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We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the…
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute…
Sensitivity analysis plays an important role in searching for constitutive parameters (e.g. permeability) subsurface flow simulations. The mathematics behind is to solve a dynamic constrained optimization problem. Traditional methods like…
In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take the advantage of the exact solution of (generalised) geometric Brownian motion. We examine both Euler and Milstein…
Systems of reaction-diffusion partial differential equations (RD-PDEs) are widely applied for modelling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the…
The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several…
The optimization of the latents and parameters of diffusion models with respect to some differentiable metric defined on the output of the model is a challenging and complex problem. The sampling for diffusion models is done by solving…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave…
We introduce a class of explicit exponential Rosenbrock methods for the time integration of large systems of stiff differential equations. Their application with respect to simulation tasks in the field of visual computing is discussed…
Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and…
In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier…
Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models…
The adjoint method is an efficient way to numerically compute gradients in optimization problems with constraints, but is only formulated to differentiable cost and constraint functions on real variables. With the introduction of complex…
Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or…
Matrix differential Riccati equation (DRE) typically exhibits transient and steady-state phases, posing challenges for fixed-step time integration methods, which may lack accuracy during transients or oversample in steady regimes. In this…
Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati…
With this contribution, we shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a…
A C++ library for sensitivity analysis of optimisation problems involving ordinary differential equations (ODEs) enabled by automatic differentiation (AD) and SIMD (Single Instruction, Multiple data) vectorization is presented. The discrete…
We study the risk-sensitive exponential cost MDP formulation and develop a trajectory-based gradient algorithm to find the stationary point of the cost associated with a set of parameterized policies. We derive a formula that can be used to…