PARAMETER AND UNCERTAINTY ESTIMATION FOR DYNAMICAL SYSTEMS USING SURROGATE STOCHASTIC PROCESSES.

Inference on unknown quantities in dynamical systems via observational data is essential for providing meaningful insight, furnishing accurate predictions, enabling robust control, and establishing appropriate designs for future experiments. Merging mathematical theory with empirical measurements in a statistically coherent way is critical and challenges abound, e.g., ill-posedness of the parameter estimation problem, proper regularization and incorporation of prior knowledge, and computational limitations. To address these issues, we propose a new method for learning parameterized dynamical systems from data. We first customize and fit a surrogate stochastic process directly to observational data, front-loading with statistical learning to respect prior knowledge (e.g., smoothness), cope with challenging data features like heteroskedasticity, heavy tails, and censoring. Then, samples of the stochastic process are used as "surrogate data" and point estimates are computed via ordinary point estimation methods in a modular fashion. Attractive features of this two-step approach include modularity and trivial parallelizability. We demonstrate its advantages on a predator-prey simulation study and on a real-world application involving within-host influenza virus infection data paired with a viral kinetic model, with comparisons to a more conventional Markov chain Monte Carlo (MCMC) based Bayesian approach.

An efficient phase and object estimation scheme for phase-diversity time series data.

We present a two-stage method for obtaining both phase and object estimates from phase-diversity time series data. In the first stage, the phases are estimated for each time frame using the limited memory BFGS method. In the second stage, an algorithm that incorporates a nonnegativity constraint as well as prior knowledge of data noise statistics is used to obtain an estimate of the object being observed. The approach is tested on real phase-diversity data with 32 time frames, and a comparison is made between it and a previously developed approach. Also, the image deblurring algorithm in stage two is tested against other standard methods and is shown to be the best for our problem.

A limited-memory, quasi-Newton preconditioner for nonnegatively constrained image reconstruction.

Image reconstruction gives rise to some challenging large-scale constrained optimization problems. We consider a convex minimization problem with nonnegativity constraints that arises in astronomical imaging. To solve this problem, we use an efficient hybrid gradient projection-reduced Newton (active-set) method. By "reduced Newton," we mean that we take Newton steps only in the inactive variables. Owing to the large size of our problem, we compute approximate reduced Newton steps by using the conjugate gradient (CG) iteration. We introduce a limited-memory, quasi-Newton preconditioner that speeds up CG convergence. A numerical comparison is presented that demonstrates the effectiveness of this preconditioner.