Related papers: High order approximations and simulation schemes f…

We adapt the quasi-monotone method from [2] for composite convex minimization in the stochastic setting. For the proposed numerical scheme we derive the optimal convergence rate in terms of the last iterate, rather than on average as it is…

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 design an algorithm which finds an $\epsilon$-approximate stationary point (with $\|\nabla F(x)\|\le \epsilon$) using $O(\epsilon^{-3})$ stochastic gradient and Hessian-vector products, matching guarantees that were previously available…

A novel dynamical inertial Newton system, which is called Hessian-driven Nesterov accelerated gradient (H-NAG) flow is proposed. Convergence of the continuous trajectory are established via tailored Lyapunov function, and new first-order…

We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging…

We develop a two-dimensional high-order numerical scheme that exactly preserves and captures the moving steady states of the shallow water equations with topography or Manning friction. The high-order accuracy relies on a suitable…

In this paper we describe a numerical algorithm for integrating the multicomponent, reacting, compressible Navier-Stokes equations, targeted for direct numerical simulation of combustion phenomena. The algorithm addresses two shortcomings…

Approximate second-order optimization methods often exhibit poorer generalization compared to first-order approaches. In this work, we look into this issue through the lens of the loss landscape and find that existing second-order methods…

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by L\'evy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the…

In this work, we consider bilevel optimization when the lower-level problem is strongly convex. Recent works show that with a Hessian-vector product (HVP) oracle, one can provably find an $\epsilon$-stationary point within…

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…

We propose a quasi-Monte Carlo algorithm for pricing knock-out and knock-in barrier options under the Heston (1993) stochastic volatility model. This is done by modifying the LT method from Imai and Tan (2006) for the Heston model such that…

In this paper, we propose a high-order extension of the multiscale method introduced by the authors in [SIAM J. Numer. Anal., 63(4) (2025), pp. 1617--1641] for heterogeneous Stokes problems, while also providing several other improvements,…

We present a unified convergence analysis for first order convex optimization methods using the concept of strong Lyapunov conditions. Combining this with suitable time scaling factors, we are able to handle both convex and strong convex…

We introduce an affine extension of the Heston model where the instantaneous variance process contains a jump part driven by $\alpha$-stable processes with $\alpha\in(1,2]$. In this framework, we examine the implied volatility and its…

We consider the numerical solution of Hamilton-Jacobi-Bellman equations arising in stochastic control theory. We introduce a class of monotone approximation schemes relying on monotone interpolation. These schemes converge under very weak…

We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that…

We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according…

A new method to represent and approximate rotation matrices is introduced. The method represents approximations of a rotation matrix $Q$ with linearithmic complexity, i.e. with $\frac{1}{2}n\lg(n)$ rotations over pairs of coordinates,…

Large-scale optimization problems require algorithms both effective and efficient. One such popular and proven algorithm is Stochastic Gradient Descent which uses first-order gradient information to solve these problems. This paper studies…