The main goal of this work is to develop a data-driven Reduced Order Model (ROM) strategy from high-fidelity simulation result data of a Full Order Model (FOM). The goal is to predict at lower computational cost the time evolution of solutions of Fluid-Structure Interaction (FSI) problems. For some FSI applications, the elastic solid FOM (often chosen as quasi-static) can take far more computational time than the fluid one. In this context, for the sake of performance one could only derive a ROM for the structure and try to achieve a partitioned FOM fluid solver coupled with a ROM solid one. In this paper, we present a data-driven partitioned ROM on two study cases: (i) a simplified 1D-1D FSI problem representing an axisymmetric elastic model of an arterial vessel, coupled with an incompressible fluid flow; (ii) an incompressible 2D wake flow over a cylinder facing an elastic solid with two flaps. We evaluate the accuracy and performance of the proposed ROM-FOM strategy on these cases while investigating the effects of the model's hyperparameters. We demonstrate a high prediction accuracy and significant speedup achievements using this strategy.
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Generalized linear models (GLMs) are popular for data-analysis in almost all quantitative sciences, but the choice of likelihood family and link function is often difficult. This motivates the search for likelihoods and links that minimize the impact of potential misspecification. We perform a large-scale simulation study on double-bounded and lower-bounded response data where we systematically vary both true and assumed likelihoods and links. In contrast to previous studies, we also study posterior calibration and uncertainty metrics in addition to point-estimate accuracy. Our results indicate that certain likelihoods and links can be remarkably robust to misspecification, performing almost on par with their respective true counterparts. Additionally, normal likelihood models with identity link (i.e., linear regression) often achieve calibration comparable to the more structurally faithful alternatives, at least in the studied scenarios. On the basis of our findings, we provide practical suggestions for robust likelihood and link choices in GLMs.
Variational autoencoder (VAE) architectures have the potential to develop reduced-order models (ROMs) for chaotic fluid flows. We propose a method for learning compact and near-orthogonal ROMs using a combination of a $\beta$-VAE and a transformer, tested on numerical data from a two-dimensional viscous flow in both periodic and chaotic regimes. The $\beta$-VAE is trained to learn a compact latent representation of the flow velocity, and the transformer is trained to predict the temporal dynamics in latent space. Using the $\beta$-VAE to learn disentangled representations in latent-space, we obtain a more interpretable flow model with features that resemble those observed in the proper orthogonal decomposition, but with a more efficient representation. Using Poincar\'e maps, the results show that our method can capture the underlying dynamics of the flow outperforming other prediction models. The proposed method has potential applications in other fields such as weather forecasting, structural dynamics or biomedical engineering.
This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.
Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering.Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and conditional independence for functional variables. Our method not only extends the applicability to functional data of the HSIC and its d-variate version (d-HSIC), but also allows us to introduce a test for conditional independence by defining a novel statistic for the CPT based on the HSCIC, with optimised regularisation strength estimated through an evaluation rejection rate. Our empirical results of the size and power of these tests on synthetic functional data show good performance, and we then exemplify their application to several constraint- and regression-based causal structure learning problems, including both synthetic examples and real socio-economic data.
We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes also known as testing by betting, as well as tests based on combining p-values. Simulation and empirical studies are conducted for a few settings of the null hypotheses, and they show that methods based on e-processes are efficient.
The convolution quadrature method originally developed for the Riemann-Liouville fractional calculus is extended in this work to the Hadamard fractional calculus by using the exponential type meshes. Local truncation error analysis is presented for singular solutions. By adopting the fractional BDF-$p(1\leq p \leq 6)$ for the Caputo-Hadamard fractional derivative in solving subdiffusion problem with singular source terms, and using the finite element method to discretize the space variable, we carry out the sharp error analysis rigorously and obtain the optimal accuracy by the novel correction technique. Our correction method is a natural generalization of the one developed for subdiffusion problems with smooth source terms. Numerical tests confirm the correctness of our theoretical results.
Traditional perturbative statistical disclosure control (SDC) approaches such as microaggregation, noise addition, rank swapping, etc, perturb the data in an ``ad-hoc" way in the sense that while they manage to preserve some particular aspects of the data, they end up modifying others. Synthetic data approaches based on the fully conditional specification data synthesis paradigm, on the other hand, aim to generate new datasets that follow the same joint probability distribution as the original data. These synthetic data approaches, however, rely either on parametric statistical models, or non-parametric machine learning models, which need to fit well the original data in order to generate credible and useful synthetic data. Another important drawback is that they tend to perform better when the variables are synthesized in the correct causal order (i.e., in the same order as the true data generating process), which is often unknown in practice. To circumvent these issues, we propose a fully non-parametric and model free perturbative SDC approach that approximates the joint distribution of the original data via sequential applications of restricted permutations to the numerical microdata (where the restricted permutations are guided by the joint distribution of a discretized version of the data). Empirical comparisons against popular SDC approaches, using both real and simulated datasets, suggest that the proposed approach is competitive in terms of the trade-off between confidentiality and data utility.
The joint modeling of multiple longitudinal biomarkers together with a time-to-event outcome is a challenging modeling task of continued scientific interest. In particular, the computational complexity of high dimensional (generalized) mixed effects models often restricts the flexibility of shared parameter joint models, even when the subject-specific marker trajectories follow highly nonlinear courses. We propose a parsimonious multivariate functional principal components representation of the shared random effects. This allows better scalability, as the dimension of the random effects does not directly increase with the number of markers, only with the chosen number of principal component basis functions used in the approximation of the random effects. The functional principal component representation additionally allows to estimate highly flexible subject-specific random trajectories without parametric assumptions. The modeled trajectories can thus be distinctly different for each biomarker. We build on the framework of flexible Bayesian additive joint models implemented in the R-package 'bamlss', which also supports estimation of nonlinear covariate effects via Bayesian P-splines. The flexible yet parsimonious functional principal components basis used in the estimation of the joint model is first estimated in a preliminary step. We validate our approach in a simulation study and illustrate its advantages by analyzing a study on primary biliary cholangitis.
A classic result by Stockmeyer gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of chop under the homogeneity assumption. In this paper, we study the complexity of the satisfiability problem for suitable weakenings of the chop interval temporal logic, that can be equivalently viewed as fragments of Halpern and Shoham interval logic. We first consider the logic $\mathsf{BD}_{hom}$ featuring modalities $B$, for \emph{begins}, corresponding to the prefix relation on pairs of intervals, and $D$, for \emph{during}, corresponding to the infix relation. The homogeneous models of $\mathsf{BD}_{hom}$ naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations. Such a fragment has been recently shown to be PSPACE-complete . In this paper, we study the extension $\mathsf{BD}_{hom}$ with the temporal neighborhood modality $A$ (corresponding to the Allen relation \emph{Meets}), and prove that it increases both its expressiveness and complexity. In particular, we show that the resulting logic $\mathsf{BDA}_{hom}$ is EXPSPACE-complete.
The ability to synthesize realistic data in a parametrizable way is valuable for a number of reasons, including privacy, missing data imputation, and evaluating the performance of statistical and computational methods. When the underlying data generating process is complex, data synthesis requires approaches that balance realism and simplicity. In this paper, we address the problem of synthesizing sequential categorical data of the type that is increasingly available from mobile applications and sensors that record participant status continuously over the course of multiple days and weeks. We propose the paired Markov Chain (paired-MC) method, a flexible framework that produces sequences that closely mimic real data while providing a straightforward mechanism for modifying characteristics of the synthesized sequences. We demonstrate the paired-MC method on two datasets, one reflecting daily human activity patterns collected via a smartphone application, and one encoding the intensities of physical activity measured by wearable accelerometers. In both settings, sequences synthesized by paired-MC better capture key characteristics of the real data than alternative approaches.
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