Robert Dan Berry

Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System

Projection onto polynomial chaos (PC) basis functions is often used to reformulate a system of ordinary differential equations (ODEs) with uncertain parameters and initial conditions as a deterministic ODE system that describes the evolution of the PC modes. The deterministic Jacobian of this projected system is different and typically much larger than the random Jacobian of the original ODE system. This paper shows that the location of the eigenvalues of the projected Jacobian is largely determined by the eigenvalues of the original Jacobian, regardless of PC order or choice of orthogonal polynomials. Specifically, the eigenvalues of the projected Jacobian always lie in the convex hull of the numerical range of the Jacobian of the original system.

Efficient uncertainty quantification methodologies for high-dimensional climate land models.

In this report, we proposed, examined and implemented approaches for performing efficient uncertainty quantification (UQ) in climate land models. Specifically, we applied Bayesian compressive sensing framework to a polynomial chaos spectral expansions, enhanced it with an iterative algorithm of basis reduction, and investigated the results on test models as well as on the community land model (CLM). Furthermore, we discussed construction of efficient quadrature rules for forward propagation of uncertainties from high-dimensional, constrained input space to output quantities of interest. The work lays grounds for efficient forward UQ for high-dimensional, strongly non-linear and computationally costly climate models. Moreover, to investigate parameter inference approaches, we have applied two variants of the Markov chain Monte Carlo (MCMC) method to a soil moisture dynamics submodel of the CLM. The evaluation of these algorithms gave us a good foundation for further building out the Bayesian calibration framework towards the goal of robust component-wise calibration.