A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds

Abstract

In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm.

A chart based splitting integrator

Following the same lines as in [15], we justify how the chart based method introduced in Sect. 3.1.2 can be interpreted as a splitting scheme relying on the projection decomposition (3.4) as the sum of three contributions $P_{T_Z(t)}= P_1+ P_2 + P_3$. One integration step of the splitting method starting from $t_0$ to $t_1$ with initial guess $Z(t_0) = U(t_0)G(t_0)V(t_0)^T$ proceeds as follows.

• (S1) Find $Z \in \mathcal{U}_{Z(t_0)}$ on $[t_0,t_1]$ such that $\dot{Z} = P_{U} F(Z) P_{V}^T$ with initial condition $Z(t_0)$.

• (S2) Find $Z \in \mathcal{U}_{Z(t_0)}$ on $[t_0,t_1]$ such that $\dot{Z} = P_{U}^\perp F(Z) P_{V}^T$ with initial condition given by final condition of step (S1).

• (S3) Find $Z \in \mathcal{U}_{Z(t_0)}$ on $[t_0,t_1]$ such that $\dot{Z} = P_{U} F(Z) (P^\perp _{V})^T$ with initial condition given by final condition of step (S2).

At each step (Si) of the splitting, Z belongs to the neighborhood of $Z(t_0)$. Thus it is given by $Z(t) = U(t)H(t)V(t)^T$ with $U(t) = U(t_0)+U(t_0)_\perp X(t)$, $Y(t)= V(t_0)+V(t_0)_\perp Y(t)$ provided by the ODE solved at Step i of the chart based splitting, as stated in the following proposition.

Proposition 5.1

The solution of (S1) is given by Z with

$$\begin{aligned} \dot{H}= U^+ F (Z)V^T, \quad \dot{X} =0,\quad \dot{Y} = 0. \end{aligned}$$

(A.1)

with $H(t_0)=G(t_0)$, $X(t_0)=0$ and $Y(t_0)=0$. Set

• Letting $H_1$ be the final condition of H from (S1), the solution of (S2) is given by Z with

  $$\begin{aligned} \dot{X} H = {U}_\perp ^+ F (Z)(V^+)^T, \quad \dot{Y}= 0,\quad \dot{H}=0, \end{aligned}$$

  (A.2)

  with $H(t_0)=H_1$, $X(t_0)=0$ and $Y(t_0)=0$. Set $X_1 = X(t_1)$.

• Letting $X_1$ be the final condition of X from (S2), the solution of (S3) is given by Z with

  $$\begin{aligned} \dot{Y} H^T = {V}_\perp ^+ F (Z) (U^+)^T, \quad \dot{X}= 0,\quad \dot{H}=0, \end{aligned}$$

  (A.3)

  with $H(t_0)=H_1$, $X(t_0)=X_1$ and $Y(t_0)=0$.

Proof

For each step (Si), Z admits the decomposition

$$\begin{aligned} Z = (U(t_0)+U(t_0)_\perp X) H(V(t_0)+V(t_0)_\perp Y)^T \end{aligned}$$

with derivative

$$\begin{aligned} \dot{Z} =U(t_0)_{\perp } \dot{X} H(V(t_0)+V(t_0)_{\perp }Y)^T + (U(t_0)+U(t_0)_{\perp }X) \dot{H}(V(t_0)+ V(t_0)_\perp Y)^T\nonumber \\ + (U(t_0)+U(t_0)_{\perp }X) H(V(t_0)_{\perp }\dot{Y})^T. \end{aligned}$$

For (S1), the derivative satisfies $\dot{Z} = P_{U} F(Z) P_{V}^T$. Then, multiplying on the left by $U(t_0)^+$ and on the right by $(V(t_0)^+)^T$ the matrix $\dot{Z}$ in both expressions leads to $\dot{H}= U^+ F(Z) (V^+)^T$ and $\dot{X} =0, \dot{Y} = 0.$ Now let us turn to (S2). The derivative satisfies $\dot{Z} = P_{U}^\perp F(Z) P_{V}^T$. By multiplying on the right by $(V(t_0)^+)^T$, the equality is satisfied if $\dot{X} H = U_\perp ^+ F(Z) (V^+)^T$ and $\dot{Y}= 0, \dot{H} =0.$ The third point of the lemma is obtained from (S3) in the same manner, by multiplying the equation $\dot{Z} = P_{U} F(Z) (P_{V}^\perp )^T$ on the left by $(U(t_0)+U(t_0)_\perp X_1)^+$ and setting $\dot{X} =0$, $\dot{H}=0$. $\square$