Solving Eigenvalue Problems in High Dimensions Using Contour Integration and Tensor Train Format

Abstract. In high-dimensional settings, solving eigenvalue problems is hindered by the curse of dimensionality, particularly when only a subset of eigenpairs within a prescribed spectral interval is sought. In this work, we investigate an adaptation of the FEAST algorithm, originally developed for symmetric eigenproblems based on contour integration, to computations where both operators and vectors are represented in the tensor train (TT) format. This representation drastically reduces memory and computational demands. We introduce an adaptive scheme for determining the projection subspace dimension by incorporating a rank-revealing Modified Gram–Schmidt procedure with pivoting tailored to TT-vectors. A perturbation-based analysis provides explicit bounds on the attainable residual accuracy, from which we derive a robust stopping criterion for the proposed TT-FEAST algorithm. Moreover, we design a continuation strategy that gradually refines convergence and rounding tolerances to effectively control memory growth during iterations. To demonstrate the effectiveness of TT-FEAST as a viable alternative to existing high-dimensional eigensolvers when a few eigenvalues are required, we present numerical experiments on problems up to 12 dimensions, including the Laplacian and a vibrational Hamiltonian operator.