Related papers: PDE-Based Multidimensional Extrapolation of Scalar…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
We present a class of diffusion-based algorithms to draw samples from high-dimensional probability distributions given their unnormalized densities. Ideally, our methods can transport samples from a Gaussian distribution to a specified…
We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of…
Functions with jumps and kinks typically arising from parameter dependent or stochastic hyperbolic PDEs are notoriously difficult to approximate. If the jump location in physical space is parameter dependent or random, standard…
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations…
This paper describes applications of extrapolation for the computation of coefficients in an expansion of infrared divergent integrals. An extrapolation procedure is performed with respect to a parameter introduced by dimensional…
Conventional sampling and interpolation commonly rely on discrete measurements. In this paper, we develop a theoretical framework for extrapolation of signals in higher dimensions from knowledge of the continuous waveform on bounded…
We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
The scalar normal modes of higher dimensional gravitating kink solutions are derived. By perturbing to second order the gravity and matter parts of the action in the background of a five-dimensional kink, the effective Lagrangian of the…
An efficient proximal-gradient-based method, called proximal extrapolated gradient method, is designed for solving monotone variational inequality in Hilbert space. The proposed method extends the acceptable range of parameters to obtain…
Scale selection methods based on local extrema over scale of scale-normalized derivatives have been primarily developed to be applied sparsely --- at image points where the magnitude of a scale-normalized differential expression…
Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains…
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE…
Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using…
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably…
We present a general framework for accurately evaluating finite difference operators in the presence of known discontinuities across an interface. Using these techniques, we develop simple-to-implement, second-order accurate methods for…
Deep learning inspired by differential equations is a recent research trend and has marked the state of the art performance for many machine learning tasks. Among them, time-series modeling with neural controlled differential equations…
Richardson extrapolation is a classical technique from numerical analysis that can improve the approximation error of an estimation method by combining linearly several estimates obtained from different values of one of its hyperparameters,…
Diffusion models have shown remarkable empirical success in sampling from rich multi-modal distributions. Their inference relies on numerically solving a certain differential equation. This differential equation cannot be solved in closed…