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We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
We have proposed new algorithms for the numerical integration of the equations of motion for classical spin systems. In close analogy to symplectic integrators for Hamiltonian equations of motion used in Molecular Dynamics these algorithms…
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high…
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization…
A transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the…
The rapid advancements in high-dimensional statistics and machine learning have increased the use of first-order methods. Many of these methods can be regarded as instances of the proximal point algorithm. Given the importance of the…
Accelerated gradient methods have had significant impact in machine learning -- in particular the theoretical side of machine learning -- due to their ability to achieve oracle lower bounds. But their heuristic construction has hindered…
Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical…
Using Suzuki-Trotter decompositions of exponential operators we describe new algorithms for the numerical integration of the equations of motion for classical spin systems. These techniques conserve spin length exactly and, in special…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
We propose a novel second-order ODE as the continuous-time limit of a Riemannian accelerated gradient-based method on a manifold with curvature bounded from below. This ODE can be seen as a generalization of the ODE derived for Euclidean…
This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to…
A wide range of implicit time integration methods, including multi-step, implicit Runge-Kutta, and Galerkin finite-time element schemes, is evaluated in the context of chaotic dynamical systems. The schemes are applied to solve the Lorenz…
Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility…
This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in…
We present a practical algorithm based on symplectic splitting methods to integrate numerically in time the Schr\"odinger equation. When discretized in space, the Schr\"odinger equation can be recast as a classical Hamiltonian system…
We present a new efficient computational approach for time-dependent first-order Hamilton-Jacobi-Bellman PDEs. Since our method is based on a time-implicit Eulerian discretization, the numerical scheme is unconditionally stable, but…
Modern N-body techniques for planetary dynamics are generally based on symplectic algorithms specially adapted to the Kepler problem. These methods have proven very useful in studying planet formation, but typically require the timestep for…
We describe an algorithm for long-term planetary orbit integrations, including the dominant post-Newtonian effects, that employs individual timesteps for each planet. The algorithm is symplectic and exhibits short-term errors that are…