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In this study, two classes of methods including statistical and variational data assimilation algorithms will be described. In statistical methods, the model state is updated sequentially based on the previous estimate. Variational methods,…

Modeling and predicting the dynamics of complex multiscale systems remains a significant challenge due to their inherent nonlinearities and sensitivity to initial conditions, as well as limitations of traditional machine learning methods…

Accurately representing surface weather at the sub-kilometer scale is crucial for optimal decision-making in a wide range of applications. This motivates the use of statistical techniques to provide accurate and calibrated probabilistic…

An efficient method, preconditioned conjugate gradient method with a filtering function (PCG-F), is proposed for solving iteratively the Dirac equation in 3D lattice space for nuclear systems. The filtering function is adopted to avoid the…

This paper introduces a novel variational Bayesian method that integrates Tucker decomposition for efficient high-dimensional inverse problem solving. The method reduces computational complexity by transforming variational inference from a…

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is…

We study the generalization error of randomized learning algorithms -- focusing on stochastic gradient descent (SGD) -- using a novel combination of PAC-Bayes and algorithmic stability. Importantly, our generalization bounds hold for all…

We propose $\textsf{ScaledGD($\lambda$)}$, a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor…

We introduce a novel Bayesian approach for both covariate selection and sparse precision matrix estimation in the context of high-dimensional Gaussian graphical models involving multiple responses. Our approach provides a sparse estimation…

Classical radar detection techniques rely on adaptive detectors that estimate the noise covariance matrix from target-free secondary data. While effective in Gaussian environments, these methods degrade in the presence of clutter, which is…

We develop two novel stochastic variance-reduction methods to approximate solutions of a class of nonmonotone [generalized] equations. Our algorithms leverage a new combination of ideas from the forward-reflected-backward splitting method…

Stochastic gradient descent (SGD) is commonly used for optimization in large-scale machine learning problems. Langford et al. (2009) introduce a sparse online learning method to induce sparsity via truncated gradient. With high-dimensional…

We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical…

We present preconditioned stochastic gradient descent (SGD) algorithms for the $\ell_1$ minimization problem $\min_{x}\|A x - b\|_1$ in the overdetermined case, where there are far more constraints than variables. Specifically, we have $A…

Provably solving stochastic convex optimization problems with constraints is essential for various problems in science, business, and statistics. Recently proposed XOR-Stochastic Gradient Descent (XOR-SGD) provides a convergence rate…

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and…

Statistical downscaling of global climate models (GCMs) allows researchers to study local climate change effects decades into the future. A wide range of statistical models have been applied to downscaling GCMs but recent advances in…

The projected gradient descent (PGD) method has shown to be effective in recovering compressed signals described in a data-driven way by a generative model, i.e., a generator which has learned the data distribution. Further reconstruction…

Recent variational Bayes methods for geospatial regression, proposed as an alternative to computationally expensive Markov chain Monte Carlo (MCMC) sampling, have leveraged Nearest Neighbor Gaussian processes (NNGP) to achieve scalability.…

The spatio-temporal interpolation of large geophysical datasets has historically been addressed by Optimal Interpolation (OI) and more sophisticated equation-based or data-driven Data Assimilation (DA) techniques. Recent advances in the…