Related papers: Randomized Complexity of Parametric Integration an…
We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for $r,d_1,d_2\in{\mathbb N}$, $1\le p,q\le \infty$, $D_1= [0,1]^{d_1}$, and $D_2= [0,1]^{d_2}$ we are…
Recently the adaption problem of Information-Based Complexity (IBC) for linear problems in the randomized setting was solved in Heinrich (J. Complexity 82, 2024, 101821). Several papers treating further aspects of this problem followed.…
We study the randomized $n$-th minimal errors (and hence the complexity) of vector valued approximation. In a recent paper by the author [Randomized complexity of parametric integration and the role of adaption I. Finite dimensional case…
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance…
We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
Gradient information is widely useful and available in applications, and is therefore natural to include in the training of neural networks. Yet little is known theoretically about the impact of Sobolev training -- regression with both…
When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with…
We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…
In this work, we introduce a machine/deep learning methodology to solve parametric integrals. Besides classical machine learning approaches, we consider a differential learning framework that incorporates derivative information during…
Random smoothing data augmentation is a unique form of regularization that can prevent overfitting by introducing noise to the input data, encouraging the model to learn more generalized features. Despite its success in various…
We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration…
We analyze univariate oscillatory integrals for the standard Sobolev spaces $H^s$ of periodic and non-periodic functions with an arbitrary integer $s\ge1$. We find matching lower and upper bounds on the minimal worst case error of…
We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral…
We consider the non-parametric Poisson regression problem where the integer valued response $Y$ is the realization of a Poisson random variable with parameter $\lambda(X)$. The aim is to estimate the functional parameter $\lambda$ from…
A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits…
We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing…
In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error…
We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of $s$-variate functions. Here $s$ is large including $s=\infty$. Under the assumption of sufficiently fast decaying weights, we prove in a…