Tensor networks and large-scale eigenvalue problems
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Abstract
The present thesis is concerned with the numerical solution of large-scale eigenvalue problems via methods based on tensor networks. We address the computation of an eigenvector associated with the minimal eigenvalue of a Hamilton operator modeling a quantum mechanical many-particle spin system. If d denotes the number of particles and q the number of degrees of freedom for each particle, then the size of this symmetric eigenvalue problem scales exponentially in d to the base q. A well-known strategy with regard to the numerical treatment is to reshape those components of a method whose size scales exponentially into a tensor, i.e. a higher-dimensional array of numbers. This opens up the possibility of representation in a low-rank tensor format with non-exponentially growing ranks. In this way, a numerical method may be formulated by means of tensor networks which are a convenient tool when working with low-rank tensor formats. Our focus is on the q-XYZ and the q-Potts model. We introduce sets of vectors with a certain sparsity pattern whose definition is based on the q-ary representation of the indices of the potential nonzero entries. We show that for each eigenspace of a Hamilton operator of the q-XYZ or q-Potts model, there exists an orthonormal basis all of whose elements have a sparsity pattern suitable for the model. Moreover we show that for the q-XYZ model, the characterization of the sparsity pattern depends on the coupling parameters determining the Hamilton operator. We consider the hierarchical Tucker format and the tensor train format and describe how to represent the given Hamilton operators in these formats. In order to solve the eigenvalue problem numerically, we employ on the one hand the locally optimal conjugate gradient method expressed in the hierarchical Tucker format and on the other hand the modified alternating linear scheme, also called density matrix renormalization group, built on the representation in the tensor train format. A key aspect of this thesis is the construction of an initial guess for a numerical eigensolver. We propose a strategy which utilizes information about the relation of solutions for two different small problem sizes to construct an approximation of a solution for the original large problem which then may be used as an initial guess in an iterative method. This particular way of construction is enabled by the specific sparsity patterns of the eigenvectors and we show that the result of the construction has a sparsity pattern suitable for an eigenvector as well. In this context we discuss for the q-XYZ model two alternative construction strategies depending on the coupling parameters. We explain how to carry out the construction of the initial guess in the hierarchical Tucker format and in the tensor train format and that the ranks of the initial guess are independent of d or scale only linearly in d. To demonstrate the effect of our contribution on the computational cost, we conduct extensive numerical tests. We compare the constructed initial guesses with random initial guesses in terms of an acceleration of the methods.