Solving differential Riccati equations: A nonlinear space-time method using tensor trains

Abstract

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton-Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

Symmetries and Zero Modes in Sample Path Large Deviations

“…With the general treatment of zero modes completed, it is now theoretically possible to compute leading order large deviation prefactors even for multi-dimensional SPDEs such as the two-dimensional or three-dimensional Navier-Stokes equations where spontaneous symmetry breaking of the rotational symmetry of instantons has indeed been observed [47,48]. The remaining complication for numerical computations is the high dimensionality of the involved Riccati matrices, and it would be interesting future work to consider low-rank approximations of the Riccati differential equations [77] in this regard, that could e.g. make use of the sparsity of the large-scale forcing typically used in turbulence simulations.…”

Gel’fand–Yaglom type equations for calculating fluctuations around instantons in stochastic systems

“…The linearized system (98) would have the enormous advantage of being ideally suited for parallel calculations, but difficulties may arise due to the appearance of the backward heat equation hidden in the term ∇N . It should be mentioned, however, that the matrix Riccati equation ( 119) is also amenable to a massive parallel approach [38,50] or tensor network techniques [51]. The ultimate challenge would be the application of our approach to the full three-dimensional Navier-Stokes equations.…”

Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

“…The linearized system (98) would have the enormous advantage of being ideally suited for parallel calculations, but difficulties may arise due to the appearance of the backward heat equation hidden in the term ∇N . It should be mentioned, however, that the matrix Riccati equation ( 119) is also amenable to a massive parallel approach [34,43] or tensor network techniques [44]. The ultimate challenge would be the application of our approach to the full three-dimensional Navier-Stokes equations.…”

“…resulting from recursive Gaussian integration as discussed previously. Again using (44) for the ddimensional δ -function, this can be rewritten as…”