scholarly journals The back-and-forth method for Wasserstein gradient flows

2021 ◽  
Vol 27 ◽  
pp. 28
Author(s):  
Matt Jacobs ◽  
Wonjun Lee ◽  
Flavien Léger
Keyword(s):  
Large Class ◽  
Large Scale ◽  
Dual Problem ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Primal Problem ◽  
Transport Problems ◽  
Wasserstein Gradient Flows ◽  
Numer Math

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in Jacobs and Léger [Numer. Math. 146 (2020) 513–544.]. to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

scholarly journals A Novel Lagrangian Multiplier Update Algorithm for Short-Term Hydro-Thermal Coordination

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6621
Author(s):  
P. M. R. Bento ◽  
S. J. P. S. Mariano ◽  
M. R. A. Calado ◽  
L. A. F. M. Ferreira
Keyword(s):  
Lagrange Multipliers ◽  
Large Scale ◽  
Dual Problem ◽  
Electrical Power ◽  
Decomposition Methods ◽  
Subgradient Methods ◽  
Lagrangian Multiplier ◽  
Step Size ◽  
Primal Problem ◽  
Require Time

The backbone of a conventional electrical power generation system relies on hydro-thermal coordination. Due to its intrinsic complex, large-scale and constrained nature, the feasibility of a direct approach is reduced. With this limitation in mind, decomposition methods, particularly Lagrangian relaxation, constitutes a consolidated choice to “simplify” the problem. Thus, translating a relaxed problem approach indirectly leads to solutions of the primal problem. In turn, the dual problem is solved iteratively, and Lagrange multipliers are updated between each iteration using subgradient methods. However, this class of methods presents a set of sensitive aspects that often require time-consuming tuning tasks or to rely on the dispatchers’ own expertise and experience. Hence, to tackle these shortcomings, a novel Lagrangian multiplier update adaptative algorithm is proposed, with the aim of automatically adjust the step-size used to update Lagrange multipliers, therefore avoiding the need to pre-select a set of parameters. A results comparison is made against two traditionally employed step-size update heuristics, using a real hydrothermal scenario derived from the Portuguese power system. The proposed adaptive algorithm managed to obtain improved performances in terms of the dual problem, thereby reducing the duality gap with the optimal primal problem.

scholarly journals Nonlinear systems coupled through multi-marginal transport problems

2019 ◽  
Vol 31 (3) ◽  
pp. 450-469
Author(s):  
M. LABORDE
Keyword(s):  
Nonlinear Systems ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Uniqueness Result ◽  
System Of Nonlinear Equations ◽  
Planning Model ◽  
Highly Nonlinear ◽  
Transport Problems ◽  
Dimension One ◽  
Monge Ampere

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

scholarly journals Short-term Hydro-thermal Coordination By Lagrangian Relaxation: A New Algorithm for the Solution of the Dual Problem

2020 ◽  
Author(s):  
Pedro Bento ◽  
Filipe Pina ◽  
Sílvio Mariano ◽  
Maria do Rosario Calado
Keyword(s):  
Lagrangian Relaxation ◽  
Lagrange Multipliers ◽  
Large Scale ◽  
Dual Problem ◽  
Unit Commitment ◽  
Subgradient Methods ◽  
Step Size ◽  
Coordination Problem ◽  
Unit Commitment Problem ◽  
Primal Problem

For decades, researchers have been studying the unit commitment problem in electrical power generation. To solve this complex, large scale and constrained optimization (primal) problem in a direct manner is not a feasible approach, which is where Lagrangian relaxation comes in as the answer. The dual Lagrangian problem translates a relaxed problem approach, that indirectly leads to solutions of the original (primal) problem. In the coordination problem, a decomposition takes place where the global solution is achieved by coordinating between the respective subproblems solutions. This dual problem is solved iteratively, and Lagrange multipliers are updated between each iteration using subgradient methods. To tackle, time-consuming tuning tasks  or user related experience, a new adaptative algorithm, is proposed to better adjust the step-size used to update Lagrange multipliers, i.e., avoid the need to pre-select  a set of parameters. A results comparison against a traditionally employed step-size update mechanism, showed that the adaptive algorithm manages to obtain improved performances in terms of the targeted primal problem. Keywords: Hydro-Thermal coordination, Lagrangian relaxation, Lagrangian dual problem, Lagrange multipliers, Subgradient methods

scholarly journals Domain decomposition for entropy regularized optimal transport

2021 ◽  
Author(s):  
Mauro Bonafini ◽  
Bernhard Schmitzer
Keyword(s):  
Domain Decomposition ◽  
Large Scale ◽  
Optimal Transport ◽  
Optimal Solution ◽  
Linear Convergence ◽  
Efficient Implementation ◽  
Computationally Efficient ◽  
Solution Quality ◽  
Leibler Divergence ◽  
Transport Problems

AbstractWe study Benamou’s domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback–Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient implementation, such as adaptive sparsity, a coarse-to-fine scheme and parallelization, paving the way to numerically solving large-scale optimal transport problems. We demonstrate efficient numerical performance for computing the Wasserstein-2 distance between 2D images and observe that, even without parallelization, domain decomposition compares favorably to applying a single efficient implementation of the Sinkhorn algorithm in terms of runtime, memory and solution quality.

SCOTCluster: Deep Clustering with Optimal Transport for Large-scale Single-cell RNA-seq Data

2021 ◽  
Author(s):  
Faning Long ◽  
Xiaojun Ding ◽  
Xiaoqing Peng ◽  
Jianxin Wang ◽  
Xiaoshu Zhu
Keyword(s):  
Single Cell ◽  
Large Scale ◽  
Optimal Transport ◽  
Rna Seq

scholarly journals A Flow Perspective on Nonlinear Least-Squares Problems

2020 ◽  
Vol 48 (4) ◽  
pp. 987-1003
Author(s):  
Hans Georg Bock ◽  
Jürgen Gutekunst ◽  
Andreas Potschka ◽  
María Elena Suaréz Garcés
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Large Scale ◽  
Gradient Flow ◽  
Nonlinear Least Squares ◽  
Bundle Adjustment ◽  
Least Squares Problems ◽  
Flow Equations ◽  
Backward Euler ◽  
Forward Euler

AbstractJust as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

scholarly journals ROBUST EXPONENTIAL HEDGING AND INDIFFERENCE VALUATION

2010 ◽  
Vol 13 (07) ◽  
pp. 1075-1101 ◽  
Author(s):  
KEITA OWARI
Keyword(s):  
Dual Problem ◽  
Maximization Problem ◽  
Probability Measures ◽  
Primal Problem ◽  
Robust Version ◽  
Indifference Valuation ◽  
Robust Utility Maximization ◽  
Indifference Price ◽  
Utility Indifference ◽  
Utility Maximization Problem

We discuss the problem of exponential hedging in the presence of model uncertainty expressed by a set of probability measures. This is a robust utility maximization problem with a contingent claim. We first consider the dual problem which is the minimization of penalized relative entropy over a product set of probability measures, showing the existence and variational characterizations of the solution. These results are applied to the primal problem. Then we consider the robust version of exponential utility indifference valuation, giving the representation of indifference price using a duality result.

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