Related papers: Mixed-Precision in adaptive Runge-Kutta method for…

The family of PDE-constrained LDDMM methods is emerging as a particularly interesting approach for physically meaningful diffeomorphic transformations. The original combination of Gauss--Newton--Krylov optimization and Runge--Kutta…

Mixed-precision computing has become increasingly important in modern high-performance computing and machine learning applications. When implementing custom mixed-precision functions -- such as fused operators, optimized GPU kernels, or…

Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…

The GMRES method is used to solve sparse, non-symmetric systems of linear equations arising from many scientific applications. The solver performance within a single node is memory bound, due to the low arithmetic intensity of its…

Finite differences and Runge-Kutta time stepping schemes used in Computational AeroAcoustics simulations are often optimized for low dispersion and dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems in order to…

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order…

There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to specific problems. Some of them have interesting properties, such as stability for stiff problems or symplectic capability for problems with energy…

Low precision arithmetic, in particular half precision floating point arithmetic, is now available in commercial hardware. Using lower precision can offer significant savings in computation and communication costs with proportional savings…

We present a new method for developing time step controllers based on a technique from the field of machine learning. This method is applicable to stable time integrators that have an embedded scheme, i.e., that have local error estimation…

Neural ordinary differential equations (neural ODE) are powerful continuous-time machine learning models for depicting the behavior of complex dynamical systems, but their verification remains challenging due to limited reachability…

In this work, we introduce high-order Basis-Update & Galerkin (BUG) integrators based on explicit Runge-Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator to arbitrary…

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…

Finite precision computations using digital computers involve the following inherent errors: (1) Round-off error of finite precision computations (2) Binary computer arithmetic precludes exact number representation of traditional decimal…

This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric…

Multirate integration is an increasingly relevant tool that enables scientists to simulate multiphysics systems. Existing multirate methods are designed for equations whose fast and slow variables can be linearly separated using additive or…

This study was aimed at simultaneously achieving sufficient accuracy and high performance for general matrix multiplications. Recent architectures, such as NVIDIA GPUs, feature high-performance units designed for low-precision matrix…

Precision tuning or customized precision number representations is emerging, in these recent years, as one of the most promising techniques that has a positive impact on the footprint of programs concerning energy consumption, bandwidth…

Using lower precision in algorithms can be beneficial in terms of reducing both computation and communication costs. Motivated by this, we aim to further the state-of-the-art in developing and analyzing mixed precision variants of iterative…

This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions…

An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes.…