Related papers: Generalized explicit pseudo two-step Runge-Kutta-N…
In this paper, we propose a class of super-schemes for efficiently solving nonlinear unconstrained optimization problems. The proposed approach introduces two novel choices of step-size parameters, leading to efficient descent directions…
We introduce new multilevel methods for solving large-scale unconstrained optimization problems. Specifically, the philosophy of multilevel methods is applied to Newton-type methods that regularize the Newton sub-problem using second order…
Graph Neural Networks (GNNs) have made significant advancements in node classification, but their success relies on sufficient labeled nodes per class in the training data. Real-world graph data often exhibits a long-tail distribution with…
We present MDIRK: a Multifluid second-order Diagonally-Implicit Runge-Kutta method to study momentum transfer between gas and an arbitrary number ($N$) of dust species. The method integrates the equations of hydrodynamics with an Implicit…
In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which…
Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size,…
A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some…
A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations by general one step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second…
Worst-case complexity guarantees for nonconvex optimization algorithms have been a topic of growing interest. Multiple frameworks that achieve the best known complexity bounds among a broad class of first- and second-order strategies have…
This paper investigates the performance of a subclass of exponential integrators, specifically explicit exponential Runge--Kutta methods. It is well known that third-order methods can suffer from order reduction when applied to linearized…
We present novel entropy-conservative and entropy-stable multirate Runge-Kutta methods based on Paired Explicit Runge-Kutta (P-ERK) schemes with relaxation for conservation laws and related systems of partial differential equations.…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
We propose a new method to prove the partitioned Runge--Kutta methods with symplectic conditions for determinate and stochastic Hamiltonian systems are symplectic. We utilize Gr\"obner basis technology which is the one of symbolic…
This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK…
Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized…
Historically speaking, it is hard to balance the global and local efficiency of second-order optimization algorithms. For instance, the classical Newton's method possesses excellent local convergence but lacks global guarantees, often…
Many machine learning models involve solving optimization problems. Thus, it is important to deal with a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention…
This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T.…
We present the formulation and optimization of a Runge-Kutta-type time-stepping scheme for solving the shallow water equations, aimed at substantially increasing the effective allowable time-step over that of comparable methods. This…
This paper introduces a novel paradigm for constructing linearly implicit and high-order unconditionally energy-stable schemes for general gradient flows, utilizing the scalar auxiliary variable (SAV) approach and the additive Runge-Kutta…