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The study of multiphase flow is essential for understanding the complex interactions of various materials. In particular, when designing chemical reactors such as fluidized bed reactors (FBR), a detailed understanding of the hydrodynamics is critical for optimizing reactor performance and stability. An FBR allows experts to conduct different types of chemical reactions involving multiphase materials, especially interaction between gas and solids. During such complex chemical processes, formation of void regions in the reactor, generally termed as bubbles, is an important phenomenon. Study of these bubbles has a deep implication in predicting the reactor's overall efficiency. But physical experiments needed to understand bubble dynamics are costly and non-trivial. Therefore, to study such chemical processes and bubble dynamics, a state-of-the-art massively parallel computational fluid dynamics discrete element model (CFD-DEM), MFIX-Exa is being developed for simulating multiphase flows. Despite the proven accuracy of MFIX-Exa in modeling bubbling phenomena, the very-large size of the output data prohibits the use of traditional post hoc analysis capabilities in both storage and I/O time. To address these issues and allow the application scientists to explore the bubble dynamics in an efficient and timely manner, we have developed an end-to-end visual analytics pipeline that enables in situ detection of bubbles using statistical techniques, followed by a flexible and interactive visual exploration of bubble dynamics in the post hoc analysis phase. Positive feedback from the experts has indicated the efficacy of the proposed approach for exploring bubble dynamics in very-large scale multiphase flow simulations.

A functional dynamic factor model for time-dependent functional data is proposed. We decompose a functional time series into a predictive low-dimensional common component consisting of a finite number of factors and an infinite-dimensional idiosyncratic component that has no predictive power. The conditions under which all model parameters, including the number of factors, become identifiable are discussed. Our identification results lead to a simple-to-use two-stage estimation procedure based on functional principal components. As part of our estimation procedure, we solve the separation problem between the common and idiosyncratic functional components. In particular, we obtain a consistent information criterion that provides joint estimates of the number of factors and dynamic lags of the common component. Finally, we illustrate the applicability of our method in a simulation study and to the problem of modeling and predicting yield curves. In an out-of-sample experiment, we demonstrate that our model performs well compared to the widely used term structure Nelson-Siegel model for yield curves.

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple Eulerian and Lagrangian variables. In practice, discretizations of these integral transforms use regularized delta function kernels, and although a number of different types of regularized delta functions have been proposed, there has been limited prior work to investigate the impact of the choice of kernel function on the accuracy of the methodology. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses finite element structural discretizations combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Further, many IB-type methods evaluate the delta functions at the nodes of the structural mesh, and this requires the Lagrangian mesh to be relatively fine compared to the background Eulerian grid to avoid leaks. The IFED formulation offers the possibility to avoid leaks with relatively coarse structural meshes by evaluating the delta function on a denser collection of interaction points. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations. Although this study is done within the context of the IFED method, the effect of different kernels could be important not just for this method, but also for other IB-type methods more generally.

We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradient step to minimize its own cost function while sharing and retrieving information locally among its neighbors. The existing methods impose strong assumptions such as balancedness of the mixing matrices and global knowledge of the network communication structure, including Perron-Frobenius eigenvector of the adjacency matrix and other graph connectivity constants. In contrast, our approach relies only on a reasonable and widely-used assumption of row-stochasticity of the mixing matrices. We analyze the algorithm for time-varying directed graphs and prove its convergence to the NE, when the agents' cost functions are strongly convex and have Lipschitz continuous gradients. Numerical simulations are performed for a Nash-Cournot game to illustrate the efficacy of the proposed algorithm.

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, is simpler to implement. It solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

When assessing the performance of wireless communication systems operating over fading channels, one often encounters the problem of computing expectations of some functional of sums of independent random variables (RVs). The outage probability (OP) at the output of Equal Gain Combining (EGC) and Maximum Ratio Combining (MRC) receivers is among the most important performance metrics that falls within this framework. In general, closed form expressions of expectations of functionals applied to sums of RVs are out of reach. A naive Monte Carlo (MC) simulation is of course an alternative approach. However, this method requires a large number of samples for rare event problems (small OP values for instance). Therefore, it is of paramount importance to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), being known for its efficiency in requiring less computations for achieving the same accuracy requirement. In this line, we propose a state-dependent IS scheme based on a stochastic optimal control (SOC) formulation to calculate rare events quantities that could be written in a form of an expectation of some functional of sums of independent RVs. Our proposed algorithm is generic and can be applicable without any restriction on the univariate distributions of the different fading envelops/gains or on the functional that is applied to the sum. We apply our approach to the Log-Normal distribution to compute the OP at the output of diversity receivers with and without co-channel interference. For each case, we show numerically that the proposed state-dependent IS algorithm compares favorably to most of the well-known estimators dealing with similar problems.

Reaction networks are often used to model interacting species in fields such as biochemistry and ecology. When the counts of the species are sufficiently large, the dynamics of their concentrations are typically modeled via a system of differential equations. However, when the counts of some species are small, the dynamics of the counts are typically modeled stochastically via a discrete state, continuous time Markov chain. A key quantity of interest for such models is the probability mass function of the process at some fixed time. Since paths of such models are relatively straightforward to simulate, we can estimate the probabilities by constructing an empirical distribution. However, the support of the distribution is often diffuse across a high-dimensional state space, where the dimension is equal to the number of species. Therefore generating an accurate empirical distribution can come with a large computational cost. We present a new Monte Carlo estimator that fundamentally improves on the "classical" Monte Carlo estimator described above. It also preserves much of classical Monte Carlo's simplicity. The idea is basically one of conditional Monte Carlo. Our conditional Monte Carlo estimator has two parameters, and their choice critically affects the performance of the algorithm. Hence, a key contribution of the present work is that we demonstrate how to approximate optimal values for these parameters in an efficient manner. Moreover, we provide a central limit theorem for our estimator, which leads to approximate confidence intervals for its error.

Modelling and forecasting homogeneous age-specific mortality rates of multiple countries could lead to improvements in long-term forecasting. Data fed into joint models are often grouped according to nominal attributes, such as geographic regions, ethnic groups, and socioeconomic status, which may still contain heterogeneity and deteriorate the forecast results. Our paper proposes a novel clustering technique to pursue homogeneity among multiple functional time series based on functional panel data modelling to address this issue. Using a functional panel data model with fixed effects, we can extract common functional time series features. These common features could be decomposed into two components: the functional time trend and the mode of variations of functions (functional pattern). The functional time trend reflects the dynamics across time, while the functional pattern captures the fluctuations within curves. The proposed clustering method searches for homogeneous age-specific mortality rates of multiple countries by accounting for both the modes of variations and the temporal dynamics among curves. We demonstrate that the proposed clustering technique outperforms other existing methods through a Monte Carlo simulation and could handle complicated cases with slow decaying eigenvalues. In empirical data analysis, we find that the clustering results of age-specific mortality rates can be explained by the combination of geographic region, ethnic groups, and socioeconomic status. We further show that our model produces more accurate forecasts than several benchmark methods in forecasting age-specific mortality rates.

This study presents a generalized multiscale nonlocal elasticity theory that leverages distributed order fractional calculus to accurately capture coexisting multiscale and nonlocal effects within a macroscopic continuum. The nonlocal multiscale behavior is captured via distributed order fractional constitutive relations derived from a nonlocal thermodynamic formulation. The governing equations of the inhomogeneous continuum are obtained via the Hamilton principle. As a generalization of the constant order fractional continuum theory, the distributed order theory can model complex media characterized by inhomogeneous nonlocality and multiscale effects. In order to understand the correspondence between microscopic effects and the properties of the continuum, an equivalent mass-spring lattice model is also developed by direct discretization of the distributed order elastic continuum. Detailed theoretical arguments are provided to show the equivalence between the discrete and the continuum distributed order models in terms of internal nonlocal forces, potential energy distribution, and boundary conditions. These theoretical arguments facilitate the physical interpretation of the role played by the distributed order framework within nonlocal elasticity theories. They also highlight the outstanding potential and opportunities offered by this methodology to account for multiscale nonlocal effects. The capabilities of the methodology are also illustrated via a numerical study that highlights the excellent agreement between the displacement profiles and the total potential energy predicted by the two models under various order distributions. Remarkably, multiscale effects such as displacement distortion, material softening, and energy concentration are well captured at continuum level by the distributed order theory.

Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.