted. The numerical results suggest that the proposed model is able to provide low rank predictions of complex statistically stationary data and to provide insights into the evolution of phenomena characterized by specific range of frequencies. The comparison between POD and SPOD surrogate strategies highlights the need for further work on the characterization of the error interplay between data reduction techniques and neural network forecasts.

We propose a non-intrusive Deep Learning-based Reduced Order Model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity snapshots are used to generate a POD-Galerkin ROM which is subsequently exploited to generate the data, covering the whole parameter range, used in the training phase of the DL-ROM. A convolutional autoencoder is employed to map the system response onto a low-dimensional representation and, in parallel, to model the reduced nonlinear trial manifold. The system dynamics on the manifold is described by means of a deep feedforward neural network that is trained together with the autoencoder. The strategy is benchmarked against high fidelity solutions on a clamped-clamped beam and on a real micromirror with softening response and multiplicity of solutions. By comparing the different computational costs, we discuss the impressive gain in performance and show that the DL-ROM truly represents a real-time tool which can be profitably and efficiently employed in complex system-level simulation procedures for design and optimisation purposes.

Joint modeling of a large number of variables often requires dimension reduction strategies that lead to structural assumptions of the underlying correlation matrix, such as equal pair-wise correlations within subsets of variables. The underlying correlation matrix is thus of interest for both model specification and model validation. In this paper, we develop tests of the hypothesis that the entries of the Kendall rank correlation matrix are linear combinations of a smaller number of parameters. The asymptotic behavior of the proposed test statistics is investigated both when the dimension is fixed and when it grows with the sample size. We pay special attention to the restricted hypothesis of partial exchangeability, which contains full exchangeability as a special case. We show that under partial exchangeability, the test statistics and their large-sample distributions simplify, which leads to computational advantages and better performance of the tests. We propose various scalable numerical strategies for implementation of the proposed procedures, investigate their behavior through simulations and power calculations under local alternatives, and demonstrate their use on a real dataset of mean sea levels at various geographical locations.