Related papers: On the Ideal Interpolation Operator in Algebraic M…
A free semigroupoid algebra is the weak operator topology closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a…
Optimization decomposition methods are a fundamental tool to develop distributed solution algorithms for large scale optimization problems arising in fields such as machine learning and optimal control. In this paper, we present an…
We present a multigrid iterative algorithm for solving a system of coupled free boundary problems for pricing American put options with regime-switching. The algorithm is based on our recently developed compact finite difference scheme…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Given a pair of smooth transversally intersecting manifolds in some ambient manifold, we construct an operator algebra generated by pseudodifferential operators and the (co)boundary operators associated with the submanifolds. We show that…
We introduce and develop the notion of hyper-ideals of multilinear operators between Banach spaces. While the well studied notion of ideals of multilinear operators (multi-ideals) relies on the composition with linear operators, the notion…
Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently…
In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as…
Let $G$ be a commutative algebraic group embedded in projective space and $\Gamma$ a finitely generated subgroup of $G$. From these data we construct a chain of algebraic subgroups of $G$ which is intimately related to obstructions to…
Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural…
This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a…
The Floater--Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of…
Algebraic curve interpolation is described by specifying the location of N points in the plane and constructing an algebraic curve of a function f that should pass through them. In this paper, we propose a novel approach to construct the…
Model Order Reduction is a key technology for industrial applications in the context of digital twins. Key requirements are non-intrusiveness, physics-awareness, as well as robustness and usability. Operator inference based on least-squares…
Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…
In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
In the era of deep learning, understanding over-fitting phenomenon becomes increasingly important. It is observed that carefully designed deep neural networks achieve small testing error even when the training error is close to zero. One…
Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely…
To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional…