Lowrank Riemannian eigensolver for highdimensional Hamiltonians Abstract   Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as highdimensional eigenvalue problems, i.e., when the eigenvector can be naturally represented as a multidimensional tensor. Tensor methods have proven to be an efficient tool for the approximation of solutions of highdimensional eigenvalue problems, however, their performance deteriorates quickly when the number of eigenstates to be computed increases. We address this issue by designing a new algorithm motivated by the ideas of Riemannian optimization (optimization on smooth manifolds) for the approximation of multiple eigenstates in the tensortrain format, which is also known as matrix product state representation. The proposed algorithm is implemented in TensorFlow, which allows for both CPU and GPU parallelization.

Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming model We present three methods for distributed memory parallel inverse factorization of blocksparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quadtree matrix representation and parallel matrixmatrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasilinear Glutamic AcidAlanine molecules and threedimensional water clusters discretized using the standard Gaussian basis set STO3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.