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We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used…
We present PetIGA, a code framework to approximate the solution of partial differential equations using isogeometric analysis. PetIGA can be used to assemble matrices and vectors which come from a Galerkin weak form, discretized with…
The demand for substantial increases in the spatial resolution of global weather- and climate- prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale…
As one of open-source codes widely used in computational ocean acoustics, FOR3D can provide a very good estimate for underwater acoustic propagation. In this paper, we propose a performance optimization and parallelization to speed up the…
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in…
This paper presents the implementation of a HLLC finite volume solver using GPU technology for the solution of shallow water problems in two dimensions. It compares both CPU and GPU approaches for implementing all the solver's steps. The…
The study deals with the parallelization of finite element based Navier-Stokes codes using domain decomposition and state-ofart sparse direct solvers. There has been significant improvement in the performance of sparse direct solvers.…
The ISO C++17 standard introduces \emph{parallel algorithms}, a parallel programming model promising portability across a wide variety of parallel hardware including multi-core CPUs, GPUs, and FPGAs. Since 2019, the NVIDIA HPC SDK compiler…
We describe a new library named picasso, which implements a unified framework of pathwise coordinate optimization for a variety of sparse learning problems (e.g., sparse linear regression, sparse logistic regression, sparse Poisson…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
Identifying unknown differential equations from a given set of discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable, and nonlinearity and differential equations with varying…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
While recent AI-for-math has made strides in pure mathematics, areas of applied mathematics, particularly PDEs, remain underexplored despite their significant real-world applications. We present PDE-Controller, a framework that enables…
This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by…