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The paper presents a computational algorithm for solving problems formulated in terms of Mean Field Game theory with restriction on the control resource. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman ones. The article is devoted to the discrete approximation of these equations. In addition, contrary to other studies, the algorithm constructed here is applicable to problem with more general control function. This algorithm is applied to the problem of heating insulation. [ABSTRACT FROM AUTHOR]

The paper presents a computational algorithm for solving the problem of pricing an American put-option described by the Black-Sholes equation. Firstly, two changes of variables are presented. First change is used to eliminate a discontinuity of solution derivatives. The second one transforms the equation on the domain with free boundary to the equation with an unknown coeffcient on a domain with fixed boundary. Secondly, the finite-difference approximation of the obtained equation is fulfilled including boundary condition at infinity. Finally, the computational algorithm is presented for solving discrete analogue of parabolic equation with iteration of the secant method at each time layer to found an unknown coeffcient with the help of the additional boundary condition. [ABSTRACT FROM AUTHOR]

The paper presents a computational algorithm for solving problems formulated in terms of Mean Field Game theory with diffusion expressed by fractional derivative. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: modified Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman equations with the highest derivative in space of fractional order less than 2. The article is devoted to the discrete approximation of these equations and the computational algorithm for solving discrete analogues. [ABSTRACT FROM AUTHOR]

The numerical methods are presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium (i.e., the Nash equilibrium for an infinite number of players) leads to the coupled system of two parabolic partial differential equations: the Hamilton-Jacobi-Bellman-Isaacs equation and the Fokker-Planck-Kolmogorov one. The description is focused on the discrete approximation of these equations and on the application of the MFG theory directly at discrete level. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to difference schemes with directed differences used by other authors, here the semi-Lagrangian approximation is applied which improves some properties of a discrete problem of this type. This implies the fast convergence of an iterative algorithm for the monotone minimization of the cost functional. [ABSTRACT FROM AUTHOR]

We present the semi-Lagrangian approximation of transfer operator for three-dimensional convection-diffusion problem with corresponding initial and boundary conditions. This problem describes, for instance, a transfer of a substance with diffusion. To construct numerical method, we decompose operator of this equation into two parts. The first one is the transfer operator. The second part is the elliptic diffusion terms. To approximate the first part, we use conservative semi-Lagrangian approximation connecting two integrals of solution at neighboring time levels. To compute integral at the previous time level, we approximate integral domain by 48 tetrahedrons and interpolate solution by trilinear functions. The second part is approximated by conventional finite differences. Finally, we combine approximations of two parts to get a system of linear equations at each time level. Matrix of this system at each time level is the symmetric M-matrix. The proposed approximation has the first-order convergence that is confirmed by computational experiments. [ABSTRACT FROM AUTHOR]