Josef Dick
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View article: Special Issue of the Journal of Complexity
Special Issue of the Journal of Complexity Open
View article: A lattice algorithm with multiple shifts for function approximation in Korobov spaces
A lattice algorithm with multiple shifts for function approximation in Korobov spaces Open
In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ $\mat…
View article: A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity
A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity Open
View article: Bayesian inference calibration of the modulus of elasticity
Bayesian inference calibration of the modulus of elasticity Open
This work uses the Bayesian inference technique to infer the Young modulus from the stochastic linear elasticity equation. The Young modulus is modeled by a finite Karhunen Loéve expansion, while the solution to the linear elasticity equat…
View article: A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity
A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity Open
Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lamé coefficient $λ\to\infty$, or equivalently as the Poisson ratio $ν\to1/2$. Th…
View article: Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values
Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values Open
In this work, we explore a time-fractional diffusion equation of order $$\alpha \in (0,1)$$ with a stochastic diffusivity coefficient $${\kappa }$$ . We focus on efficient estimation of the expected values of linear functionals …
View article: On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences
On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences Open
The discrepancy of a point set quantifies how well the points are distributed, with low-discrepancy point sets demonstrating exceptional uniform distribution properties. Such sets are integral to quasi-Monte Carlo methods, which approximat…
View article: Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation
Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation Open
We study numerical integration over bounded regions in $$\mathbb {R}^s$$ , $$s \ge 1$$ , with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is deriv…
View article: On the quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences
On the quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences Open
We study the quasi-uniformity properties of digital nets, a class of quasi-Monte Carlo point sets. Quasi-uniformity is a space-filling property used for instance in experimental designs and radial basis function approximation. However, it …
View article: Locking-Free Training of Physics-Informed Neural Network for Solving Nearly Incompressible Elasticity Equations
Locking-Free Training of Physics-Informed Neural Network for Solving Nearly Incompressible Elasticity Equations Open
View article: Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence
Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence Open
In this short note, we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important r…
View article: Quasi-Monte Carlo integration over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg" display="inline" id="d1e534"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math> based on digital nets
Quasi-Monte Carlo integration over based on digital nets Open
View article: Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence
Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence Open
In this short note we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important ro…
View article: Some tractability results for multivariate integration in subspaces of the Wiener algebra
Some tractability results for multivariate integration in subspaces of the Wiener algebra Open
In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the $d$-dimensional unit cube. We show that intractability holds for multivariate integration in the st…
View article: QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces
QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces Open
We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavi…
View article: Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values
Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values Open
In this work, we explore a time-fractional diffusion equation of order $α\in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the par…
View article: A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity
A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity Open
Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter $λ\to\infty$, or equivalently as the Poisson ratio $ν\t…
View article: An $$\alpha $$-Robust and Second-Order Accurate Scheme for a Subdiffusion Equation
An $$\alpha $$-Robust and Second-Order Accurate Scheme for a Subdiffusion Equation Open
We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order $$\alpha \in (0,1)$$ . The basic idea of our scheme is based on local integration follo…
View article: High-order QMC nonconforming FEMs for nearly incompressible planar stochastic elasticity equations
High-order QMC nonconforming FEMs for nearly incompressible planar stochastic elasticity equations Open
In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lamé parameters which are parameterized by a countably infinite number of terms in separate expansions. We estimated the exp…
View article: An $α$-robust and second-order accurate scheme for a subdiffusion equation
An $α$-robust and second-order accurate scheme for a subdiffusion equation Open
We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$α\in (0,1)$. The basic idea of our scheme is based on local integration followed by linear inte…
View article: Sparse grid approximation of nonlinear SPDEs: The Landau--Lifshitz--Gilbert equation
Sparse grid approximation of nonlinear SPDEs: The Landau--Lifshitz--Gilbert equation Open
We show convergence rates for a sparse grid approximation of the distribution of solutions of the stochastic Landau-Lifshitz-Gilbert equation. Beyond being a frequently studied equation in engineering and physics, the stochastic Landau-Lif…
View article: Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties
Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties Open
We explore a linear inhomogeneous elasticity equation with random Lamé parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (cons…
View article: The fast reduced QMC matrix-vector product
The fast reduced QMC matrix-vector product Open
We study the approximation of integrals $\int_D f(\boldsymbol{x}^\top A) \mathrm{d} μ(\boldsymbol{x})$, where $A$ is a matrix, by quasi-Monte Carlo (QMC) rules $N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$. We are interested in case…
View article: Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation
Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation Open
We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$ with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic …
View article: Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold
Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold Open
View article: Deep Learning Extraction of the Temperature-Dependent Parameters of Bulk Defects
Deep Learning Extraction of the Temperature-Dependent Parameters of Bulk Defects Open
Bulk defects in silicon solar cells are a key contributor to loss of efficiency. To detect and identify those defects, temperature- and injection-dependent lifetime spectroscopy is usually used, and the defect parameters are traditionally …
View article: Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate
Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate Open
We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that…
View article: Elasticity equations with random domains—the shape derivative approach
Elasticity equations with random domains—the shape derivative approach Open
In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries and we apply these equations to modelling human corneas. References R. C. Augustyn, D. Nankivil, A. Mohamed, B. Maceo, F. Pierre, and J.-M. Pa…
View article: A new regularization for sparse optimization
A new regularization for sparse optimization Open
Several numerical studies have shown that non-convex sparsity-induced regularization can outperform the convex ℓ1-penalty. In this article, we introduce a new non-convex and non-smooth regularization. This new regularization is a continuou…
View article: Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate
Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate Open
We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randoml…