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 Solving the Monge–Ampère equations for the inverse reflector problem

2015 ◽  
Vol 25 (05) ◽  
pp. 803-837 ◽  
Author(s):  
Kolja Brix ◽  
Yasemin Hafizogullari ◽  
Andreas Platen
Keyword(s):  
Optimal Transport ◽  
Nonlinear Partial Differential Equation ◽  
Problem Formulation ◽  
Free Form ◽  
Benchmark Problems ◽  
Discrete Problem ◽  
Nonimaging Optics ◽  
Nonlinear Solver ◽  
Highly Nonlinear ◽  
Monge Ampere

The inverse reflector problem arises in geometrical nonimaging optics: given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g. a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge–Ampère equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge–Ampère equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: it enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge–Ampère equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.

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 Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

2020 ◽  
Author(s):  
Mohit Bansil ◽  
Jun Kitagawa
Keyword(s):  
Optimal Transport ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Uniform Norm ◽  
Quantitative Stability ◽  
Wirtinger Inequality ◽  
Dual Variables ◽  
Transport Problems ◽  
Monge Ampere ◽  
Stability Results

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

scholarly journals Optimal transport over nonlinear systems via infinitesimal generators on graphs

2018 ◽  
Vol 0 (0) ◽  
pp. 1-32 ◽  
Author(s):  
Karthik Elamvazhuthi ◽  
◽  
Piyush Grover ◽  
Keyword(s):  
Nonlinear Systems ◽  
Optimal Transport ◽  
Infinitesimal Generators

Quasi-parallelized Multicanonical Monte Carlo Method for Highly Nonlinear Systems, with Application to All-optical Regeneration

2011 ◽  
Author(s):  
Taras I. Lakoba ◽  
Michael Vasilyev ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Nonlinear Systems ◽  
Highly Nonlinear ◽  
All Optical ◽  
Optical Regeneration

Weak Instability of Chen–Ricles Explicit Method for Structural Dynamics

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
High Frequency ◽  
Explicit Method ◽  
Order Accuracy ◽  
Practical Applications ◽  
Correction Scheme ◽  
Highly Nonlinear ◽  
Steady State Responses ◽  
State Responses

Although the Chen–Ricles (CR) explicit method (CRM) (proposed by Chen and Ricles) has been claimed to have desired numerical properties, such as unconditional stability, explicit formulation, and second-order accuracy, it also shows some unusual properties, such as a less accuracy of solving highly nonlinear systems, a high-frequency overshoot in steady-state responses, and a weak instability. A correction scheme by adjusting the displacement difference equation with a loading term can be employed to extinguish the high-frequency overshoot in steady-state responses. However, there is still no way to get rid of the weak instability and to improve the less accuracy of solving highly nonlinear systems. It is recognized that a weak instability might result in inaccurate solutions or numerical explosions. Hence, the practical applications of CRM are strictly limited.

scholarly journals Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1168 ◽  
Author(s):  
Ligang Sun ◽  
Hamza Alkhatib ◽  
Boris Kargoll ◽  
Vladik Kreinovich ◽  
Ingo Neumann
Keyword(s):  
Kalman Filter ◽  
Nonlinear Systems ◽  
State Estimation ◽  
Discrete Time ◽  
Probability Distributions ◽  
Nonlinear Problems ◽  
The State ◽  
Filter Model ◽  
Highly Nonlinear ◽  
Correction Step

In this paper, we propose a new technique—called Ellipsoidal and Gaussian Kalman filter—for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system’s state (and the optimal ellipsoid for describing the systems’s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better—the state estimation technique usually applied to such nonlinear problems.

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