Abstract
This paper introduces and analyzes the new grid-based tensor approach
to approximate solutions of the elliptic eigenvalue problem
for the 3D lattice-structured systems.
We consider the linearized Hartree–Fock equation over a spatial
A Appendix
A.1 Overview on Block Circulant Matrices
We recall that a one-level block circulant matrix
where
where
and
In the case
so that (A.2) simplifies to
Let
the unitary matrix of Fourier transform. Since the shift matrix
the same holds for any circulant matrix,
where
Conventionally, we denote by
For example, the first block column in (A.1)
has the form
It is known that similarly to the case of circulant matrices (A.5),
a block circulant matrix in
so that the
Proposition A.1.
For
where
can be recognized as the j-th
A set of eigenvalues λ of A is then given by
The eigenvectors corresponding to the spectral sets
can be represented in the form
with
Proof.
We combine representations (A.2) and (A.4) to obtain
where the final step follows by the definition of FT matrix and by the construction of
The next
statement describes the block-diagonal form for a class of symmetric BC
matrices,
Corollary A.2.
Let
where
Corollary A.2 combined with Proposition A.1 describes
a simplified structure of spectral data in the symmetric case.
Notice that the above representation imposes the symmetry of each
real-valued diagonal block
A.2 Multilevel Block Circulant/Toeplitz Matrices
We describe the extension of (one-level) block circulant matrices to multilevel structure.
First, we recall the main notions of multilevel block circulant (MBC) matrices
with the particular focus on the three-level case.
Given a multi-index
Definition A.3.
For
where
Likewise to the case of one-level BC matrices, it can be seen that a matrix
Recall that a symmetric block Toeplitz matrix
where
A matrix class
Definition A.4.
For
A similar recursion applies to the case
A.3 Rank-Structured Tensor Formats
We consider a tensor of order
The rank-
where
Given the rank parameter
completely specified by a set of orthonormal vectors
The remarkable approximating properties of the Tucker and canonical tensor decomposition
applied to the wide class of function-related tensors were revealed in
[39, 27, 42],
promoting using tensor tools for the numerical treatment of the multidimensional PDEs.
It was proved for some classes of function related tensors that the rank-
In the case of many spacial dimensions, the product type tensor formats provide
the stable rank-structured approximation.
The matrix-product states (MPS) decomposition has been for a long time used
in quantum chemistry and quantum information theory, see the survey paper [57].
The particular case of MPS representation is called a tensor train (TT) format
[51, 50]. The quantics-TT (QTT) tensor approximation method
for functional
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