Tensor-Train Formulation and Numerical Study of the Peaceman–Rachford and Douglas–Rachford Splitting Schemes for Multidimensional Parabolic Evolution Equations

We formulate the Peaceman–Rachford (PR) and Douglas–Rachford (DR) operator splitting schemes for multidimensional parabolic evolution equations in the tensor-train (TT) format. The resulting TT-PR and TT-DR algorithms maintain the evolving state and the split operators in TT form, solve the implicit subproblems approximately with the Alternating Minimal Energy (AMEn) method, and control TT ranks through preventive rounding and a rank cap enforced within the inner solves. We analyze per-step computational complexity, practical rank behavior, and total error decomposition, and we validate the algorithms on benchmark problems in two to eight spatial dimensions, with additional higher-dimensional scaling illustrations.