The Iso-Inverse: A Contravariant Sparse Approximate Inverse Matrix.

Area of Science:

• Computational Chemistry
• Numerical Analysis
• Linear Algebra

Background:

• Sparse matrices are fundamental in computational science, particularly in electronic structure calculations.
• Approximating the inverse of sparse matrices (sparse inverse approximation) is crucial for efficiency.
• Existing methods often minimize residuals, which can lead to approximations that differ in sparsity structure.

Purpose of the Study:

• To introduce and define a novel sparse inverse approximation, termed the iso-inverse.
• To establish that the iso-inverse shares the same sparsity structure as the original matrix.
• To explore the potential applications of the iso-inverse in scientific computations.

Main Methods:

• Definition of the iso-inverse matrix (B) approximating the inverse of a sparse matrix (A).
• The iso-inverse is constructed by enforcing the condition AB = I for a specific subset of identity matrix elements.
• Analysis of the iso-inverse's properties, including its sparsity structure and contravariant variation with A.

Main Results:

• The proposed iso-inverse (B) accurately approximates the inverse of matrix A (A^-1).
• The iso-inverse B possesses the identical sparsity structure as the original matrix A.
• The construction method differs from residual minimization, enforcing exactness on a subset of the identity matrix.

Conclusions:

• The iso-inverse offers a new approach to sparse inverse approximation with preserved sparsity.
• This method provides an alternative to residual-minimizing approximations.
• The iso-inverse shows potential utility in advanced computational tasks like electronic structure calculations using nonorthogonal localized molecular orbitals.