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Hidden Markov models (HMMs) are widely used statistical models for modeling sequential data. The parameter estimation for HMMs from time series data is an important learning problem. The predominant methods for parameter estimation are…
In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential…
This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $\alpha$ ($0 < \alpha < 1$). The solution regularity in the Sobolev space is revisited, and new regularity results in the…
In many time-harmonic electromagnetic wave problems, the considered geometry exhibits an axial symmetry. In this case, by exploiting a Fourier expansion along the azimuthal direction, fully three-dimensional (3D) calculations can be carried…
We propose a new computational framework that combines the recently developed time-parallel (TP) and the compound wavelet matrix (CWM) methods. The framework, termed tpCWM, offers significant computational acceleration by making…
We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time…
Quantifying uncertainties in hyperbolic equations is a source of several challenges. First, the solution forms shocks leading to oscillatory behaviour in the numerical approximation of the solution. Second, the number of unknowns required…
We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss--Markov prior and…
We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP)…
Gaussian processes (GP) for machine learning have been studied systematically over the past two decades and they are by now widely used in a number of diverse applications. However, GP kernel design and the associated hyper-parameter…
Systems governed by partial differential equations (PDEs) require computationally intensive numerical solvers to predict spatiotemporal field evolution. While machine learning (ML) surrogates offer faster solutions, autoregressive inference…
A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this…
Stochastic gradient methods (SGMs) have been widely used for solving stochastic optimization problems. A majority of existing works assume no constraints or easy-to-project constraints. In this paper, we consider convex stochastic…
In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite…
The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using…
Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank…
Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference…
This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Gr\"onwall--Wendroff type that…
The linear primal-dual hybrid gradient (PDHG) method is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Unlike those obtained in most splitting methods, these subproblems…
We consider large linear and nonlinear fixed point problems, and solution with proximal algorithms. We show that there is a close connection between two seemingly different types of methods from distinct fields: 1) Proximal iterations for…