Related papers: A new weak approximation scheme of stochastic diff…

We study Euler-type discrete-time schemes for the rough Heston model, which can be described by a stochastic Volterra equation (with non-Lipschtiz coefficient functions), or by an equivalent integrated variance formulation. Using weak…

For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore…

We study the connections between ordinary differential equations and optimization algorithms in a non-Euclidean setting. We propose a novel accelerated algorithm for minimising convex functions over a convex constrained set. This algorithm…

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the…

We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient…

Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…

This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the…

The existing discrete variational derivative method is only second-order accurate and fully implicit. In this paper, we propose a framework to construct an arbitrary high-order implicit (original) energy stable scheme and a second-order…

We propose a straightforward and effective method for discretizing multi-dimensional diffusion processes as an extension of Milstein scheme. The new scheme is explicitly given and can be simulated using Gaussian variates, requiring the same…

We introduce a family of stochastic optimization methods based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their…

We present novel algorithms for simulation optimization using random directions stochastic approximation (RDSA). These include first-order (gradient) as well as second-order (Newton) schemes. We incorporate both continuous-valued as well as…

We propose a new financial model, the stochastic volatility model with sticky drawdown and drawup processes (SVSDU model), which enables us to capture the features of winning and losing streaks that are common across financial markets but…

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC)…

This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step and thus the…

This paper aims to investigate the distributed stochastic optimization problems on compact embedded submanifolds (in the Euclidean space) for multi-agent network systems. To address the manifold structure, we propose a distributed…

We study the construction and convergence of semi-explicit and iterative decoupling schemes for an elliptic-parabolic problem using higher-order Runge-Kutta methods. For the semi-explicit schemes, which are constructed using a nearby delay…

This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical…

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…

This work focuses on the numerical study of a recently published class of Runge-Kutta methods designed for mixed-precision arithmetic. We employ the methods in solving partial differential equations on modern hardware. In particular we…

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in…