Dynamics of structures

Single subject prediction of brain disorders in neuroimaging: Promises and pitfalls.

Mathematical programming

We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

Some modified matrix eigenvalue problems.

We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large-scale SDP problems with the matrix dimension $n$ up to $4,110$ and the number of equality constraints $m$ up to $2,156,544$ show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with $n=4,110$ and $m=1,154,467$) in the Seventh DIMACS Implementation Challenge much more accurately than in previous attempts.

/pdf/a-newton-cg-augmented-lagrangian-method-for-semidefinite-3m8u9dwsas.pdf

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

Identifying the patterns of large data sets is a key requirement in data mining. A powerful technique for this purpose is the principal component analysis (PCA). PCA-based clustering algorithms are effective when the data sets are found in the same location. In applications where the large data sets are physically far apart, moving huge amounts of data to a single location can become an impractical, or even impossible, task. A way around this problem was proposed in [10], where truncated singular value decompositions (SVDs) are computed locally and used to reduce the communication costs. Unfortunately, truncated SVDs introduce local approximation errors that could add up and would adversely affect the accuracy of the final PCA. In this paper, we introduce a new method to compute the PCA without incurring local approximation errors. In addition, we consider the situation of updating the PCA when new data arrive at the various locations.

/pdf/principal-component-analysis-for-distributed-data-sets-with-2nmrqq5i7y.pdf

Principal component analysis for distributed data sets with updating

http://www.math.niu.edu/~dattab/psfiles/BAI.PROOF.pdf

Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.

A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenproblem defined as follows: given a set of complex $n$-vectors $\{\bx_i\}_{i=1}^{m}$ and a set of complex numbers $\{\la_i\}_{i=1}^m$, and an $s$-by-$s$ real matrix $C_0$, find an $n$-by-$n$ real centrosymmetric matrix $C$ such that the $s$-by-$s$ leading principal submatrix of $C$ is $C_0$, and $\{\bx_i\}_{i=1}^{m}$ and $\{\la_i\}_{i=1}^m$ are the eigenvectors and eigenvalues of $C$, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real $n$-by-$n$ matrix $\tilde{C}$, find a matrix $C$ which is the solution to the constrained inverse problem such that the distance between $C$ and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. Some illustrative experiments are also presented.

The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation

The Cayley transform method is a Newton-like method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and it improves the efficiency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeoff between the required inner and outer iterations can be obtained.

/pdf/an-inexact-cayley-transform-method-for-inverse-eigenvalue-3d564psndl.pdf

Zheng-Jian Bai

Papers

Principal component analysis for distributed data sets with updating

Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach

A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems

The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation

An Inexact Cayley Transform Method For Inverse Eigenvalue Problems