A Density Matrix-based Algorithm for Solving Eigenvalue Problems (0901.2665v1)

Abstract: A new numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques, and takes its inspiration from the contour integration and density matrix representation in quantum mechanics. It will be shown that this new algorithm - named FEAST - exhibits high efficiency, robustness, accuracy and scalability on parallel architectures. Examples from electronic structure calculations of Carbon nanotubes (CNT) are presented, and numerical performances and capabilities are discussed.

• The paper presents FEAST, a novel algorithm that leverages density matrix and contour integration methods to efficiently solve symmetric eigenvalue problems.
• Its iterative refinement and Gauss-Legendre quadrature approach deliver robust performance and linear scalability, as demonstrated in electronic structure calculations for carbon nanotubes.
• The algorithm decouples computational expense from system size, making it ideal for parallel architectures and large-scale simulations.

Overview of the FEAST Algorithm for Symmetric Eigenvalue Problems

The paper presents a novel numerical algorithm, FEAST, designed for tackling the symmetric eigenvalue problem. Unlike traditional methods such as the Krylov subspace iteration techniques (Arnoldi and Lanczos algorithms) or Davidson-Jacobi techniques, FEAST draws inspiration from quantum mechanics via contour integration and density matrix representation. The algorithm exhibits efficiency, robustness, accuracy, and scalability, particularly beneficial for parallel architectures. FEAST's performance is demonstrated through electronic structure calculations of Carbon nanotubes (CNTs), offering a detailed discussion on its numerical capabilities.

Theoretical Insights

FEAST deviates from standard paradigms by leveraging the density matrix and contour integration methods inherent in quantum mechanics. The fundamental idea is based on a precise mathematical factorization of the contour integration of the Green's function matrix. The density matrix—which in quantum mechanics represents electron density—is computed via contour integration, offering an alternative approach to traditional eigenvalue problems. In essence, FEAST can compute the electron density efficiently, decoupling the number of necessary computations from the system size.

Algorithmic Structure

The algorithm formulates the eigenvalue problem $A x = \lambda B x$, focusing on a specified interval. FEAST determines all eigenpairs within this interval as it naturally captures the multiplicity of eigenvalues and circumvents the need for repeated orthogonalization processes. It combines problem-solving capability with numerical efficiency, executing a contour integration via a Gauss-Legendre quadrature. The computational infrastructure is characterized by the following principal elements:

• Numerical Integration: Solving linear systems along a complex contour.
• Ritz-Galerkin Projection: A reduced generalized eigenvalue problem is formed, minimizing the initial problem size.
• Iterative Refinement: The subspace is iteratively refined, using spectral solutions as initial guesses for successive approximations.

Numerical Results

The paper provides compelling numerical evidence of FEAST's efficiency and scalability. It was tested extensively on electronic structure calculations for CNTs, performing favorably against ARPACK, a well-established library for eigenvalue problems. FEAST maintained efficient scaling even as system size and eigenvalue multiplicity increased, showcasing the algorithm's resilience and adaptability. Particularly, it displayed impressive linear scale-up capabilities, supporting numerous eigenvalue computations simultaneously without increased computational expense.

Performance and Practical Implications

FEAST is tailored for both direct and iterative solution strategies for linear systems necessary in its process. Utilizing iterative solvers allows FEAST to be compatible with modern scalable applications, managing large-scale simulations efficiently. Performance tests revealed FEAST's potential in handling large, sparse systems, making it a valuable tool for scientific and engineering applications requiring high accuracy and parallel computational support.

Future Directions

The implications of FEAST stretch beyond current applications. Its framework might be expanded to address non-symmetric eigenvalue problems, potentially using complex contour regions. With advancements in parallel computing architectures, the FEAST algorithm could become instrumental in solving vast numbers of eigenpairs for large symmetric eigenvalue problems. Its architecture opens possibilities for even larger scientific computations, crossing into millions of eigenpairs with precision and efficiency.

The algorithm's capability to utilize previously computed subspaces as initial guesses not only optimizes computations for sequential problems but also enhances accuracy—an essential feature for iterative processes in emerging computational fields.