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This thesis investigates the design of algorithms for solving min-max optimization problems, which form the mathematical foundation of many modern applications in machine learning, game theory, and optimization. This work offers new…
We design new differentially private algorithms for the Euclidean k-means problem, both in the centralized model and in the local model of differential privacy. In both models, our algorithms achieve significantly improved error guarantees…
We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modeled by differential equations. The algorithm relies…
Fast matrix multiplication algorithms are asymptotically faster than the classical cubic-time algorithm, but they are often slower in practice. One important obstacle is the use of complex coefficients, which increases arithmetic overhead…
Consider the collection of all binary matrices having a specific sequence of row and column sums and consider sampling binary matrices uniformly from this collection. Practical algorithms for exact uniform sampling are not known, but there…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite…
This paper presents a new mixed finite element method for the Cahn-Hilliard equation. The well-posedness of the mixed formulation is established and the error estimates for its linearized fully discrete scheme are provided. The new mixed…
Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement…
Consider a polynomial optimization problem. Adding polynomial equations generated by the Fritz John conditions to the constraint set does not change the optimal value. As proved in [arXiv:2205.04254 (2022)], the objective polynomial has…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
A finite element solution of an ion channel dielectric continuum model such as Poisson-Boltzmann equation (PBE) and a system of Poisson-Nernst-Planck equations (PNP) requires tetrahedral meshes for an ion channel protein region, a membrane…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D…
Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residues mappings and residues. Basic ideas of our approach are the use of…
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…
Various specifiable combinatorial structures, with d extensive parameters, can be exactly sampled both by the recursive method, with linear arithmetic complexity if a heavy preprocessing is performed, or by the Boltzmann method, with…
Kaltofen has proposed a new approach in [Kaltofen 1992] for computing matrix determinants. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a…
We develop a tool that enables domain experts to quickly generate numerical solvers for emerging multi-physics phenomena starting from a high-level description based on ordinary/partial differential equations and their initial and boundary…
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…