Related papers: A probabilistic model for the numerical solution o…

Diffusion processes are a class of stochastic differential equations (SDEs) providing a rich family of expressive models that arise naturally in dynamic modelling tasks. Probabilistic inference and learning under generative models with…

This paper presents a probabilistic perspective on iterative methods for approximating the solution $\mathbf{x}_* \in \mathbb{R}^d$ of a nonsingular linear system $\mathbf{A} \mathbf{x}_* = \mathbf{b}$. In the approach a standard iterative…

We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We…

In this work we study the initial value problem (IVP) for the fifth order KdV equations, \begin{align*} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0,\text{} & \quad x,t\in \mathbb R, \quad k=1,2, \end{align*} in weighted Sobolev…

This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's…

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary…

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigma u(t),\,…

In partial differential equations-based (PDE-based) inverse problems with many measurements, many large-scale discretized PDEs must be solved for each evaluation of the misfit or objective function. In the nonlinear case, evaluating the…

Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of…

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with…

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then…

Variational inference methods for latent variable statistical models have gained popularity because they are relatively fast, can handle large data sets, and have deterministic convergence guarantees. However, in practice it is unclear…

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems,…

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…

In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard…

Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some…

Gaussian processes are a versatile probabilistic machine learning model whose effectiveness often depends on good hyperparameters, which are typically learned by maximising the marginal likelihood. In this work, we consider iterative…

We consider the problem of parameter estimation in dynamic systems described by ordinary differential equations. A review of the existing literature emphasizes the need for deterministic global optimization methods due to the nonconvex…

Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical…

We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem…