Related papers: Parallelizing Explicit and Implicit Extrapolation …
Many problems in computational science and engineering involve partial differential equations and thus require the numerical solution of large, sparse (non)linear systems of equations. Multigrid is known to be one of the most efficient…
We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the…
A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…
The objective of this paper is to prove the convergence of a linear implicit multi-step numerical method for ordinary differential equations. The algorithm is obtained via Taylor approximations. The convergence is proved following the…
A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…
Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions…
In this paper, we evaluate the performance of various parallel optimization methods for Kernel Support Vector Machines on multicore CPUs and GPUs. In particular, we provide the first comparison of algorithms with explicit and implicit…
Large-scale deep learning models contribute to significant performance improvements on varieties of downstream tasks. Current data and model parallelism approaches utilize model replication and partition techniques to support the…
In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal…
Multiscale methods for second order elliptic equations based on non-overlapping domain decomposition schemes have great potential to take advantage of multi-core, state-of-the-art parallel computers. These methods typically involve solving…
We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for…
Linear-scaling electronic-structure techniques, also called O(N) techniques, rely heavily on the multiplication of sparse matrices, where the sparsity arises from spatial cut-offs. In order to treat very large systems, the calculations must…
The past years have seen widening efforts at increasing Prolog's declarativeness and expressiveness. Tabling has proved to be a viable technique to efficiently overcome SLD's susceptibility to infinite loops and redundant subcomputations.…
Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…
Solving large, sparse linear systems is a fundamental workload in scientific computing and engineering simulations, often dominating runtime and energy consumption in high-performance computing (HPC) applications. In this work, we explore…
We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which,…
Current state-of-the-art discrete optimization methods struggle behind when it comes to challenging contrast-enhancing discrete energies (i.e., favoring different labels for neighboring variables). This work suggests a multiscale approach…
We describe a parallel solver for the discretized weakly singular space-time boundary integral equation of the spatially two-dimensional heat equation. The global space-time nature of the system matrices leads to improved parallel…
Whilst there have been great advances in HPC hardware and software in recent years, the languages and models that we use to program these machines have remained much more static. This is not from a lack of effort, but instead by virtue of…
The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As…