Related papers: Solving differential Riccati equations: A nonlinea…
In this work we propose an efficient black-box solver for two-dimensional stationary diffusion equations, which is based on a new robust discretization scheme. The idea is to formulate an equation in a certain form without derivatives with…
In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such…
This paper considers a stochastic linear quadratic problem for discrete-time systems with multiplicative noises over an infinite horizon. To obtain the optimal solution, we propose an online iterative algorithm of reinforcement learning…
We consider the Cauchy problem for the 1D generalized Schr\"odinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transparent boundary conditions (TBCs)…
Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models…
We present a novel method to approximate optimal feedback laws for nonlinear optimal control based on low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variational principle with the modification that the…
We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in…
This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving…
A novel hybrid algorithm is presented for the Boltzmann-BGK equation, in which a low-rank decomposition is applied solely in the velocity subspace, while a full-rank representation is maintained in the physical (position) space. This…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau…
Non-hydrostatic atmospheric models often use semi-implicit temporal discretisations in order to negate the time step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision…
In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme…
This paper proposes a novel iterative algorithm to compute the stabilizing solution of regime-switching stochastic game-theoretic Riccati differential equations with periodic coefficients. The method decomposes the original complex…
We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical…
We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local…
Recent advances in nonlinear dynamical systems theory provide a new insight into numerical properties of discrete algorithms developed to solve nonlinear initial value problems. Basic features like accuracy and stability are well pointed…
The efficient numerical integration of large-scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications. Standard numerical methods applied to such problems require an unduly…
We present high order explicit geometric integrators to solve linear-quadratic optimal control problems and $N$-player differential games. These problems are described by a system coupled non-linear differential equations with boundary…
Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial…