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We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded…
We present a local interpolation method in four dimensions utilising cubic splines. An extension of the three-dimensional tricubic method, the interpolated function has C$^1$ continuity and its partial derivatives are analytically…
Consider a random graph process with $n$ vertices corresponding to points $v_{i} \sim {Unif}[0,1]$ embedded randomly in the interval, and where edges are inserted between $v_{i}, v_{j}$ independently with probability given by the graphon…
Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness…
In STOC'95 [ADMSS'95] Arya et al. showed that any set of $n$ points in $\mathbb R^d$ admits a $(1+\epsilon)$-spanner with hop-diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges). They also gave a…
We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$ and $\alpha > 1/p$, where…
We develop two new ideas for interpolation on $\mathbb{S}^2$. In this first part, we will introduce a simple interpolation method named \textit{Spherical Interpolation of orDER} $n$ (SIDER-$n$) that gives a $C^{n}$ interpolant given $n \geq…
When modeling propagation and scattering phenomena using integral equations discretized by the boundary element method, it is common practice to approximate the boundary of the scatterer with a mesh comprising elements of size approximately…
We define a discrete version of the bilinear spherical maximal function, and show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$,…
The iodine-cell technique, which is known to be efficient in precisely establishing Doppler velocity shifts, was once applied by the author to measuring the solar differential rotation based on full-disk spectroscopic observations (Takeda…
Radial basis functions are a common mathematical tool used to construct a smooth interpolating function from a set of data points. A spatial prior based on thin-plate spline radial basis functions can be easily implemented resulting in a…
Standard interpolation techniques are implicitly based on the assumption that the signal lies on a single homogeneous domain. In contrast, many naturally occurring signals lie on an inhomogeneous domain, such as brain activity associated to…
We present error estimates of the fully semi-Lagrangian scheme with high-order interpolation operators, solving the initial value problems for the one-dimensional nonlinear diffusive conservation laws, including the Burgers equations. We…
We obtain the spline recovery method on a $d$-dimensional simplex $T$ that uses as information values and gradients of a function $f$ at the vertices of $T$ and is optimal for recovery of $f({\bf w})$ at every point ${\bf w}$ of an…
This paper establishes the generalization error of pooled min-$\ell_2$-norm interpolation in transfer learning where data from diverse distributions are available. Min-norm interpolators emerge naturally as implicit regularized limits of…
Radial basis functions are typically used when discretization sche-mes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in…
We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger equation we analyze…
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…
We consider the problem of interpolating a function given on scattered points using Hermite-Birkhoff formulas on the sphere and other manifolds. We express each proposed interpolant as a linear combination of basis functions, the…
Periodic splines are a special kind of splines that are defined over a set of knots over a circle and are adequate for solving interpolation problems related to closed curves. This paper presents a method of implementing the objects…