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Data Assimilation: A Mathematical Introduction
This book provides a systematic treatment of the mathematical underpinnings of work in data assimilation, covering both theoretical and computational approaches. Specifically the authors develop a…
Exploration of the (Non-)Asymptotic Bias and Variance of Stochastic Gradient Langevin Dynamics
A modified SGLD which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived and bounds on the finite-time bias, variance and mean squared error are obtained.
Statistical analysis of differential equations: introducing probability measures on numerical solutions
- Patrick R. Conrad, M. Girolami, S. Särkkä, A. Stuart, K. Zygalakis
- MathematicsStat. Comput.
- 2 June 2016
It is shown that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of ordinary and partial differential equation models, and the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis are provided.
High Order Numerical Approximation of the Invariant Measure of Ergodic SDEs
A systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method.
Long Time Accuracy of Lie-Trotter Splitting Methods for Langevin Dynamics
A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed.…
On the Existence and the Applications of Modified Equations for Stochastic Differential Equations
- K. Zygalakis
- MathematicsSIAM J. Sci. Comput.
- 1 February 2011
This paper describes a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence using the Gaussianity of the underlying solutions.
Adsorption and desorption dynamics of citric acid anions in soil
The functional role of organic acid anions in soil has been intensively investigated, with special focus on (i) microbial respiration and soil carbon dynamics, (ii) nutrient solubilization or (iii)…
Probabilistic Linear Multistep Methods
A rigorous proof of the convergence of the Adams-Bashforth and Adams-Moulton family of linear multistep methods, as well as an empirical investigation demonstrating their convergence rates in practice.
The True Cost of Stochastic Gradient Langevin Dynamics
- Tigran Nagapetyan, A. Duncan, Leonard Hasenclever, S. Vollmer, L. Szpruch, K. Zygalakis
- Computer Science, Mathematics
- 8 June 2017
The mean squared error of Lipschitz functionals in strongly log- concave models with i.i.d. data of growing data set size is studied and it is shown that, given a batchsize, to control the bias of SGLD the stepsize has to be chosen so small that the computational cost of reaching a target accuracy is roughly the same for all batchsizes.
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a Stochastic dynamical system.