2.5 Numerical experiments
2.5.4 A stiff differential equation
Since the error analysis of the projector-splitting integrator given in Section 2.3 relies on both, the boundedness and Lipschitz continuity of F , it does not transfer directly to stiff problems such as spatially discretized partial differential equations. Numerical evidence however suggests that the projector-splitting scheme is robust and accurate also in this case. We consider the time-dependent Schrödinger equation in two dimensions with a harmonic potential, iut(x, t) =− 1 2∆u(x, t) + 1 2x >A x u(x, t), x ∈ R2, t > 0, u(x, 0) = π−1/2exp1 2x 2 1+ 1 2(x2− 1) 2, with A = 2 −1 −1 3 ! .
As the right-hand side contains a second order differential operator, its spatial semidis- cretization scales as ∆x−2, where ∆x is the spatial gridsize and therefore the Lipschitz constant of the right-hand side is of the same magnitude. While the initial data is of rank 1, the non-diagonal potential will increase the effective rank of the solution during time evolution. We discretize the problem using Fourier collocation with m× m grid points on Ω = [−7.5, 7.5]2. The spatially localized solution is essentially supported within Ω. The
2. Error analysis of the matrix projector-splitting integrator
solve low-rank approximations to the problem with ranks r = 1, 2, . . . , 20 using the Lie– Trotter splitting scheme, integrating up to the timet = 5. We use m = 64 and m = 128, and time steps of length h = 0.02 and h = 0.01. The subproblems are solved to high accuracy by approximating the action of the matrix exponential in a Krylov subspace, see, e.g., [HL97, Saa92], generated by the Arnoldi process [Arn51]. We compare the low-rank approximation to a full-rank reference solution computed by standard Fourier collocation, see, e.g., [Boy01], and Arnoldi time stepping withm = 128 and h = 0.01. The error is mea- sured in the Frobenius norm, scaled such that it approximates the continuousL2(Ω)-norm.
We depict it in the following figure:
2 4 6 8 10 12 14 16 18 20 10−10 10−8 10−6 10−4 10−2 100
Figure 2.8: Error at different approximation ranks when solving the Schrödinger equation on an m× m spatial grid. We use m = 64 (dashed) and m = 128 (solid) grid points per dimension, and the time stepsh = 0.02 (×) and h = 0.01 (plain).
The error decreases exponentially with the rank, which indicates that the method is robust with respect to small singular values also for stiff problems. For the time step h = 0.02 we see how the error at high approximation ranks is slightly larger for the finer spatial grid, suggesting a dependence on the Lipschitz constant. The dependence is however mild, and the method much more robust with respect to stiffness than explained by the theory presented in Theorem 2.4.
3
A low-rank splitting integrator
for stiff matrix differential equations
We have ended the previous chapter with a numerical example that illustrates a good performance of the projector-splitting integrator also for a stiff matrix differential equation in order to compute low-rank approximations of matrix ODEs. To our best knowledge, it is not known or shown why the projector-splitting integrator has this favorable behavior with regard to stiff differential equations. The difficulty in its error analysis is the usage of the Lipschitz constant of the right-hand side. To omit this difficulty, we will propose an integration method for stiff matrix differential equations, which in contrast can be analyzed without introducing the Lipschitz constant of the right-hand side.We are dealing with a specific right-hand side of the differential equation, of which the solution needs to be approximated in terms of low rank. The class of stiff and semi-linear matrix differential equations we consider in the time intervalt0 ≤ t ≤ T is given by
.
A(t) = B A(t) + A(t) B>+G(t, A(t)), A(t0) = A0,
where A(t) ∈ Rn1×n1 is the unknown matrix that satisfies the differential equation and
G : [t0,∞) × Rn1×n1 → Rn1×n1 is a given function. We aim to find a low-rank approxi-
mationY(t)∈ M = {Y(t) ∈ Rn1×n1 : rank Y(t) = r} to the solution of the above matrix
differential equation. We propose an integration scheme, which handles the stiff part of the differential equation by simply splitting it off and consider two subproblems, which can be treated separately. For both arising subproblems we compute a low-rank solution by following the dynamical low-rank approximation proposed in [KL07]. The subproblem for the linear stiff part can be integrated explicitly and efficiently by use of exponential integrators. For the nonlinear non-stiff subproblem we apply the projector-splitting inte- grator proposed in [LO14]. This procedure is derived in Section 3.1, where we also give a practical algorithm that is simple and efficient. Afterwards, we perform an error analysis in Section 3.2 that does not require the Lipschitz constant of the stiff right-hand side of the above matrix differential equation and further shows the key property of the method being robust with respect to small singular values, which might appear in case of over- approximation, i.e., when choosing the approximation rank rather large. The method we propose proves to be of first order. In Section 3.3 we discuss the order of convergence when applying a second order method, such as the Strang splitting scheme. As a special case of the above stiff matrix differential equation, we consider differential Lyapunov equa- tions (DLEs) in Section 3.4, which are of crucial importance in many applications, e.g.,
Kalman filtering [Kal60, KB61, AG15] or model reduction of linear time-varying systems [LSS16, San04]. Another important class of matrix differential equations that follow the structure of the above given stiff matrix differential equation are differential Riccati equa- tions (DREs) [Rei72, BL18]. They play an essential role in optimal and robust control problems [Men07], optimal filtering [DS87] and differential games [Baş91, BM17]. Finally, we illustrate the favorable behavior of the proposed method for stiff matrix ODEs with the help of numerical examples given in Section 3.6.
This chapter is based on a joint work of the author with A. Ostermann and C. Piazzola, see [OPW18].
3.1
The low-rank Lie–Trotter splitting integrator
We consider the following matrix differential equation
.
A(t) = B A(t) + A(t) B>+G(t, A(t)), A(t0) = A0, (3.1)
for t0 ≤ t ≤ T , where A(t) ∈ Rn1×n1 is the unknown solution matrix and the function
G : [t0,∞)×Rn1×n1 → Rn1×n1 is supposed to be nonlinear. The matrixB∈ Rn1×n1 is time-
independent. In many cases, where we have to deal with a stiff matrix differential equation of type (3.1), there is a parabolic partial differential equation underlying, where the elliptic operator is assumed to generate a strongly continuous semigroup. After discretizing this parabolic partial differential equation in space, we obtain the matrixB, which is the spatial discretization of the elliptic differential operator. Therefore, the stiffness of (3.1), which is given by the linear partB A(t) + A(t) B>, is induced by the matrixB. The exact full-rank solution of the above differential equation can be represented by the variation-of-constants formula as A(t) = e(t−t0)BA(t 0)e(t−t0)B > + Z t t0 e(t−s)BG(s, A(s))e(t−s)B>ds.
We aim to compute an approximate solution Y(t) ∈ M to A(t), which is of low rank r withr n1. The difficulty is the stiffness of the right-hand side of (3.1), which makes a
direct application of the dynamical low-rank approximation by employing the projector- splitting integrator unfeasible. We have seen in the chapter before that in this case, the error bound of the projector-splitting integrator depends on the Lipschitz constant of the full right-hand side. In order to elude difficulties with the Lipschitz constant of stiff matrix differential equations of the class (3.1), the key idea is to separate the ODE into a stiff and a non-stiff subproblem, respectively. For both subproblems we compute a low-rank solution.