Optimal transport over a linear dynamical system

Chen, Yongxin; Georgiou, Tryphon T.; Pavon, Michele

doi:10.48550/arxiv.1502.01265

Cited by 11 publications

(20 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If ε ց 0 it appears that in the limit we get the Benamou-Brenier formulation of OMT (2). This is indeed the case, see [46,47,[55][56][57] and [16,17] for the case with prior. Alternatively, since both OMT and SBP admit a static formulation [46], the regularization of OMT as in (1) can be obtained by adding a term proportional to the Shannon entropy on…”

Section: Schr öDinger Bridges and Entropic Interpolationmentioning

confidence: 68%

“…It allows to view the optimal transport problem as an (atypical) optimal control problem [13]- [17]. It provides a ground on which the connection with the Schrödinger bridge problem appears as a regularization of the former [16,17,46,47,[55][56][57]. Similarly with Nelson's stochastic mechanics, see below.…”

Section: Background On Optimal Mass Transportmentioning

confidence: 99%

“…Here v and v P are the current velocity fields [60] of the trial and prior measure, respectively. Comparing (4) to the OMT with prior formulated and studied in [16], see also [17] for the Gauss-Markov case, we see that the essential difference is that there is here an extra term in the action functional which has the form of a relative Fisher information. To find the connection to the classical OMT, let us specialize to the situation where the prior P = W ε .…”

Section: Schr öDinger Bridges and Entropic Interpolationmentioning

confidence: 99%

“…Within Nelson's stochastic mechanics, this would correspond to the variational mechanism leading to the so-called collapse of the wavefunction after a position measurement [67,68]. The zero-noise limit of a general Schrödinger bridge, instead, is given by an OMT problem with prior formulated and studied in [16], see also [17].…”

Section: B Remarks On Stochastic Mechanicsmentioning

confidence: 99%

“…Other examples are the flows associated to the Schrödinger Bridge Problem (SBP) [75,76] and the quantum evolution of a nonrelativistic particle in Nelson's Stochastic Mechanics (NSM) [60,61]. It is known that SBP may be viewed as a "regularization" of OMT , the latter problem being recovered through a "zero-noise limit" [16,17,46,47,[55][56][57]. Recently, it was shown by von Renesse [81], for the quantum mechanical Madelung fluid [36], and by one of us [20] for the Schrödinger bridge, that their flows satisfy a suitable Newton-like law in Wasserstein space.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Extremal flows on Wasserstein space

Conforti¹,

Pavon²

2017

Preprint

Self Cite

View full text Add to dashboard Cite

We develop an intrinsic geometric approach to calculus of variations on Wasserstein space. We show that the flows associated to the Schrödinger bridge with general prior, to Optimal Mass Transport and to the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm's and Nelson's stochastic mechanics.

show abstract

Section: Schr öDinger Bridges and Entropic Interpolationmentioning

confidence: 68%

Section: Background On Optimal Mass Transportmentioning

confidence: 99%

Section: Schr öDinger Bridges and Entropic Interpolationmentioning

confidence: 99%

Section: B Remarks On Stochastic Mechanicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extremal flows on Wasserstein space

Conforti¹,

Pavon²

2017

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Optimal Mass Transport over Bridges

Chen

Georgiou

Pavon

2015

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

Optimal perturbations for nonlinear systems using graph-based optimal transport

Grover

Elamvazhuthi

2018

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge-Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

show abstract

Optimal transport over a linear dynamical system

Cited by 11 publications

References 17 publications

Extremal flows on Wasserstein space

Extremal flows on Wasserstein space

Optimal Mass Transport over Bridges

Optimal perturbations for nonlinear systems using graph-based optimal transport

Contact Info

Product

Resources

About