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This paper provides the theoretical foundation for the construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. Our construction…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly…
Gaussian processes are powerful non-parametric probabilistic models for stochastic functions. However, the direct implementation entails a complexity that is computationally intractable when the number of observations is large, especially…
For some $\epsilon > 10^{-36}$ we give a randomized $3/2-\epsilon$ approximation algorithm for metric TSP.
Matrix multiplication computation acceleration has been a research hotspot across various domains. Due to the characteristics of some applications, approximate matrix multiplication can achieve significant performance improvements without…
We study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum…
Statistical applications often involve the calculation of intractable multidimensional integrals. The Laplace formula is widely used to approximate such integrals. However, in high-dimensional or small sample size problems, the shape of the…
Consider a family of distributions $\{\pi_{\beta}\}$ where $X\sim\pi_{\beta}$ means that $\mathbb{P}(X=x)=\exp(-\beta H(x))/Z(\beta)$. Here $Z(\beta)$ is the proper normalizing constant, equal to $\sum_x\exp(-\beta H(x))$. Then…
We describe an algorithm for arbitrary-precision computation of the elementary functions (exp, log, sin, atan, etc.) which, after a cheap precomputation, gives roughly a factor-two speedup over previous state-of-the-art algorithms at…
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(\log\log n)$.…
In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let $K$ be an algebraic number field. In the first half of the paper, we calculate the running time and…
Linear approximation approaches suffer from Gibbs oscillations when approximating functions with singularities. ENO-SR resolution is a local approach avoiding oscillations and with a full order of accuracy, but a loss of regularity of the…
We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the…
An explicit solution for the generating functional of n-point functions in the planar approximation is given in terms of two sets of free-algebraic annihilation and creation operators.
Accuracy and Availability computations for a GNSS System - or combination of Systems - through Service Volume Simulations take considerable time. Therefore, the computation of the accuracy in 2D and 3D are often simplified by an approximate…
This paper presents a polynomial-time $1/2$-approximation algorithm for maximizing nonnegative $k$-submodular functions. This improves upon the previous $\max\{1/3, 1/(1+a)\}$-approximation by Ward and \v{Z}ivn\'y~(SODA'14), where…
Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function $w\left(z \right)…
We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete three-point function converges to a simple…
This work explores the neural network approximation capabilities for functions within the spectral Barron space $\mathscr{B}^s$, where $s$ is the smoothness index. We demonstrate that for functions in $\mathscr{B}^{1/2}$, a shallow neural…