Collocation methods for nonlinear differential equations on low-rank manifolds

Abstract

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

Figures (8)

• Figure 1: A sketch of the low-rank manifold $\mathcal{M}_{\bm r}$ and its tangent space $T_{Y}\mathcal{M}_{\bm r}$ at the point $Y \in \mathcal{M}_{\bm r}$. Also depicted is $Z \in \mathbb{R}^{n_1 \times \cdots \times n_d}$ and its orthogonal projection $\widehat{P}_{Y} Z$ and oblique projection $P_{Y} Z$ onto the tangent space $T_{Y}\mathcal{M}_{\bm r}$. The orthogonal projection is the best approximation of $Z$ on the tangent space with respect to the Frobenius norm but is impractical compute when $G$ lacks rank structure. The oblique projection is a quasi-optimal approximation of $Z$ on the tangent space that is efficient to compute for any $G$ that can be evaluated entry-wise. Such oblique projectors allow us to efficiently apply dynamical low-rank methods to a broad class of nonlinear differential equations.
• Figure 2: Tensor network diagrams of a $d=6$ dimensional tensor in the TT format. (a) No orthogonalization with the TT-core partial products $C_{\leq 2}$ and $C_{>2}$ indicated. (b) Orthogonalized TT \ref{['orth_TT']} with $k=2$ and left orthogonal TT-cores $U_{\leq 2}$ and right orthogonal TT-cores $V_{>2}$ indicated.
• Figure 3: Illustration of the TT-cross-DEIM left-to-right sweep that computes multi-indices $\mathcal{I}^{\leq j}$ sequentially for $j = 1,2,\ldots,d-1$ with $d=4$.
• Figure 4: Low-rank approximations to the solution of the two-dimensional Vlasov-Poisson equation \ref{['VP']}. (a) Relative error versus time of TT-cross and ST-SVD solutions. The TT-cross solution was computed using rank-adaptive singular value threshold $\epsilon_l = 10^{-7}$ and the ST-SVD solution was computed using truncation threshold $\delta = 10^{-7}$. (b) Rank versus time of the rank-adaptive TT-cross and ST-SVD solutions and the numerical rank of the reference RK4 solution with singular value threshold $10^{-7}$. (c) Relative error versus time of solutions computed with interpolatory and orthogonal projector-splitting using rank-adaptive singular value threshold $\epsilon_l = 10^{-7}$. (d) Rank versus time of the rank-adaptive solutions computed with interpolatory and orthogonal projector splitting integrators and the numerical rank of the reference RK4 solution with singular value threshold $10^{-7}$.
• Figure 5: Low-rank approximations to the solution of the three-dimensional Allen-Cahn equation \ref{['Allen-Cahn']} computed with the TT-cross and ST-SVD methods. The ranks were determined using different truncation tolerances $\delta = 10^{-4},10^{-6},10^{-10}$ in the ST-SVD method. (a) Relative error in the Frobenius norm versus time. (b) 1-norm of the TT-rank vector versus time. (c) Relative error of interpolatory (i-proj) and orthogonal (o-proj) projections onto the tensor manifold tangent space versus time. (d) Condition number of the matrices non-orthogonalized matrices in \ref{['TT_cross_time']} and the corresponding orthogonalized matrices in \ref{['TT_cross_time2']} used to construct the TT-cross solution at each time step.