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In this paper we develop the first fine-grained rounding error analysis of finite element (FE) cell kernels and assembly. The theory includes mixed-precision implementations and accounts for hardware-acceleration via matrix multiplication…

An efficient framework is conceived for fractional matrix programming (FMP) optimization problems (OPs) namely for minimization and maximization. In each generic OP, either the objective or the constraints are functions of multiple…

In this paper, we present a spherical Fast Multipole Method (sFMM) for ray tracing simulation of gravitational lensing (GL) on a curved sky. The sFMM is a non-trivial extension of the Fast Multiple Method (FMM) to sphere $\mathbb S^2$, and…

We introduce $\textsf{gradOL}$, the first gradient-based optimization framework for solving Chebyshev center problems, a fundamental challenge in optimal function learning and geometric optimization. $\textsf{gradOL}$ hinges on…

First-order methods in convex optimization offer low per-iteration cost but often suffer from slow convergence, while second-order methods achieve fast local convergence at the expense of costly Hessian inversions. In this paper, we…

Series expansions have been a cornerstone of applied mathematics and engineering for centuries. In this paper, we revisit the Taylor series expansion from a modern Machine Learning perspective. Specifically, we introduce the Fast Continuous…

Kernel methods are extensively employed for nonlinear data clustering, yet their effectiveness heavily relies on selecting suitable kernels and associated parameters, posing challenges in advance determination. In response, Multiple Kernel…

This work presents a multi-layered methodology for efficiently accelerating multimodal foundation models (MFMs). It combines hardware and software co-design of transformer blocks with an optimization pipeline that reduces computational and…

Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some…

Fast Multipole Methods (FMM) are a fundamental operation for the simulation of many physical problems. The high performance design of such methods usually requires to carefully tune the algorithm for both the targeted physics and the…

In this paper, we will introduce a new heterogeneous fast multipole method (H-FMM) for 2-D Helmholtz equation in layered media. To illustrate the main algorithm ideas, we focus on the case of two and three layers in this work. The key…

We introduce two new approximation methods for the numerical evaluation of the long-range Coulomb potential and the approximation of the resulting high dimensional Two-Electron Integrals tensor (TEI) with long-range interactions arising in…

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal…

Machine learning (ML) provides access to fast and accurate quantum chemistry (QC) calculations for various properties of interest such as excitation energies. It is often the case that high accuracy in prediction using an ML model, demands…

Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank…

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and…

Kernel methods represent some of the most popular machine learning tools for data analysis. Since exact kernel methods can be prohibitively expensive for large problems, reliable low-rank matrix approximations and high-performance…

The Fast Multipole Method (FMM) for the Poisson equation is extended to the case of non-axisymmetric problems in an axisymmetric domain, described by cylindrical coordinates. The method is based on a Fourier decomposition of the source into…

We present a novel approach for accelerating convolutions during inference for CPU-based architectures. The most common method of computation involves packing the image into the columns of a matrix (im2col) and performing general matrix…

The rapid growth of microcontroller-based IoT devices has opened up numerous applications, from smart manufacturing to personalized healthcare. Despite the widespread adoption of energy-efficient microcontroller units (MCUs) in the Tiny…