Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progresson using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.

## Type of Work

Article (44) Book (0) Theses (0) Multimedia (0)

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Peer-reviewed only (36)

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## Campus

UC Berkeley (5) UC Davis (3) UC Irvine (4) UCLA (4) UC Merced (1) UC Riverside (2) UC San Diego (2) UCSF (0) UC Santa Barbara (0) UC Santa Cruz (0) UC Office of the President (1) Lawrence Berkeley National Laboratory (33) UC Agriculture & Natural Resources (0)

## Department

Donald Bren School of Information and Computer Sciences (3) Department of Computer Science (3)

Samueli School of Engineering (3) Electrical Engineering and Computer Science (3)

Bourns College of Engineering (1) Department of Mathematics (1) Research Grants Program Office (RGPO) (1) School of Medicine (1)

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Dermatology Online Journal (1)

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Engineering (2) Physical Sciences and Mathematics (1)

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## Scholarly Works (44 results)

The three-dimensional reconstruction of macromolecules from two-dimensional single-particle electron images requires determination and correction of the contrast transfer function (CTF) and envelope function. A computational algorithm based on constrained non-linear optimization is developed to estimate the essential parameters in the CTF and envelope function model simultaneously and automatically. The application of this estimation method is demonstrated with focal series images of amorphous carbon film as well as images of ice-embedded icosahedral virus particles suspended across holes.

Ptychography promises diffraction limited resolution without the need for high resolution lenses. To achieve high resolution one has to solve the phase problem for many partially overlapping frames. Here we review some of the existing methods for solving ptychographic phase retrieval problem from a numerical analysis point of view, and propose alternative methods based on numerical optimization.

We present a practical approach to calculate the complex band structure of an electrode for quantum transport calculations. This method is designed for plane wave based Hamiltonian with nonlocal pseudopotentials and the auxiliary periodic boundary condition transport calculation approach. Currently there is no direct method to calculate all the evanescent states for a given energy for systems with nonlocal pseudopotentials. On the other hand, in the auxiliary periodic boundary condition transport calculation, there is no need for all the evanescent states at a given energy. The current method fills this niche. The method has been used to study copper and gold nanowires and bulk electrodes.

A framework for constructing circulant and block circulant preconditioners ($C$) for a symmetric linear system $Ax=b$ arising from certain signal and image processing applications is presented in this paper. The proposed scheme does not make explicit use of matrix elements of $A$. It is ideal for applications in which $A$ only exists in the form of a matrix vector multiplication routine, and in which the process of extracting matrix elements of $A$ is costly. The proposed algorithm takes advantage of the fact that for many linear systems arising from signal or image processing applications, eigenvectors of $A$ can be well represented by a small number of Fourier modes. Therefore, the construction of $C$ can be carried out in the frequency domain by carefully choosing its eigenvalues so that the condition number of $C^TAC$ can be reduced significantly. We illustrate how to construct the spectrum of $C$ in a way such that the smallest eigenvalues of $C^TAC$ overlaps with those of $A$ extremely well while the largest eigenvalues of $C^TAC$ are smaller than those of $A$ by several orders of magnitude. Numerical examples are provided to demonstrate the effectiveness of the preconditioner on accelerating the solution of linear systems arising from image reconstruction application.