Solving the Monge–Ampère equations for the inverse reflector problem

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.

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Nonlinear systems coupled through multi-marginal transport problems

European Journal of Applied Mathematics ◽

10.1017/s0956792519000123 ◽

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.

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Mathematical study for peristaltic flow of Williamson fluid in a curved channel

International Journal of Biomathematics ◽

10.1142/s1793524515500059 ◽

2015 ◽

Vol 08 (01) ◽

pp. 1550005 ◽

Cited By ~ 5

Author(s):

E. N. Maraj ◽

Noreen Sher Akbar ◽

S. Nadeem

Keyword(s):

Fluid Model ◽

Nonlinear Partial Differential Equations ◽

Nonlinear Partial Differential Equation ◽

Pressure Rise ◽

Peristaltic Flow ◽

Curved Channel ◽

Partial Differential ◽

Williamson Fluid ◽

Highly Nonlinear ◽

Frame Transformation

In this paper, we have investigated the peristaltic flow of Williamson fluid in a curved channel. The governing equations of Williamson fluid model for curved channel are derived including the effects of curvature. The highly nonlinear partial differential equations are simplified by using the wave frame transformation, long wavelength and low Reynolds number assumptions. The reduced nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method. The physical features of pertinent parameters have been discussed by plotting the graphs of pressure rise, velocity profile and stream functions.

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Lecture 7: The Monge-Ampére Equation and Optimal Transport on Riemannian Manifolds

10.1007/978-3-030-72162-6_7 ◽

2021 ◽

pp. 65-76

Author(s):

Luigi Ambrosio ◽

Elia Brué ◽

Daniele Semola

Keyword(s):

Riemannian Manifolds ◽

Optimal Transport ◽

Monge Ampere

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Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.13-13s01 ◽

2013 ◽

Vol 5 (04) ◽

pp. 407-422 ◽

Cited By ~ 2

Author(s):

Matthew A. Beauregard ◽

Qin Sheng

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Finite Difference ◽

Nonlinear Partial Differential Equation ◽

Partial Differential ◽

Spatial Adaptation ◽

Monitor Function ◽

Monitoring Strategies ◽

Highly Nonlinear ◽

Equidistribution Principle

AbstractFinite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-calledprimitiveregime, the rest belong to a later category of themodifiedtype, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

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Application of GDQ method in nonlinear analysis of a flexible manipulator undergoing large deformation

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406213478541 ◽

2013 ◽

Vol 227 (12) ◽

pp. 2671-2685 ◽

Cited By ~ 1

Author(s):

Mohammad R Fazel ◽

Majid M Moghaddam ◽

Javad Poshtan

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Equations Of Motion ◽

Nonlinear Partial Differential Equation ◽

Flexible Manipulator ◽

Runge Kutta Method ◽

Highly Nonlinear ◽

Frechet Derivatives ◽

Fréchet Derivatives ◽

Generalized Differential

Analysis of a flexible manipulator as an initial value problem, due to its large deformations, involves nonlinear ordinary differential equations of motion. In the present work, these equations are solved through the general Frechet derivatives and the generalized differential quadrature (GDQ) method directly. The results so obtained are compared with those of the fourth-order Runge–Kutta method. It is seen that both the results match each other well. Further considering the same manipulator as a boundary value problem, its governing equation is a highly nonlinear partial differential equation. Again applying the general Frechet derivatives and the GDQ method, it is seen that the results are in good match with the linear theory. In both cases, the general Frechet derivatives are introduced and successfully used for linearization. The results of the present study indicate that the GDQ method combined with the general Frechet derivatives can be successfully used for the solution of nonlinear differential equations.

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A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

Mathematical Problems in Engineering ◽

10.1155/2013/609536 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

I. Rusagara ◽

C. Harley

Keyword(s):

Heat Transfer ◽

Differential Equation ◽

Partial Differential Equation ◽

Nonlinear Partial Differential Equation ◽

Transfer Coefficients ◽

Partial Differential ◽

One Dimensional ◽

Heat Transfer Coefficients ◽

Highly Nonlinear ◽

Triangular Fin

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

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An Optimal Transport Approach to Monge–Ampère Equations on Compact Hessian Manifolds

Journal of Geometric Analysis ◽

10.1007/s12220-018-0068-5 ◽

2018 ◽

Vol 29 (3) ◽

pp. 1953-1990

Author(s):

Jakob Hultgren ◽

Magnus Önnheim

Keyword(s):

Optimal Transport ◽

Monge Ampere ◽

Transport Approach

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Deep Learning for Constrained Utility Maximisation

Methodology And Computing In Applied Probability ◽

10.1007/s11009-021-09912-3 ◽

2021 ◽

Author(s):

Ashley Davey ◽

Harry Zheng

Keyword(s):

Differential Equation ◽

Deep Learning ◽

Stochastic Control ◽

Backward Stochastic Differential Equation ◽

Nonlinear Partial Differential Equation ◽

Computational Cost ◽

Utility Maximisation ◽

Highly Nonlinear ◽

Path Dependent ◽

Hamilton Jacobi Bellman

AbstractThis paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

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