High-dimensional and incomplete (HDI) matrix contains many complex interactions between numerous nodes. A stochastic gradient descent (SGD)-based latent factor analysis (LFA) model is remarkably effective in extracting valuable information from an HDI matrix. However, such a model commonly encounters the problem of slow convergence because a standard SGD algorithm only considers the current learning error to compute the stochastic gradient without considering the historical and future state of the learning error. To address this critical issue, this paper innovatively proposes an ADRC-incorporated SGD (ADS) algorithm by refining the instance learning error by considering the historical and future state by following the principle of an ADRC controller. With it, an ADS-based LFA model is further achieved for fast and accurate latent factor analysis on an HDI matrix. Empirical studies on two HDI datasets demonstrate that the proposed model outperforms the state-of-the-art LFA models in terms of computational efficiency and accuracy for predicting the missing data of an HDI matrix.
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Maximum mean discrepancy (MMD) is a particularly useful distance metric for differentially private data generation: when used with finite-dimensional features it allows us to summarize and privatize the data distribution once, which we can repeatedly use during generator training without further privacy loss. An important question in this framework is, then, what features are useful to distinguish between real and synthetic data distributions, and whether those enable us to generate quality synthetic data. This work considers the using the features of $\textit{neural tangent kernels (NTKs)}$, more precisely $\textit{empirical}$ NTKs (e-NTKs). We find that, perhaps surprisingly, the expressiveness of the untrained e-NTK features is comparable to that of the features taken from pre-trained perceptual features using public data. As a result, our method improves the privacy-accuracy trade-off compared to other state-of-the-art methods, without relying on any public data, as demonstrated on several tabular and image benchmark datasets.
Uncertainty quantification (UQ) to detect samples with large expected errors (outliers) is applied to reactive molecular potential energy surfaces (PESs). Three methods - Ensembles, Deep Evidential Regression (DER), and Gaussian Mixture Models (GMM) - were applied to the H-transfer reaction between ${\it syn-}$Criegee and vinyl hydroxyperoxide. The results indicate that ensemble models provide the best results for detecting outliers, followed by GMM. For example, from a pool of 1000 structures with the largest uncertainty, the detection quality for outliers is $\sim 90$ \% and $\sim 50$ \%, respectively, if 25 or 1000 structures with large errors are sought. On the contrary, the limitations of the statistical assumptions of DER greatly impacted its prediction capabilities. Finally, a structure-based indicator was found to be correlated with large average error, which may help to rapidly classify new structures into those that provide an advantage for refining the neural network.
A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found in non-uniformly refined meshes and different wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N storage many-stage Runge-Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. The methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.
The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.
Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.
We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.
Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of a multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite-difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme, and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.
This letter studies the performance of reconfigurable intelligent surface (RIS)-aided communications for a fluid antenna system (FAS) enabled receiver. Specifically, a fixed singleantenna base station (BS) transmits information through a RIS to a mobile user (MU) which is equipped with a planar fluid antenna in the absence of a direct link.We first analyze the spatial correlation structures among the positions (or ports) in the planar FAS, and then derive the joint distribution of the equivalent channel gain at the user by exploiting the central limit theorem. Furthermore, we obtain compact analytical expressions for the outage probability (OP) and delay outage rate (DOR). Numerical results illustrate that using FAS with only one activated port into the RIS-aided communication network can greatly enhance the performance, when compared to traditional antenna systems (TAS).
The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps across mesh interfaces from the FEM stiffness bilinear form while maintaining the system's coercivity. Herein, we present two generalizations for SoftFEM that aim to improve the approximation accuracy and further reduce the discrete systems' stiffness. Firstly and most naturally, we generalize SoftFEM by adding a least-square term to the mass bilinear form. Superconvergent results of rates $h^6$ and $h^8$ for eigenvalues are established for linear uniform elements; $h^8$ is the highest order of convergence known in the literature. Secondly, we generalize SoftFEM by applying the blended Gaussian-type quadratures. We demonstrate further reductions in stiffness compared to traditional FEM and SoftFEM. The coercivity and analysis of the optimal error convergences follow the work of SoftFEM. Thus, this paper focuses on the numerical study of these generalizations. For linear and uniform elements, analytical eigenpairs, exact eigenvalue errors, and superconvergent error analysis are established. Various numerical examples demonstrate the potential of generalized SoftFEMs for spectral approximation, particularly in high-frequency regimes.
We propose a novel data-driven linear inverse model, called Colored-LIM, to extract the linear dynamics and diffusion matrix that define a linear stochastic process driven by an Ornstein-Uhlenbeck colored-noise. The Colored-LIM is a new variant of the classical linear inverse model (LIM) which relies on the white noise assumption. Similar to LIM, the Colored-LIM approximates the linear dynamics from a finite realization of a stochastic process and then solves the diffusion matrix based on, for instance, a generalized fluctuation-dissipation relation, which can be done by solving a system of linear equations. The main difficulty is that in practice, the colored-noise process can be hardly observed while it is correlated to the stochastic process of interest. Nevertheless, we show that the local behavior of the correlation function of the observable encodes the dynamics of the stochastic process and the diffusive behavior of the colored-noise. In this article, we review the classical LIM and develop Colored-LIM with a mathematical background and rigorous derivations. In the numerical experiments, we examine the performance of both LIM and Colored-LIM. Finally, we discuss some false attempts to build a linear inverse model for colored-noise driven processes, and investigate the potential misuse and its consequence of LIM in the appendices.
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