Abstract-We apply first-and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.

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 to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in an efficient way by utilizing a lowrank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used, e.g., for time step adaptivity, and our numerical experiments in this direction show promising results.

Summary The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large‐scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.

Abstract. We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert-Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis.

We consider operator-valued differential Lyapunov and Riccati equations, where the operators B and C may be relatively unbounded with respect to A (in the standard notation). In this setting, we prove that the singular values of the solutions decay fast under certain conditions. In fact, the decay is exponential in the negative square root if A generates an analytic semigroup and the range of C has finite dimension. This extends previous similar results for algebraic equations to the differential case. When the initial condition is zero, we also show that the singular values converge to zero as time goes to zero, with a certain rate that depends on the degree of unboundedness of C. A fast decay of the singular values corresponds to a low numerical rank, which is a critical feature in large-scale applications. The results reported here provide a theoretical foundation for the observation that, in practice, a low-rank factorization usually exists.

We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the L 2 operator norm, and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations, and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.

We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and firstorder in time. The theoretical results are verified by several numerical experiments in two and three dimensions.

Abstract. We present a convergence analysis for the implicit-explicit (IMEX) Euler discretization of nonlinear evolution equations. The governing vector field of such an equation is assumed to be the sum of an unbounded dissipative operator and a Lipschitz continuous perturbation. By employing the theory of dissipative operators on Banach spaces, we prove that the IMEX Euler and the implicit Euler schemes have the same convergence order, i.e., between one half and one depending on the initial values and the vector fields. Concrete applications include the discretization of diffusion-reaction systems, with fully nonlinear and degenerate diffusion terms. The convergence and efficiency of the IMEX Euler scheme are also illustrated by a set of numerical experiments.