The Davidson Method

How do you know if what you feel is real, or is simply the natural result of a modified belief/desire? Does it even matter? In this work of philosophical short story fiction, Susan and Richard are getting a divorce, but their son is struggling to come to terms with it. They head to the clinic to learn about using the “Davidson Method” to modify their child’s brain so he will be more accepting of their divorce. Alison, the person in charge of explaining their options to them, explains that long term brain modifications in children are not permitted. However, she suggests, as adults, they could use the method to make a long-term modification so that they are happy in their marriage. They reject this idea. The next option, she suggests, is to modify them so that they don’t care that their child is unhappy about the divorce. They likewise, reject this idea. Alison explains to them the desire to make their child happy too, can be modified. The couple rejects the idea and leaves the office, determined to work harder on their marriage, rather than face the horrible prospect of programming themselves not to love their child.

A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method

The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

A SIMPLIFIED TWO-NODE COARSE-MESH FINITE DIFFERENCE METHOD FOR PIN-WISE CALCULATION WITH SP3

For accurate and efficient pin-by-pin core calculation of SP3 equations, a simplified two-node Coarse Mesh Finite Difference (CMFD) method with the nonlinear iterative strategy is proposed. In this study, the two-node method is only used for discretization of Laplace operator of the 0th moment in the first equation, while the fine mesh finite difference (FMFD) is used for the 2nd moment flux and the second equation. In the two-node problem, transverse flux is expanded to second-order Legendre polynomials. In addition, the associated transverse leakage is approximated with flat distribution. Then the current coupling coefficients are updated in nonlinear iterations. The generalized eigenvalue problem from CMFD is solved using Jacobi-Davidson method. A protype code CORCA-PIN is developed. FMFD scheme is implemented in CORCA-PIN as well. The 2D KAIST 3A benchmark problem and extended 3D problem, which are cell homogenized problems with strong absorber, are tested. Numerical results show that the solution of the simplified two-node method with 1×1 mesh per cell has comparable accuracy of FMFD with 4×4 meshes per cell, but cost less time. The method is suitable for whole core pin-wise calculation.

A Jacobi–Davidson Method for Large Scale Canonical Correlation Analysis

In the large scale canonical correlation analysis arising from multi-view learning applications, one needs to compute canonical weight vectors corresponding to a few of largest canonical correlations. For such a task, we propose a Jacobi–Davidson type algorithm to calculate canonical weight vectors by transforming it into the so-called canonical correlation generalized eigenvalue problem. Convergence results are established and reveal the accuracy of the approximate canonical weight vectors. Numerical examples are presented to support the effectiveness of the proposed method.