Related papers: $L^p$ Error Estimates for Approximation by Sobolev…
We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates…
Abundant literature has been published on approximation methods for the forward initial margin. The most popular ones being the family of regression methods. This paper describes the mathematical foundations on which these regression…
The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…
When solving partial differential equations on scattered nodes using the Radial Basis Function-generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs…
We propose a novel parameter estimation procedure that works efficiently for conditional random fields (CRF). This algorithm is an extension to the maximum likelihood estimation (MLE), using loss functions defined by Bregman divergences…
We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise…
Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…
The popular, stable, robust and computationally inexpensive cubic spline interpolation algorithm is adopted and used for finite temperature Green's function calculations of realistic systems. We demonstrate that with appropriate…
We show that, for the purpose of pricing Swaptions, the Swap rate and the corresponding Forward rates can be considered lognormal under a single martingale measure. Swaptions can then be priced as options on a basket of lognormal assets and…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
We establish precise convergence rates for semi-discrete schemes based on Radial Basis Function interpolation, as well as approximate approximation results for such schemes. Our schemes use stationary interpolation on regular grids, with…
In this article, we describe a function fitting method that has potential applications in machine learning and also prove relevant theorems. The described function fitting method is a convex minimization problem and can be solved using a…
In the equation of motion approach to the two-time Green's functions, conventional Tyablikov-type truncation of the chain of equations is rather arbitrary and apt to violate the analytical structure of Green's functions. Here, we propose a…
This paper studies approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces…
Because of the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that…
First, we extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Second, We develop a class of fast approximate generalized linear regression algorithms with respect to the spectral norm.…
Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to the geometry of the computational domain, they can…
In this paper we introduce a generalization of the classical $\Leb_2(\Rd)$-based Sobolev spaces with the help of a vector differential operator $\mathbf{P}$ which consists of finitely or countably many differential operators $P_n$ which…
In formal scattering theory, Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm-Liouville theory to compute Green functions within a rigorous mathematical theory. We shall show that both…
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…