Related papers: Reducing triangular systems of ODEs with rational …
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…
High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's…
This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
We propose a sufficient condition of the convergence of a power-log series that formally satisfies an algebraic ordinary differential equation (ODE) of arbitrary order. A general form and properties of the functional coefficients of such a…
Computing reduced-order models using non-intrusive methods is particularly attractive for systems that are simulated using black-box solvers. However, obtaining accurate data-driven models can be challenging, especially if the underlying…
There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to…
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise,…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
A Schwarzschild type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for…
This paper derives the analytic and practicable expression of general solution of vacuum Regge-Wheeler equation via Homotopy Analysis Method.
Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been…
In this short note, we review several one-dimensional problems such as those involving linear Schroedinger equation, variable-coefficient Helmholtz equation, Zakharov-Shabat system and Kubelka-Munk equations. We show that they all can be…
Reconsideration of the Regge-Wheeler equation is processed by using the Painlev\'{e} coordinate and "good" timelier to define the initial time. We find that: the Regge-Wheeler equation could has positive imaginary frequency. Because the…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the…