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Journal articles on the topic 'Hamilton-Jacobi-Bellman and Fokker-Planck equations'

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Author: Grafiati

Published: 13 July 2024

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Bensoussan, Alain, and Sheung Chi Phillip Yam. "Mean field approach to stochastic control with partial information." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 89. http://dx.doi.org/10.1051/cocv/2021085.

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In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandini et al. [Stochastic Process. Appl. 129 (2019) 674–711], which is fundamentally different from our present proposed framework.

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Cortés, Emilio, and J. I. Jiménez-Aquino. "Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator." Physica A: Statistical Mechanics and its Applications 411 (October 2014): 1–11. http://dx.doi.org/10.1016/j.physa.2014.05.064.

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Tottori, Takehiro, and Tetsuya J. Kobayashi. "Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic Control." Entropy 25, no. 2 (January 21, 2023): 208. http://dx.doi.org/10.3390/e25020208.

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Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. To obtain the optimal control function of ML-POSC, a system of the forward Fokker–Planck (FP) equation and the backward Hamilton–Jacobi–Bellman (HJB) equation needs to be solved. In this work, we first show that the system of HJB-FP equations can be interpreted via Pontryagin’s minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM) for ML-POSC. FBSM is one of the most basic algorithms for Pontryagin’s minimum principle, which alternately computes the forward FP equation and the backward HJB equation in ML-POSC. Although the convergence of FBSM is generally not guaranteed in deterministic control and mean-field stochastic control, it is guaranteed in ML-POSC because the coupling of the HJB-FP equations is limited to the optimal control function in ML-POSC.

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Bakaryan, Tigran, Rita Ferreira, and Diogo Gomes. "A potential approach for planning mean-field games in one dimension." Communications on Pure and Applied Analysis 21, no. 6 (2022): 2147. http://dx.doi.org/10.3934/cpaa.2022054.

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<p style='text-indent:20px;'>This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.</p><p style='text-indent:20px;'>We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.</p>

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Jiménez-Aquino, J. I., and Emilio Cortés. "Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator in the inertial regime." Physica A: Statistical Mechanics and its Applications 422 (March 2015): 203–9. http://dx.doi.org/10.1016/j.physa.2014.12.012.

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Mollai, Maedeh, and Seyed Majid Saberi Fathi. "An Application of the Madelung Formalism for Dissipating and Decaying Systems." Symmetry 13, no. 5 (May 6, 2021): 812. http://dx.doi.org/10.3390/sym13050812.

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This paper is concerned with the modeling and analysis of quantum dissipation and diffusion phenomena in the Schrödinger picture. We derive and investigate in detail the Schrödinger-type equations accounting for dissipation and diffusion effects. From a mathematical viewpoint, this equation allows one to achieve and analyze all aspects of the quantum dissipative systems, regarding the wave equation, Hamilton–Jacobi and continuity equations. This simplification requires the performance of “the Madelung decomposition” of “the wave function”, which is rigorously attained under the general Lagrangian justification for this modification of quantum mechanics. It is proved that most of the important equations of dissipative quantum physics, such as convection-diffusion, Fokker–Planck and quantum Boltzmann, have a common origin and can be unified in one equation.

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Корниенко, Виктория Сергеевна, Владимир Викторович Шайдуров, and Евгения Дмитриевна Карепова. "A finite difference analogue of the “mean field” equilibrium problem." Вычислительные технологии, no. 4(25) (September 16, 2020): 31–44. http://dx.doi.org/10.25743/ict.2020.25.4.004.

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Представлен конечно-разностный аналог дифференциальной задачи, сформулированной в терминах теории “игр среднего поля” (mean field games). Задачи оптимизации такого типа формулируются как связанные системы параболических дифференциальных уравнений в частных производных типа Фоккера - Планка и Гамильтона - Якоби - Беллмана. Предложенный конечно-разностный аналог обладает основными свойствами оптимизационной дифференциальной задачи непосредственно на дискретном уровне. В итоге он может служить как приближение, сходящееся к исходной дифференциальной задаче при стремлении шагов дискретизации к нулю, так и как самостоятельная оптимизационная задача с конечным числом участников. Для предложенного аналога построен алгоритм монотонной минимизации функционала стоимости, проиллюстрированный на модельной экономической задаче In most forecasting problems, overstating or understating forecast leads to various losses. Traditionally, in the theory of “mean field games”, the functional responsible for the costs of implementing the interaction of the continuum of agents between each other is supposed to be dependent on the squared function of control of the system. Since additional external factors can influence the player’s strategy, the control function of a dynamic system is more complex. Therefore, the purpose of this article is to develop a computational algorithm applicable for more general set of control functions. As a research method, a computational experiment and proof of the stability of the constructed computational scheme are used in this study. As a result, the numerical algorithm was applied on the problem of economic interaction in the presence of alternative resources. We consider the model, in which a continuum of consumer agents consists of households deciding on heating, having a choice between the cost of installing and maintaining the thermal insulation or the additional cost of electricity. In the framework of the problem, the convergence of the method is numerically demonstrated. Conclusions. The article considers a model of the strategic interaction of continuum of agents, the interaction of which is determined by a coupled differential equations, namely, the Fokker - Planck and the Hamilton - Jacobi - Bellman one. To approximate the differential problem, difference schemes with a semi-Lagrangian approximation are used, which give a direct rule for minimizing the cost functional

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Moreno Trujillo, John Freddy. "Una nota introductoria a los juegos de campo medio. Teoría y algunas aplicaciones." ODEON, no. 22 (July 4, 2023): 159–78. http://dx.doi.org/10.18601/17941113.n22.06.

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Se presentan de forma simple los conceptos fundamentales de la teoría de juegos de campo medio, mostrando que esta se puede ver como un ingenioso acople entre ecuaciones de Hamilton-Jacobi-Bellman y Fokker-Planck-Kolmogorov para el tratamiento de sistemas complejos con un número de agentes muy grande. Se presenta también el concepto de equilibrio para este tipo de juegos y algunas aplicaciones de esta teoría en diferentes campos.

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Annunziato, Mario, Alfio Borzì, Fabio Nobile, and Raul Tempone. "On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks." Applied Mathematics 05, no. 16 (2014): 2476–84. http://dx.doi.org/10.4236/am.2014.516239.

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Fomin, Igor, and Sergey Chervon. "Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints." Universe 6, no. 11 (October 29, 2020): 199. http://dx.doi.org/10.3390/universe6110199.

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We investigate the ability of the exponential power-law inflation to be a phenomenologically correct model of the early universe. We study General Relativity (GR) scalar cosmology equations in Ivanov–Salopek–Bond (or Hamilton–Jacobi like) representation where the Hubble parameter H is the function of a scalar field ϕ. Such approach admits calculation of the potential for given H(ϕ) and consequently reconstruction of f(R) gravity in parametric form. By this manner the Starobinsky potential and non-minimal Higgs potential (and consequently the corresponding f(R) gravity) were reconstructed using constraints on the model’s parameters. We also consider methods for generalising the obtained solutions to the case of chiral cosmological models and scalar-tensor gravity. Models based on the quadratic relationship between the Hubble parameter and the function of the non-minimal interaction of the scalar field and curvature are also considered. Comparison to observation (PLANCK 2018) data shows that all models under consideration give correct values for the scalar spectral index and tensor-to-scalar ratio under a wide range of exponential-power-law model’s parameters.

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SBITNEV, VALERIY I. "BOHMIAN TRAJECTORIES AND THE PATH INTEGRAL PARADIGM: COMPLEXIFIED LAGRANGIAN MECHANICS." International Journal of Bifurcation and Chaos 19, no. 07 (July 2009): 2335–46. http://dx.doi.org/10.1142/s0218127409024104.

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David Bohm had shown that the Schrödinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions — action and probability density. The first equation is the Hamilton–Jacobi (HJ) equation, a "visiting card" of classical mechanics, is modified by the Bohmian quantum potential. This potential is a nonlinear function of the probability density. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed into two Bohmian quantum correctors. The first corrector modifies the kinetic energy term of the HJ equation, and the second one modifies the potential energy term. The unification of the quantum HJ equation and the entropy balance equation gives a complexified HJ equation containing complex kinetic and potential terms. The imaginary parts of these terms have an order of smallness about the Planck constant. The Bohmian quantum corrector is an indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to an imaginary sector. The difference between the Bohmian and Feynman's trajectories is that the former satisfies the principle of least action and they bifurcate on interfaces. The latter covers all possible paths from a source to a detector. They can split and annihilate.

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Goffi, Alessandro. "Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations." Journal of Evolution Equations, June 8, 2021. http://dx.doi.org/10.1007/s00028-021-00720-3.

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AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

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Serdyukov, Sergey I. "The Onsager–Machlup theory of fluctuations and time-dependent generalized normal distribution." Journal of Non-Equilibrium Thermodynamics, January 5, 2023. http://dx.doi.org/10.1515/jnet-2022-0071.

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Abstract Generalization of the variational formulation of the Onsager–Machlup thermodynamic theory of fluctuation is considered. Within the framework of variational theory, we introduce the time-dependent generalized normal distribution and Hamilton–Jacobi equation. The family of higher-order partial differential equations, which generalize classical Fokker–Planck equation, is considered. It is shown that proposed theory can be used for describing anomalous diffusion.

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Mannucci, Paola, Claudio Marchi, and Cristian Mendico. "Semi-linear parabolic equations on homogenous Lie groups arising from mean field games." Mathematische Annalen, March 16, 2024. http://dx.doi.org/10.1007/s00208-024-02819-7.

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AbstractThe existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker–Planck equation and the Hamilton–Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group.

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Mattos Da Silva, Leticia, Oded Stein, and Justin Solomon. "A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains." ACM Transactions on Graphics, May 28, 2024. http://dx.doi.org/10.1145/3666087.

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We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

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Trusov, Nikolai V. "Numerical study of the stock market crises based on mean field games approach." Journal of Inverse and Ill-posed Problems, April 2, 2021. http://dx.doi.org/10.1515/jiip-2020-0016.

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Abstract We present an approach to describe the stock market crises based on Mean Field Games (MFGs) and Optimal Control theory with a turnpike effect. The impact of the large amount of high-frequency traders (HFTs) can be modeled via a mean field term. We introduce the turnpike as a function that relies on the changes of the asset share price. An MFG is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time, and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The ill-posedness of this system comes from a turnpike effect. The numerical solution of an extremal problem that is dual to a PDE system is presented. We apply this approach to describe the Chinese stock market crash in 2015 considering the representative stock of CITIC Securities (ticker 600030). We consider HFTs that form a dominating bull and bear market. As a result, the bull strategy imitators do not make any profit.

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Chen, Yangang, and Justin W. L. Wan. "Artificial Viscosity Joint Spacetime Multigrid Method for Hamilton–Jacobi–Bellman and Kolmogorov–Fokker–Planck System Arising from Mean Field Games." Journal of Scientific Computing 88, no. 1 (May 26, 2021). http://dx.doi.org/10.1007/s10915-021-01520-0.

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Liu, Kang, Frédéric Bonnans, and Laurent Pfeiffer. "Error estimates of a theta-scheme for second-order mean field games." ESAIM: Mathematical Modelling and Numerical Analysis, July 18, 2023. http://dx.doi.org/10.1051/m2an/2023059.

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We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for the theta-scheme, where $h$ is the step length of the space variable and $r \in (0,1)$ is related to the H\"older continuity of the solution of the continuous problem and some of its derivatives.

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