Title: Spectral Slicing for Sparse Hermitian Matrices Based on Zolotarev's Functions
Abstract: This talk presents an efficient method for computing selected eigenpairs of sparse Hermitian matrices, using the composition of Zolotarev's functions as a rational filter. A hybrid fast algorithm based on the multifrontal method and the shift-invariance of Krylov subspaces is proposed to apply the rational filter efficiently. Assuming that the Hermitian matrix is of size $N\times N$ and contains $O(N)$ nonzero entries, the computational cost for computing $O(1)$ eigenpairs is $O(N^{1.5})$ operations. By the spectral slicing idea, the proposed method is able to compute the full diagonalization of this sparse Hermitian matrix with $O(N^{2.5})$ operations. Numerical examples of a wide range of sparse matrices shows that the proposed method is faster and more stable than existing spectral slicing algorithms based on contour-integrals.
Location: Y2E2-111
Time: 4:30-5:20 p.m.
TUESDAY, NOVEMBER 29, 2016: CME 300: First Year Seminar Series
Please join us for two speakers this week:
Speaker: James Zou, Assistant Professor of Biomedical Data Science
Bio: Dr. Zou is an Assistant Professor of Biomedical Data Science, Computer Science (courtesy) and Electrical Engineering (courtesy) at Stanford University. Dr. Zou works on a wide range of problems in machine learning (from proving mathematical properties to designing new probabilistic models) and is especially interested in applications in human genomics. He received his Ph.D. from Harvard University in May 2014 and was a member of Microsoft Research New England. Before this, he completed Part III in Mathematics at the University of Cambridge and was a Simons fellow at U.C. Berkeley. Dr. Zou joined Stanford in Fall 2016.
Speaker: Nick Henderson, ICME Research Associate
Bio: Nick Henderson is a Research Associate and Instructor at ICME. His current work includes CUDA Monte Carlo simulation of radiation therapy dosimetry based on Geant4. In collaboration with the Geant4 groups at SLAC and KEK. To learn more about Nick, visit his webpage at http://stanford.edu/~nwh/.
Location: Y2E2-101 Time: 12:30-1:20 p.m.
THURSDAY, DECEMBER 1, 2016: CME 242: Mathematical and Computational Finance Speaker: Jeff Bohn, State Street
Title: Improving portfolio-risk assessment with latent-factor-based simulation
Abstract: Forward-looking, portfolio-risk models have become increasingly important for robust risk assessment and management of financial portfolios. Latent-factor models used with a forward-looking, Monte-Carlo simulation show promise for improving portfolio-risk assessment. In this paper, we show how a latent-factor model can facilitate an integrated, forward-looking risk assessment of a financial portfolio. Further, this bottom-up framework can disentangle contribution to large downside losses at a position, sub-portfolio and factor level. When embedded in a robust risk governance process that starts with a quantified risk-appetite statement, this framework makes it possible to more effectively identify sources of portfolio concentration/correlation risk. This analysis can lead to better identification of hedging strategies, better portfolio re-allocation strategies and more robust portfolio-risk management.
Location: 200-205
Time: 4:50-5:50p.m.
THURSDAY, DECEMBER 1, 2016: CME 510: Linear Algebra and Optimization Seminar Speaker: Victor Minden, ICME PhD student
Title: Fast algorithms exploiting low-rank structure for graph clustering and integral equations
Abstract: First, I will present a new algorithm for spectral clustering of large graphs, i.e., using a k-dimensional embedding of the vertices of the graph to recover relatively well-connected components corresponding to latent community structure. Our method is based on the column-pivoted QR factorization of a matrix of eigenvectors associated with the graph, leading to a direct (non-iterative) algorithm that is simple to implement and exhibits recovery performance that tracks information-theoretic bounds for the stochastic block model, a generative model for graphs with community structure.
Second, I will discuss the strong recursive skeletonization (RS-S) factorization, a new approximate factorization for "inverting the fast multipole method" for discretized linear integral equations associated with elliptic partial differential equations in two or three dimensions.
Unlike previous skeletonization-based methods for solving such systems, the RS-S factorization uses the structure of the fast multipole method to hierarchically compress matrix blocks corresponding to far-field interactions while treating near-field interactions as full rank. This leads to an approximate factorization of the inverse operator as the product of many block unit-triangular matrices that may be used as a preconditioner or moderate-accuracy direct solver. Under suitable rank assumptions, the RS-S factorization exhibits linear computational complexity, which I will demonstrate with a number of numerical examples.
Joint work with Anil Damle, Kenneth L. Ho, and Lexing Ying.
This is the last LA/Opt seminar for 2016.
Location: Y2E2-101
Time: 4:30-5:50 p.m.
OTHER ICME RELATED SEMINARS: Applied Math Seminar Wednesday, November 30, 2016 Speaker: Yuehaw Khoo, Stanford
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