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The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schr{\"o}dinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schr{\"o}dinger case.} Numerical illustrations for the time-dependent Gross--Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods. The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross--Pitaevskii systems in real and imaginary time.

Open systems with balanced gain and loss, described by parity-time (PT-symmetric) Hamiltonians have been deeply explored over the past decade. Most explorations are limited to finite discrete models (in real or reciprocal spaces) or continuum problems in one dimension. As a result, these models do not leverage the complexity and variability of two-dimensional continuum problems on a compact support. Here, we investigate eigenvalues of non-relativistic Schrodinger equation on a disk with open boundary condition, in the presence of constant, PT-symmetric, gain-loss potential that is confined to two mirror-symmetric disks. We find a rich variety of exceptional points, re-entrant PT-symmetric phases, and a non-monotonic dependence of the PT-symmetry breaking threshold on the system parameters. By comparing results of two model variations, we show that this simple model of a multi-core fiber supports propagating modes in the presence of gain and loss.

This paper begins with a study of both the exact distribution and the asymptotic distribution of the empirical correlation of two independent AR(1) processes with Gaussian innovations. We proceed to develop rates of convergence for the distribution of the scaled empirical correlation %(i.e. the empirical correlation times the square root of the number of data points times a normalized constant) to the standard Gaussian distribution in both Wasserstein distance and in Kolmogorov distance. Given $n$ data points, we prove the convergence rate in Wasserstein distance is $n^{-1/2}$ and the convergence rate in Kolmogorov distance is $n^{-1/2} \sqrt{\ln n}$. We then compute rates of convergence of the scaled empirical correlation to the standard Gaussian distribution for two additional classes of AR(1) processes: (i) two AR(1) processes with correlated Gaussian increments and (ii) two independent AR(1) processes driven by white noise in the second Wiener chaos.

In multiple hypotheses testing it has become widely popular to make inference on the true discovery proportion (TDP) of a set $\mathcal{M}$ of null hypotheses. This approach is useful for several application fields, such as neuroimaging and genomics. Several procedures to compute simultaneous lower confidence bounds for the TDP have been suggested in prior literature. Simultaneity allows for post-hoc selection of $\mathcal{M}$. If sets of interest are specified a priori, it is possible to gain power by removing the simultaneity requirement. We present an approach to compute lower confidence bounds for the TDP if the set of null hypotheses is defined a priori. The proposed method determines the bounds using the exact distribution of the number of rejections based on a step-up multiple testing procedure under independence assumptions. We assess robustness properties of our procedure and apply it to real data from the field of functional magnetic resonance imaging.

Recurrent neural networks (RNNs) have yielded promising results for both recognizing objects in challenging conditions and modeling aspects of primate vision. However, the representational dynamics of recurrent computations remain poorly understood, especially in large-scale visual models. Here, we studied such dynamics in RNNs trained for object classification on MiniEcoset, a novel subset of ecoset. We report two main insights. First, upon inference, representations continued to evolve after correct classification, suggesting a lack of the notion of being ``done with classification''. Second, focusing on ``readout zones'' as a way to characterize the activation trajectories, we observe that misclassified representations exhibit activation patterns with lower L2 norm, and are positioned more peripherally in the readout zones. Such arrangements help the misclassified representations move into the correct zones as time progresses. Our findings generalize to networks with lateral and top-down connections, and include both additive and multiplicative interactions with the bottom-up sweep. The results therefore contribute to a general understanding of RNN dynamics in naturalistic tasks. We hope that the analysis framework will aid future investigations of other types of RNNs, including understanding of representational dynamics in primate vision.

In order to give quantitative estimates for approximating the ergodic limit, we investigate probabilistic limit behaviors of time-averaging estimators of numerical discretizations for a class of time-homogeneous Markov processes, by studying the corresponding strong law of large numbers and the central limit theorem. Verifiable general sufficient conditions are proposed to ensure these limit behaviors, which are related to the properties of strong mixing and strong convergence for numerical discretizations of Markov processes. Our results hold for test functionals with lower regularity compared with existing results, and the analysis does not require the existence of the Poisson equation associated with the underlying Markov process. Notably, our results are applicable to numerical discretizations for a large class of stochastic systems, including stochastic ordinary differential equations, infinite dimensional stochastic evolution equations, and stochastic functional differential equations.

Signal detection is one of the main challenges of data science. As it often happens in data analysis, the signal in the data may be corrupted by noise. There is a wide range of techniques aimed at extracting the relevant degrees of freedom from data. However, some problems remain difficult. It is notably the case of signal detection in almost continuous spectra when the signal-to-noise ratio is small enough. This paper follows a recent bibliographic line which tackles this issue with field-theoretical methods. Previous analysis focused on equilibrium Boltzmann distributions for some effective field representing the degrees of freedom of data. It was possible to establish a relation between signal detection and $\mathbb{Z}_2$-symmetry breaking. In this paper, we consider a stochastic field framework inspiring by the so-called "Model A", and show that the ability to reach or not an equilibrium state is correlated with the shape of the dataset. In particular, studying the renormalization group of the model, we show that the weak ergodicity prescription is always broken for signals small enough, when the data distribution is close to the Marchenko-Pastur (MP) law. This, in particular, enables the definition of a detection threshold in the regime where the signal-to-noise ratio is small enough.

G-formula is a popular approach for estimating treatment or exposure effects from longitudinal data that are subject to time-varying confounding. G-formula estimation is typically performed by Monte-Carlo simulation, with non-parametric bootstrapping used for inference. We show that G-formula can be implemented by exploiting existing methods for multiple imputation (MI) for synthetic data. This involves using an existing modified version of Rubin's variance estimator. In practice missing data is ubiquitous in longitudinal datasets. We show that such missing data can be readily accommodated as part of the MI procedure when using G-formula, and describe how MI software can be used to implement the approach. We explore its performance using a simulation study and an application from cystic fibrosis.

In this paper, we develop a multiphysics finite element method for solving the quasi-static thermo-poroelasticity model with nonlinear permeability. The model involves multiple physical processes such as deformation, pressure, diffusion and heat transfer. To reveal the multi-physical processes of deformation, diffusion and heat transfer, we reformulate the original model into a fluid coupled problem that is general Stokes equation coupled with two reaction-diffusion equations. Then, we prove the existence and uniqueness of weak solution for the original problem by the $B$-operator technique and by sequence approximation for the reformulated problem. As for the reformulated problem we propose a fully discrete finite element method which can use arbitrary finite element pairs to solve the displacement $\bu$ pressure $\tau $ and variable $\varpi,\varsigma$, and the backward Euler method for time discretization. Finally, we give the stability analysis of the above proposed method, also we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Numerical experiments show that the proposed method can achieve good results under different finite element pairs and are consistent with the theoretical analysis.

Bayesian cross-validation (CV) is a popular method for predictive model assessment that is simple to implement and broadly applicable. A wide range of CV schemes is available for time series applications, including generic leave-one-out (LOO) and K-fold methods, as well as specialized approaches intended to deal with serial dependence such as leave-future-out (LFO), h-block, and hv-block. Existing large-sample results show that both specialized and generic methods are applicable to models of serially-dependent data. However, large sample consistency results overlook the impact of sampling variability on accuracy in finite samples. Moreover, the accuracy of a CV scheme depends on many aspects of the procedure. We show that poor design choices can lead to elevated rates of adverse selection. In this paper, we consider the problem of identifying the regression component of an important class of models of data with serial dependence, autoregressions of order p with q exogenous regressors (ARX(p,q)), under the logarithmic scoring rule. We show that when serial dependence is present, scores computed using the joint (multivariate) density have lower variance and better model selection accuracy than the popular pointwise estimator. In addition, we present a detailed case study of the special case of ARX models with fixed autoregressive structure and variance. For this class, we derive the finite-sample distribution of the CV estimators and the model selection statistic. We conclude with recommendations for practitioners.