The low-rank tensor-train finite difference method for three-dimensional parabolic equations

This paper presents a numerical framework for the low-rank approximation of the solution to three-dimensional parabolic problems. The key contribution of this work is the tensorization process based on a tensor-train reformulation of the second-order accurate finite difference method. We advance the solution in time by combining the finite difference method with an explicit and implicit Euler method and with the Crank–Nicolson method. We solve the linear system arising at each time step from the implicit and semi-implicit time-marching schemes through a matrix-free preconditioned conjugate gradient (PCG) method, appositely designed to exploit the separation of variables induced by the tensor-train format. We assess the performance of our method through extensive numerical experimentation, demonstrating that the tensor-train design offers a robust and highly efficient alternative to the traditional approach. Indeed, the usage of this type of representation leads to massive time and memory savings while guaranteeing almost identical accuracy with respect to the traditional one. These features make the method particularly suitable to tackle challenging high-dimensional problems.