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Solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients has long been an open question. This paper fills this gap and in particular provides a convergence results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement.

The Fisher information matrix (FIM) is a key quantity in statistics as it is required for example for evaluating asymptotic precisions of parameter estimates, for computing test statistics or asymptotic distributions in statistical testing, for evaluating post model selection inference results or optimality criteria in experimental designs. However its exact computation is often not trivial. In particular in many latent variable models, it is intricated due to the presence of unobserved variables. Therefore the observed FIM is usually considered in this context to estimate the FIM. Several methods have been proposed to approximate the observed FIM when it can not be evaluated analytically. Among the most frequently used approaches are Monte-Carlo methods or iterative algorithms derived from the missing information principle. All these methods require to compute second derivatives of the complete data log-likelihood which leads to some disadvantages from a computational point of view. In this paper, we present a new approach to estimate the FIM in latent variable model. The advantage of our method is that only the first derivatives of the log-likelihood is needed, contrary to other approaches based on the observed FIM. Indeed we consider the empirical estimate of the covariance matrix of the score. We prove that this estimate of the Fisher information matrix is unbiased, consistent and asymptotically Gaussian. Moreover we highlight that none of both estimates is better than the other in terms of asymptotic covariance matrix. When the proposed estimate can not be directly analytically evaluated, we present a stochastic approximation estimation algorithm to compute it. This algorithm provides this estimate of the FIM as a by-product of the parameter estimates. We emphasize that the proposed algorithm only requires to compute the first derivatives of the complete data log-likelihood with respect to the parameters. We prove that the estimation algorithm is consistent and asymptotically Gaussian when the number of iterations goes to infinity. We evaluate the finite sample size properties of the proposed estimate and of the observed FIM through simulation studies in linear mixed effects models and mixture models. We also investigate the convergence properties of the estimation algorithm in non linear mixed effects models. We compare the performances of the proposed algorithm to those of other existing methods.

An old problem in multivariate statistics is that linear Gaussian models are often unidentifiable, i.e. some parameters cannot be uniquely estimated. In factor analysis, an orthogonal rotation of the factors is unidentifiable, while in linear regression, the direction of effect cannot be identified. For such linear models, non-Gaussianity of the (latent) variables has been shown to provide identifiability. In the case of factor analysis, this leads to independent component analysis, while in the case of the direction of effect, non-Gaussian versions of structural equation modelling solve the problem. More recently, we have shown how even general nonparametric nonlinear versions of such models can be estimated. Non-Gaussianity is not enough in this case, but assuming we have time series, or that the distributions are suitably modulated by some observed auxiliary variables, the models are identifiable. This paper reviews the identifiability theory for the linear and nonlinear cases, considering both factor analytic models and structural equation models.

Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, the said characterization is not (and apparently cannot be) of the inverse-zero type. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense, and convergence rates are provided. Our methodology is illustrated by simulation and two data analyses.

In this paper, we investigate the problem of system identification for autonomous Markov jump linear systems (MJS) with complete state observations. We propose switched least squares method for identification of MJS, show that this method is strongly consistent, and derive data-dependent and data-independent rates of convergence. In particular, our data-independent rate of convergence shows that, almost surely, the system identification error is $\mathcal{O}\big(\sqrt{\log(T)/T} \big)$ where $T$ is the time horizon. These results show that switched least squares method for MJS has the same rate of convergence as least squares method for autonomous linear systems. We derive our results by imposing a general stability assumption on the model called stability in the average sense. We show that stability in the average sense is a weaker form of stability compared to the stability assumptions commonly imposed in the literature. We present numerical examples to illustrate the performance of the proposed method.

Efficient structural reanalysis for high-rank modification plays an important role in engineering computations which require repeated evaluations of structural responses, such as structural optimization and probabilistic analysis. To improve the efficiency of engineering computations, a novel approximate static reanalysis method based on system reduction and iterative solution is proposed for statically indeterminate structures with high-rank modification. In this approach, a statically indeterminate structure is divided into the basis system and the additional components. Subsequently, the structural equilibrium equations are rewritten as the equation system with the stiffness matrix of the basis system and the pseudo forces derived from the additional elements. With the introduction of spectral decomposition, a reduced equation system with the element forces of the additional elements as the unknowns is established. Then, the approximate solutions of the modified structure can be obtained by solving the reduced equation system through a pre-conditioned iterative solution algorithm. The computational costs of the proposed method and the other two reanalysis methods are compared and numerical examples including static reanalysis and static nonlinear analysis are presented. The results demonstrate that the proposed method has excellent computational performance for both the structures with homogeneous material and structures composed of functionally graded beams. Meanwhile, the superiority of the proposed method indicates that the combination of system reduction and pre-conditioned iterative solution technology is an effective way to develop high-performance reanalysis methods.

Environmental epidemiologic studies routinely utilize aggregate health outcomes to estimate effects of short-term (e.g., daily) exposures that are available at increasingly fine spatial resolutions. However, areal averages are typically used to derive population-level exposure, which cannot capture the spatial variation and individual heterogeneity in exposures that may occur within the spatial and temporal unit of interest (e.g., within day or ZIP code). We propose a general modeling approach to incorporate within-unit exposure heterogeneity in health analyses via exposure quantile functions. Furthermore, by viewing the exposure quantile function as a functional covariate, our approach provides additional flexibility in characterizing associations at different quantile levels. We apply the proposed approach to an analysis of air pollution and emergency department (ED) visits in Atlanta over four years. The analysis utilizes daily ZIP code-level distributions of personal exposures to four traffic-related ambient air pollutants simulated from the Stochastic Human Exposure and Dose Simulator. Our analyses find that effects of carbon monoxide on respiratory and cardiovascular disease ED visits are more pronounced with changes in lower quantiles of the population-level exposure. Software for implement is provided in the R package nbRegQF.

In many applications that involve the inference of an unknown smooth function, the inference of its derivatives will often be just as important as that of the function itself. To make joint inferences of the function and its derivatives, a class of Gaussian processes called $p^{\text{th}}$ order Integrated Wiener's Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but have focused only on the order $p = 2$ case which does not allow appropriate inference for derivatives, and their computational feasibility relies on additional approximation to the FEM itself. In this article, we propose an alternative FEM approximation, called overlapping splines (O-spline), which pursues computational feasibility directly through the choice of test functions, and mirrors the construction of an IWP as the Ospline results from the multiple integrations of these same test functions. The O-spline approximation applies for any order $p \in \mathbb{Z}^+$, is computationally efficient and provides consistent inference for all derivatives up to order $p-1$. It is shown both theoretically, and empirically through simulation, that the O-spline approximation converges to the true IWP as the number of knots increases. We further provide a unified and interpretable way to define priors for the smoothing parameter based on the notion of predictive standard deviation (PSD), which is invariant to the order $p$ and the placement of the knot. Finally, we demonstrate the practical use of the O-spline approximation through simulation studies and an analysis of COVID death rates where the inference is carried on both the function and its derivatives where the latter has an important interpretation in terms of the course of the pandemic.

We present a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physics-informed neural networks (PINNs). In our experiments, existing versions of PINNs are seen to learn poorly in many problems, especially for complex geometries, as it becomes increasingly difficult to establish appropriate sampling strategy at the near boundary region. Overly dense sampling can adversely impede training convergence if the local gradient behaviors are too complex to be adequately modelled by PINNs. On the other hand, if the samples are too sparse, existing PINNs tend to overfit the near boundary region, leading to incorrect solution. To prevent such issues, we propose a new Boundary Connectivity (BCXN) loss function which provides linear local structure approximation (LSA) to the gradient behaviors at the boundary for PINN. Our BCXN-loss implicitly imposes local structure during training, thus facilitating fast physics-informed learning across entire problem domains with order of magnitude sparser training samples. This LSA-PINN method shows a few orders of magnitude smaller errors than existing methods in terms of the standard L2-norm metric, while using dramatically fewer training samples and iterations. Our proposed LSA-PINN does not pose any requirement on the differentiable property of the networks, and we demonstrate its benefits and ease of implementation on both multi-layer perceptron and convolutional neural network versions as commonly used in current PINN literature.

In realistic compressed sensing (CS) scenarios, the obtained measurements usually have to be quantized to a finite number of bits before transmission and/or storage, thus posing a challenge in recovery, especially for extremely coarse quantization such as 1-bit sign measurements. Recently Meng & Kabashima proposed an efficient quantized compressed sensing algorithm called QCS-SGM using the score-based generative models as an implicit prior. Thanks to the power of score-based generative models in capturing the rich structure of the prior, QCS-SGM achieves remarkably better performances than previous quantized CS methods. However, QCS-SGM is restricted to (approximately) row-orthogonal sensing matrices since otherwise the likelihood score becomes intractable. To address this challenging problem, in this paper we propose an improved version of QCS-SGM, which we call QCS-SGM+, which also works well for general matrices. The key idea is a Bayesian inference perspective of the likelihood score computation, whereby an expectation propagation algorithm is proposed to approximately compute the likelihood score. Experiments on a variety of baseline datasets demonstrate that the proposed QCS-SGM+ outperforms QCS-SGM by a large margin when sensing matrices are far from row-orthogonal.