Related papers: Mechanizing Olver's Error Arithmetic
We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions…
In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval…
This paper proves error estimates for $H^2$ conforming finite elements for equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). for an adapted scheme originally proposed…
We consider linear dynamical systems under floating-point rounding. In these systems, a matrix is repeatedly applied to a vector, but the numbers are rounded into floating-point representation after each step (i.e., stored as a…
For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of…
We explore the link between data representation and soft errors in dot products. We present an analytic model for the absolute error introduced should a soft error corrupt a bit in an IEEE-754 floating-point number. We show how this finding…
We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating…
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization…
Randomized smoothing is sound when using infinite precision. However, we show that randomized smoothing is no longer sound for limited floating-point precision. We present a simple example where randomized smoothing certifies a radius of…
Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques…
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs…
We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on…
In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these…
Large-scale numerical computations make increasing use of low-precision (LP) floating point formats and mixed precision arithmetic, which can be enhanced by the technique of stochastic rounding (SR), that is, rounding an intermediate…
A mathematical model is a function taking certain arguments and returning a theoretical prediction of a feature of a physical system. The arguments to the mathematical model can be split into two groups; (a) controllable variables of the…
Errors in floating-point programs can lead to severe consequences, particularly in critical domains such as military, aerospace, and financial systems, making their repair a crucial research problem. In practice, some errors can be fixed…
Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between of two real $n$-vectors. We derive probabilistic perturbation bounds, as well as probabilistic…
Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient,…
This paper presents a method that leverages vehicle motion constraints to refine data associations in a point-based radar odometry system. By using the strong prior on how a non-holonomic robot is constrained to move smoothly through its…
This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly…