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Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton's method on polynomial systems.…
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per…
This paper addresses the study of derivative-free smooth optimization problems, where the gradient information on the objective function is unavailable. Two novel general derivative-free methods are proposed and developed for minimizing…
Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems. To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic…
The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of…
In nonsmooth optimization, a negative subgradient is not necessarily a descent direction, making the design of convergent descent methods based on zeroth-order and first-order information a challenging task. The well-studied bundle methods…
The bilateral filter is an edge-preserving smoother that has diverse applications in image processing, computer vision, computer graphics, and computational photography. The filter uses a spatial kernel along with a range kernel to perform…
Classical theory for quasi-Newton schemes has focused on smooth deterministic unconstrained optimization while recent forays into stochastic convex optimization have largely resided in smooth, unconstrained, and strongly convex regimes.…
We develop a geometric and analytic framework for polynomial partial differential equations posed on thin annuli in the plane. Using renormalized Sobolev inner products, we construct Sobolev orthogonal polynomial bases adapted to the thin…
Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is…
A stochastic-gradient-based interior-point algorithm for minimizing a continuously differentiable objective function (that may be nonconvex) subject to bound constraints is presented, analyzed, and demonstrated through experimental results.…
This paper presents several new algorithms for the regularized reconstruction of a surface from its measured gradient field. By taking a matrix-algebraic approach, we establish general framework for the regularized reconstruction problem…
Group synchronization is a fundamental task involving the recovery of group elements from pairwise measurements. For orthogonal group synchronization, the most common approach reformulates the problem as a constrained nonconvex optimization…
Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp-Massey…
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss-Seidel iterative solution of linear equations in the precision matrix. Specifically, the…
We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert…
We present a computationally efficient algorithm for stable numerical differentiation from noisy, uniformly-sampled data on a bounded interval. The method combines multi-interval Fourier extension approximations with an adaptive domain…
We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. Where the usual CSPs from computer science and optimization have real-valued score functions, and partition functions from physics have monomials,…
We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently.…
The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and…