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Quantum trajectory methods can be used for a wide range of open quantum systems to solve the master equation by unraveling the density operator evolution into individual stochastic trajectories in Hilbert space. This C++ class library…
This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank…
Modern deep learning algorithms use variations of gradient descent as their main learning methods. Gradient descent can be understood as the simplest Ordinary Differential Equation (ODE) solver; namely, the Euler method applied to the…
This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned…
Quantum computers are known to provide an exponential advantage over classical computers for the solution of linear differential equations in high-dimensional spaces. Here, we present a quantum algorithm for the solution of nonlinear…
Nonlinear parabolic equations are central to numerous applications in science and engineering, posing significant challenges for analytical solutions and necessitating efficient numerical methods. Exponential integrators have recently…
We describe how to apply adjoint sensitivity methods to backward Monte-Carlo schemes arising from simulations of particles passing through matter. Relying on this, we demonstrate derivative based techniques for solving inverse problems for…
It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel…
We prove that Runge-Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments -- based on spectral analysis, resolvent condition or strong…
The dynamics of open quantum systems is governed by the Lindblad master equation, which provides a consistent framework for incorporating environmental effects into the evolution of the system. Since exact solutions are rarely available,…
Dynamic systems have a fundamental relevance in the description of physical phenomena. The search for more accurate and faster numerical integration methods for the resolution of such systems is, therefore, an important topic of research.…
A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…
When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order…
Obtaining exact solutions to the Schr\"odinger equation in complex quantum systems poses significant challenges. In this context, numerical methods emerge as valuable tools for analyzing such systems. This article proposes a numerical…
Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical…
Many physical, biological or chemical systems are modeled by ordinary differential equations (ODEs) and finding their solution is an every-day-task for many scientists. Here, we introduce a new C++ library dedicated to find numerical…
In this paper, we develop a higher order symmetric partitioned Runge-Kutta method for a coupled system of differential equations on Lie groups. We start with a discussion on partitioned Runge-Kutta methods on Lie groups of arbitrary order.…
We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular…