Identifying the Cause of and Fixing Ill-Conditioned Matrices in Nuclear Analysis Codes

Many of the analytical codes used in the nuclear industry, such as TRACE, RELAP5, and PARCS, approximate the equations that model the physics via a linearized system of equations. One common difficulty when solving linearized systems is that an accurately formulated system of equations may be ill-conditioned. Ill-conditioned matrices can result in significant amplification of error leading to poor, or even invalid, results. Ill-conditioned matrices lead to some challenging issues for the analytical code developers: • An ill-conditioned matrix is often solvable, and there may be no obvious indication numerically that something has gone wrong even though numerical error is large. Thus, how can ill-conditioning be effectively detected for a matrix? • When ill-conditioning is detected, how can the source of the ill-conditioning be determined so that it can be analyzed and corrected? Ill-conditioning is fundamentally a geometric problem that can be understood with geometric concepts associated with matrices and vectors. Geometric concepts and tools, useful for understanding the cause of ill-conditioning of a matrix, are presented. A geometric understanding of ill-conditioning can point to the rows or columns of the matrix that most contribute to ill-conditioning so that the source of ill-conditioning can be analyzed and understood, and leads to techniques for building matrix preconditioners to improve the solvability of the matrix.

Higher-Order Stabilized Perturbation for Recursive Eigen-Decomposition Estimation

Eigen-decomposition remains one of the most invaluable tools for signal processing algorithms. Although traditional algorithms based on QR decomposition, Jacobi rotations and block Lanczos tridiagonalization have been proposed to decompose a matrix into its eigenspace, associated computational expense typically hinders their implementation in a real-time framework. In this paper, we study recursive eigen perturbation (EP) of the symmetric eigenvalue problem of higher order (greater than one). Through a higher order perturbation approach, we improve the recently established first-order eigen perturbation (FOP) technique by creating a stabilization process for adapting to ill-conditioned matrices with close eigenvalues. Six algorithms were investigated in this regard: first-order, second-order, third-order, and their stabilized versions. The developed methods were validated and assessed on multiple structural health monitoring (SHM) problems. These were first tested on a five degrees-of-freedom (DOF) linear building model for accurate estimation of mode shapes in an automated framework. The separation of closely spaced modes was then demonstrated on a 3DOF + tuned mass damper (TMD) problem. Practical utility of the methods was probed on the Phase-I ASCE-SHM benchmark problem. The results obtained for real-time mode identification establishes the robustness of the proposed methods for a range of engineering applications.

ON THE NUMERICAL ACCURACY OF FIRST-ORDER APPROXIMATE SOLUTIONS TO DSGE MODELS

Many algorithms that provide approximate solutions for dynamic stochastic general equilibrium (DSGE) models employ the QZ factorization because it allows a flexible formulation of the model and exempts the researcher from identifying equations that give raise to infinite eigenvalues. We show, by means of an example, that the policy functions obtained by this approach may differ from both the solution of a properly reduced system and the solution obtained from solving the system of nonlinear equations that arises from applying the implicit function theorem to the model's equilibrium conditions. As a consequence, simulation results may depend on the specific algorithm used and on the numerical values of parameters that are theoretically irrelevant. The sources of this inaccuracy are ill-conditioned matrices as they emerge, e.g., in models with strong habits. Researchers should be aware of those strange effects, and we propose several ways to handle them.