Related papers: A Scala Prototype to Generate Multigrid Solver Imp…
Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a multigrid solver is the…
The problem of finding the solution of Partial Differential Equations (PDEs) plays a central role in modeling real world problems. Over the past years, Multigrid solvers have showed their robustness over other techniques, due to its high…
Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and…
The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization…
The implementation of efficient multigrid preconditioners for elliptic partial differential equations (PDEs) is a challenge due to the complexity of the resulting algorithms and corresponding computer code. For sophisticated finite element…
Elliptic partial differential equations (PDEs) frequently arise in continuum descriptions of physical processes relevant to science and engineering. Multilevel preconditioners represent a family of scalable techniques for solving discrete…
The efficient solution of sparse, linear systems resulting from the discretization of partial differential equations is crucial to the performance of many physics-based simulations. The algorithmic optimality of multilevel approaches for…
The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to…
In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling…
This paper presents a learnable solver tailored to iteratively solve sparse linear systems from discretized partial differential equations (PDEs). Unlike traditional approaches relying on specialized expertise, our solver streamlines the…
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…
In this paper, we investigate the combination of multigrid methods and neural networks, starting from a Finite Element discretization of an elliptic PDE. Multigrid methods use interpolation operators to transfer information between…
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly,…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To…
Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and…
Fault tolerant algorithms for the numerical approximation of elliptic partial differential equations on modern supercomputers play a more and more important role in the future design of exa-scale enabled iterative solvers. Here, we combine…
Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the…
A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and…
For many linear and nonlinear systems that arise from the discretization of partial differential equations the construction of an efficient multigrid solver is a challenging task. Here we present a novel approach for the optimization of…