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Generalized Benders decomposition (GBD) is a globally optimal algorithm for mixed integer nonlinear programming (MINLP) problems, which are NP-hard and can be widely found in the area of wireless resource allocation. The main idea of GBD is…

Kernelization algorithms, usually a preprocessing step before other more traditional algorithms, are very special in the sense that they return (reduced) instances, instead of final results. This characteristic excludes the freedom of…

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of…

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying…

Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices…

Gaussian graphical models are widely used to infer dependence structures. Bayesian methods are appealing to quantify uncertainty associated with structural learning, i.e., the plausibility of conditional independence statements given the…

Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high…

Cartoon-texture image decomposition is a critical preprocessing problem bottlenecked by the numerical intractability of classical variational or optimization models and the tedious manual tuning of global regularization parameters.We…

Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the…

This paper addresses the challenge of geometric quality assurance in manufacturing, particularly when human assessment is required. It proposes using Blender, an open-source simulation tool, to create synthetic datasets for machine learning…

Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and…

Although support vector machines (SVMs) are theoretically well understood, their underlying optimization problem becomes very expensive, if, for example, hundreds of thousands of samples and a non-linear kernel are considered. Several…

Satisfiability Modulo Theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulas. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints.…

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide…

In this paper, we propose a graph classification approach for automatically determining whether to use a monolithic or a decomposition-based solution method. In this approach, an optimization problem is represented as a graph that captures…

The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea…

The Shifted Boundary Method (SBM) trades some part of the burden of body-fitted meshing for increased algebraic complexity. While the resulting linear systems retain the standard $\mathcal{O}(h^{-2})$ conditioning of second-order operators,…

Calibration or parameter identification is used with computational mechanics models related to observed data of the modeled process to find model parameters such that good similarity between model prediction and observation is achieved. We…

Applications of machine learning tools to problems of physical interest are often criticized for producing sensitivity at the expense of transparency. To address this concern, we explore a data planing procedure for identifying combinations…