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We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, $\epsilon$, while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.

We introduce a novel sampler called the energy based diffusion generator for generating samples from arbitrary target distributions. The sampling model employs a structure similar to a variational autoencoder, utilizing a decoder to transform latent variables from a simple distribution into random variables approximating the target distribution, and we design an encoder based on the diffusion model. Leveraging the powerful modeling capacity of the diffusion model for complex distributions, we can obtain an accurate variational estimate of the Kullback-Leibler divergence between the distributions of the generated samples and the target. Moreover, we propose a decoder based on generalized Hamiltonian dynamics to further enhance sampling performance. Through empirical evaluation, we demonstrate the effectiveness of our method across various complex distribution functions, showcasing its superiority compared to existing methods.

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Gr\"onwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Gr\"onwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Gr\"onwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.

Multichannel convolutive blind speech source separation refers to the problem of separating different speech sources from the observed multichannel mixtures without much a priori information about the mixing system. Multichannel nonnegative matrix factorization (MNMF) has been proven to be one of the most powerful separation frameworks and the representative algorithms such as MNMF and the independent low-rank matrix analysis (ILRMA) have demonstrated great performance. However, the sparseness properties of speech source signals are not fully taken into account in such a framework. It is well known that speech signals are sparse in nature, which is considered in this work to improve the separation performance. Specifically, we utilize the Bingham and Laplace distributions to formulate a disjoint constraint regularizer, which is subsequently incorporated into both MNMF and ILRMA. We then derive majorization-minimization rules for updating parameters related to the source model, resulting in the development of two enhanced algorithms: s-MNMF and s-ILRMA. Comprehensive simulations are conducted, and the results unequivocally demonstrate the efficacy of our proposed methodologies.

We introduce ensembles of stochastic neural networks to approximate the Bayesian posterior, combining stochastic methods such as dropout with deep ensembles. The stochastic ensembles are formulated as families of distributions and trained to approximate the Bayesian posterior with variational inference. We implement stochastic ensembles based on Monte Carlo dropout, DropConnect and a novel non-parametric version of dropout and evaluate them on a toy problem and CIFAR image classification. For both tasks, we test the quality of the posteriors directly against Hamiltonian Monte Carlo simulations. Our results show that stochastic ensembles provide more accurate posterior estimates than other popular baselines for Bayesian inference.

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.

A joint mix is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence and negative association. A joint mix is not always negatively dependent in any of the above senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent, and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multi-marginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.

We consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasifrequencies. Further, we perform numerical experiments to support and illustrate our analytical results and show how periodically time-dependent material parameters affect the scattered wave field.

The increasing availability of temporal data poses a challenge to time-series and signal-processing domains due to its high numerosity and complexity. Symbolic representation outperforms raw data in a variety of engineering applications due to its storage efficiency, reduced numerosity, and noise reduction. The most recent symbolic aggregate approximation technique called ABBA demonstrates outstanding performance in preserving essential shape information of time series and enhancing the downstream applications. However, ABBA cannot handle multiple time series with consistent symbols, i.e., the same symbols from distinct time series are not identical. Also, working with appropriate ABBA digitization involves the tedious task of tuning the hyperparameters, such as the number of symbols or tolerance. Therefore, we present a joint symbolic aggregate approximation that has symbolic consistency, and show how the hyperparameter of digitization can itself be optimized alongside the compression tolerance ahead of time. Besides, we propose a novel computing paradigm that enables parallel computing of symbolic approximation. The extensive experiments demonstrate its superb performance and outstanding speed regarding symbolic approximation and reconstruction.

Datasets containing both categorical and continuous variables are frequently encountered in many areas, and with the rapid development of modern measurement technologies, the dimensions of these variables can be very high. Despite the recent progress made in modelling high-dimensional data for continuous variables, there is a scarcity of methods that can deal with a mixed set of variables. To fill this gap, this paper develops a novel approach for classifying high-dimensional observations with mixed variables. Our framework builds on a location model, in which the distributions of the continuous variables conditional on categorical ones are assumed Gaussian. We overcome the challenge of having to split data into exponentially many cells, or combinations of the categorical variables, by kernel smoothing, and provide new perspectives for its bandwidth choice to ensure an analogue of Bochner's Lemma, which is different to the usual bias-variance tradeoff. We show that the two sets of parameters in our model can be separately estimated and provide penalized likelihood for their estimation. Results on the estimation accuracy and the misclassification rates are established, and the competitive performance of the proposed classifier is illustrated by extensive simulation and real data studies.