We consider estimation of ordinary differential equation (ODE) models from noisy observations. For this problem, one conventional approach is to fit numerical solutions (e.g., Euler, Runge–Kutta) of ODEs to data. However, such a method does not account for the discretization error in numerical solutions and has limited estimation accuracy. In this study, we develop an estimation method that quantifies the discretization error based on data. The key idea is to model the discretization error as random variables and estimate their variance simultaneously with the ODE parameter. The proposed method has the form of iteratively reweighted least squares, where the discretization error variance is updated with the isotonic regression algorithm and the ODE parameter is updated by solving a weighted least squares problem using the adjoint system. Experimental results demonstrate that the proposed method improves estimation accuracy by accounting for the discretization error in a data-driven manner. This is a joint work with Yuto Miyatake (Osaka University).

The Alternating Directions Method of Multipliers (ADMM) is a widely popular first-order method for solving convex optimization problems. Its simplicity is arguably one of the main reasons for its popularity. For a broad class of problems, ADMM iterates by repeatedly solving perhaps the two most standard Linear Algebra problems: linear systems and symmetric eigenproblems. In this talk, we discuss how employing standard Krylov-subspace methods for ADMM can lead to tenfold speedups while retaining convergence guarantees.

We present analysis and computations of a non-local thin film model developed by Kalogirou et al (2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from Hoof-Bifurcation of the primary solution branch.

(This work is in collaboration with D. Papageorgiou & E. Oliveira )