A fast sequentially-decoupled matrix-decomposed algorithm for variable-order time-fractional optimal control and error estimate

Abstract

We introduce a fast sequentially decoupled matrix-decomposed approach tailored for optimal control problems constrained by a Caputo time-fractional diffusion equation featuring hidden memory and space-dependent order. Our method unveils a quasi translation-invariant structure adept at managing spatio-temporal dependencies. This structure not only slashes the computational burden of coefficients from $O\left(M{N}^2\right)$ to $O(MNlnN)$, where $M$ and $N$ denote the spatial degrees of freedom and temporal steps in discretization, respectively, but also untangles the coupling between the space-dependent order and the inner product of the finite element method. Furthermore, we derive a swift matrix-decomposed algorithm designed to tackle the first-order optimality system, yielding a marked improvement in computational cost from $O\left(M{N}^2\right)$ to $O\left(MN{ln}^{3}N\right)$ in each iteration. We substantiate our approach through rigorous numerical analysis and present numerical experiments to validate the theoretical underpinnings.