Related papers: Soft Riemann-Hilbert problems and planar orthogona…
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281-354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract…
We study a family of monic orthogonal polynomials which are orthogonal with respect to the varying, complex valued weight function, $\exp(nsz)$, over the interval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family of polynomials…
We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense…
It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our…
This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex…
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical…
We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $w(x; t) = {x^\alpha }{\mathrm e^{ - x}}{(x + t)^\lambda },\; x\in \mathbb{R}^{+} $ with parameters $\alpha > -1,\; t > 0$ and…
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks.…
We study the defocusing nonlinear Schr\"odinger equation in the quarter plane with asymptotically periodic boundary values. By studying an associated Riemann-Hilbert problem and employing nonlinear steepest descent arguments, we construct…
We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint…
The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work,…
We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the…
A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a…
This paper addresses a class of nonsmooth and nonconvex optimization problems defined on complete Riemannian manifolds. The objective function has a composite structure, combining convex, differentiable, and lower semicontinuous terms,…
Many classical and modern machine learning algorithms require solving optimization tasks under orthogonality constraints. Solving these tasks with feasible methods requires a gradient descent update followed by a retraction operation on the…
The purpose of this paper is to push forward the theory of operator-valued Riemann Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method \textit{\'{a} la} Deift-Zhou.…