Two-level method for a time-independent Fokker–Planck control problem

References

• M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes, Math. Model. Anal. 15 (2010), pp. 393–407.
   Web of Science ®Google Scholar
• M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic process, J. Comput. Appl. Math. 237 (2013), pp. 487–507.
   Web of Science ®Google Scholar
• M. Annunziato, A. Borzì, F. Nobile, and R. Tempone, On the connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck control frameworks, Appl. Math. 5 (2014), pp. 2476–2484.
   Google Scholar
• A. Brandt and N. Dinar, Multigrid solutions to elliptic flow problems numerical methods for partial differential equations, in S. Parter, ed., Academic Press, New York, 1979, pp. 53–147.
   Google Scholar
• T. Breitenbach and A. Borzì, Tim breitenbach the pontryagin maximum principle for solving Fokker-Planck optimal control problems, Comput. Optim. Appl. 76 (2020), pp. 499–533.
   Web of Science ®Google Scholar
• M.M. Butt, A multigrid solver for Stokes control problems, Int. J. Comput. Math. 94 (2017), pp. 2297–2314.
   Web of Science ®Google Scholar
• M.M. Butt and A. Borzì, Formulation and multigrid solution of Cauchy-Riemann optimal control problems, Comput. Vis. Sci. 14 (2011), pp. 79–90.
   Google Scholar
• M.M. Butt, Two-level difference scheme for the two-dimensional Fokker-Planck equation, Mathematics and Computers in Simulation, 180 (2021), pp. 276-288., 2021.
   Google Scholar
• J.S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations, J. Comput. Phys. 6 (1970), pp. 1–16.
   Google Scholar
• R. Dautray and J.-L. Lions, Problèmes d' évolution du premier ordre en temps, Chap. 18., Masson, Paris, 1984.
   Google Scholar
• I. Gihman and A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.
   Google Scholar
• C. Godrèche and J-M. Luck, Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes, J. Phys. A: Math. Theor. 52 (2019
   Web of Science ®Google Scholar
• K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, SIAM, 2008.
   Google Scholar
• D. Kalise, S. Kundu, and K. Kunisch, Robust feedback control of nonlinear PDEs by numerical approximation of high-Dimensional hamilton-Jacobi-Isaacs equations, SIAM J. Appl. Dyn. Syst. 19 (2020), pp. 1496–1524.
   Web of Science ®Google Scholar
• J.B. Krawczyk, On loss-avoiding payoff distribution in a dynamic portfolio management problem, J. Risk Financ. 9 (2008), pp. 151–172.
   Google Scholar
• J.B. Krawczyk, Delivering left-Skewed portfolio payoff distributions in the presence of transaction costs, Risks 3 (2015), pp. 318–337.
   Web of Science ®Google Scholar
• O.A. Ladyzhenskaya, R. Bellman, and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, 2016.
   Google Scholar
• J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
   Google Scholar
• N.G. Meyers, An Lp-estimate for the gradient of solutions of secondorder elliptic divergence equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze $3^e$ série, tome 17 (1963), pp. 189–206.
   Google Scholar
• M. Mohammadi and A. Borzì, Analysis of the Chang-Cooper discretization scheme for a class of Fokker- Planck equations, J. Numer. Math. 23 (2015), pp. 271–288.
   Google Scholar
• A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994.
   Google Scholar
• H. Risken, The Fokker-Planck Equation Method of Solution and Applications, Springer Verlag, Berlin, 1989.
   Google Scholar
• S. Roy, M. Annunziato, and A. Borzi, A Fokker-Planck feedback control-Constrained approach for modelling crowd motion, J. Comput. Theor. Trans. 45 (2016), pp. 442–458.
   Web of Science ®Google Scholar
• S. Roy, M. Annunziato, A. Borzi, and C. Klingenberg, A Fokker-Planck approach to control collective motion, Comput. Optim. Appl. 69 (2018), pp. 423–459.
   Web of Science ®Google Scholar
• U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid, Academic Press, London, 2001.
   Google Scholar
• F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, AMS, Providence, RI, 2010.
   Google Scholar
• D.W.K. Yeung and S.E.A. Stewart, Stationary probability distributions of some Lotka-Volterra types of stochastic predation systems, Stoch. Anal. Appl. 13 (1995), pp. 503–516.
   Web of Science ®Google Scholar