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Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems. (English) Zbl 1071.65032

It is well known that integral operators in higher dimensions can be approximated by using the so-called hierarchical Kronecker tensor-product (HKT) format. The authors construct and analyze an HKT approximation for the general class of strongly positive operators defined as a sum of low-dimensional commutative operators and combine tensor product representation that includes one-dimensional operators. They develop a data sparse HKT approximation of the inverse of an elliptic operator as well as to the solution operator of the matrix Lyapunov-Sylvester equation. The approximation can be interpreted as an extension of the widely used fast Fourier transform.

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
65F30 Other matrix algorithms (MSC2010)
46B28 Spaces of operators; tensor products; approximation properties
47A80 Tensor products of linear operators
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