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Accurate reconstruction of probability density functions (PDFs) from data is essential in engineering applications. Classical global moment-based polynomial approximations often suffer from oscillations, instability in the tails, and…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among…
The proximal point method (PPM) is a fundamental method in optimization that is often used as a building block for designing optimization algorithms. In this work, we use the PPM method to provide conceptually simple derivations along with…
Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial…
For a large class of orthogonal basis functions, there has been a recent identification of expansion methods for computing accurate, stable approximations of a quantity of interest. This paper presents, within the context of uncertainty…
Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios where second-order optimization strategies are…
In this paper we present a first-order method that admits near-optimal convergence rates for convex/concave min-max problems while requiring a simple and intuitive analysis. Similarly to the seminal work of Nemirovski and the recent…
Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and…
Position-controlled systems driving repetitive tasks are of significant importance in industrial machinery. The electric actuators used in these systems are responsible for a large part of the global energy consumption, indicating that…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
Robotic systems must be able to quickly and robustly make decisions when operating in uncertain and dynamic environments. While Reinforcement Learning (RL) can be used to compute optimal policies with little prior knowledge about the…
The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic…
Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large…
Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has…
In this paper we develop proximal methods for statistical learning. Proximal point algorithms are useful in statistics and machine learning for obtaining optimization solutions for composite functions. Our approach exploits closed-form…
Physics and equality constrained artificial neural networks (PECANN) are grounded in methods of constrained optimization to properly constrain the solution of partial differential equations (PDEs) with their boundary and initial conditions…
We solve principal component regression (PCR), up to a multiplicative accuracy $1+\gamma$, by reducing the problem to $\tilde{O}(\gamma^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit…
The semiconductor industry faces a computational crisis in extreme ultraviolet (EUV) lithography optimization, where traditional methods consume billions of CPU hours while failing to achieve sub-nanometer precision. We present a…