Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Abstract

We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.

Tensor rank reduction via coordinate flows

“…Integrating the PDE (20) forward in on a FTT tensor manifold, e.g. using rank-adaptive step-truncation methods [32,31,20] or dynamic tensor approximation methods [11,10,9,24,21], results in a FTT approximation v TT (x; ) of the function v(x; ) for all ≥ 0. The computational cost of this approach for computing the FTT expansion of a FTT ridge function is precisely the same cost as solving the hyperbolic PDE (20) in the FTT format, which in the case of step-truncation or dynamic approximation has computational complexity that scales linearly with d. Note that the accuracy of v TT (x; ) as an approximation of v(x; ) depends on the step-size and order of integration scheme used to solve the PDE (20).…”

“…It is straightforward to verify that ( 25) satisfies (26). Note that v(x; ) in ( 25) is not an FTT tensor if = πk/2 and k ∈ N. To compute the FTT representation of v TT (x; ) we can solve the PDE (26) on a tensor manifold using step-truncation or dynamic tensor approximation methods [9,11,31,33]. Given the low dimensionality of the spatial domain in this example (d = 2), we can also evaluate (25) directly, and compute its FTT decomposition by solving an eigenvalue problem.…”

“…It is convenient to use a step-truncation method [31,33,20] to integrate the initial value problem (45) on a FTT tensor manifold. This is because applying the FTT truncation operation to v(x; ) requires computing the orthogonalized tensor cores Q ≤i ( ), Q >i ( ) (i = 1, 2, .…”

“…In order to guarantee that the solution v i+1 is a low-rank FTT tensor, we apply a truncation operator to the right hand side of (47). This yields the step-truncation method [31]…”

“…From this dynamical system perspective we derive a partial differential equation (PDE) for the -dependent function v(x; ) = u(A( )x). We then integrate the PDE for v(x; ) using a step-truncation tensor integration scheme [31,33,20] to obtain v(x; ) in a low-rank tensor format for all . The quasi-optimal rankreducing coordinate map A is obtained by minimizing a rank-related cost function using an appropriate path of transformations A( ).…”