Related papers: Robust and efficient solvers for nonlinear partial…
Chaotic free surface flows are challenging problems to simulate numerically, mainly due to the significant changes in geometry and frequent topological changes. Methods that track the evolution of the fluid in a Lagrangian formulation are a…
In this work, in the context of Linear and Quadratic Programming, we interpret Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is…
In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using…
In this paper, we propose the Adaptive Physics-Informed Neural Networks (APINNs) for accurate and efficient simulation-free Bayesian parameter estimation via Markov-Chain Monte Carlo (MCMC). We specifically focus on a class of parameter…
This paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing…
The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the…
Newton-type solvers have been extensively employed for solving a variety of nonlinear system of algebraic equations. However, for some complex nonlinear system of algebraic equations, efficiently solving these systems remains a challenging…
Sampling of sharp posteriors in high dimensions is a challenging problem, especially when gradients of the likelihood are unavailable. In low to moderate dimensions, affine-invariant methods, a class of ensemble-based gradient-free methods,…
The growing demand for solving large-scale, data-intensive linear and conic optimization problems, particularly in applications such as artificial intelligence and machine learning, has highlighted the limitations of classical interior…
The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges…
The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present $\textbf{Qu-FEM}$, a fault-tolerant era quantum algorithm for the finite element method. In contrast to…
There is growing interest in learning Fourier domain sampling strategies (particularly for magnetic resonance imaging, MRI) using optimization approaches. For non-Cartesian sampling patterns, the system models typically involve non-uniform…
The use of nonlinear PDEs has led to significant advancements in various fields, such as physics, biology, ecology, and quantum mechanics. However, finding multiple solutions for nonlinear PDEs can be a challenging task, especially when…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is…
Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has…
As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we…
Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing.…
Particle breakage due to collisional interactions plays a vital role in the development of several phenomena in science and engineering. The nonlinear collisional breakage equations (NCBEs) are a significant set of equations in this…
In this paper, we propose a domain-decomposition-based deep learning (DL) framework, named transient-CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs). The framework consists of two key…