Research
Here are short summaries of some of the research projects I both haven been and currently are working on. The references, marked [X], can be found under Publications.
Computational optimal transport with applications
This work builds on recent results for efficiently solving large-scale optimal transport problems via so-called Sinkhorn iterations. In particular, we extend the Sinkhorn iterations to other optimal transport problems and optimization problems involving an optimal transport cost, and apply the methods to problems in control and estimation [J9], [C8], inverse problems [J3], and machine learning [TR1]. More precisely, in [J3] we identify the Sinkhorn iterations as coordinate ascent in the Lagrangian dual of a perturbed (surrogate) problem. Using this idea, we derive efficient numerical algorithms for more general optimization problems involving an optimal transport cost, and in particular this includes an iterative method for computing the proximal operator. The latter allows us to use variable-splitting techniques from convex optimization to solve an even larger class of problems arising, e.g., in variational regularization formulations for inverse problems in imaging.
The Sinkhorn iterations can be extended to multi-marginal optimal transport problems, however in this case the iterations only partly alleviates the computational burden. In particular, the Sinkhorn iterations build on computing the one-dimensional projections of the transport plan, represented by a nonnegative tensor, and this in general scales exponentially in the number of dimensions of the tensor. Therefore, computationally efficient methods are needed to compute these projections in order to use the algorithm. In [J8], we derive an efficient algorithm for computing the projections in the special case when the cost decouples according to a tree structure. Examples of such problems include barycenter problems, where the tree is a star graph, and problems related to estimation based on time series, where the tree is a path graph. In particular, in [C8], these methods are used in estimating the flow of an ensemble of agents. Moreover, optimal transport has recently also been linked to so-called “Schrödinger bridge problem”, and in [J8] we also extend this correspondence to Schrödinger bridges defined on the same underlying tree structure. In [J9], we show how other this can be applied to problems in systems and control, and also list a number of other cases in which the projections can be computed efficiently.
Inverse optimal control
Related publications and preprints: [J12], [J15], [C11], [P3]
Robust stability of control systems with structured uncertain
Robust stabilization of structured uncertain control systems via analytic interpolation
This line of work starts from classical methods in control theory, based on complex and functional analysis, for computing robustness measures for linear dynamical systems. In particular, in [J7], [C7] it is shown that the maximum delay margin problem can be formulated as an analytic interpolation problem with a non-classical constraint, akin to the classical maximum gain and phase margin problems. More precisely, all three problems are robust control problems with structured uncertainties, and all three can be formulated as analytic interpolation problems with a constraint that the image of the interpolant on the imaginary axis does not intersect a problem-specific family of sets. Moreover, in all three cases, exact or approximate solution methods are formulated by recasting the problem as a classical Nevanlinne-Pick interpolation problem. In [C9], these results are extended to a larger class of structured, stable perturbations.
Figure: Nyquist plot (blue) for controller designed for maximizing the delay margin (green) with specified simultaneous gain and phase margin (purple).
Robust stability of structured uncertain control systems via multipliers, and phases for MIMO systems
[P2]. This is also related to research on special types of matrices in [J10], [J11], [SJ3].
Complexity constrained multi-dimensional moment matching
This research was motivated by problems in spectral estimation and stochastic system identification for second-order stationary time series, and we build on a recently developed frame work for parameterizing low-degree analytic functions satisfying given interpolation conditions. More precisely, we consider the following problem: given a set of complex numbers, determine if there exists and if so find a nonnegative bounded measure with rational absolutely continuous part of bounded degree that has the given complex numbers as multi-dimensional trigonometric moments. In particular, in [J2] we characterize all moment sequences for which such a solution exists by formulating a primal-dual pair of convex optimization problems. Moreover, we also investigate the well-posedness of this inverse problem. In [J1], we extend the results to more general moment problems, i.e., where the basis functions are not necessarily the complex exponentials. In [C3] we consider a discrete counter-part of the problem in [J1]; the former is of interest when considering periodic stochastic process but can also be used to approximate the problem in [J1]. This also leads to a solution method for the problem in [J1] based on solving a corresponding finite-dimensional convex optimization problem. Computations for solving this problem can also be efficiently implemented by using fast computations based on FFT [C2]. Moreover, in [J4] we relax the exact moment matching condition and formulate and investigate the corresponding optimization problems.