A number of analytical codes are used in the Nuclear industry. Many of these codes, such as TRACE, RELAP5, and PARCS, reduce systems of equations to a matrix form that is then solved. A common numerical difficulty in numerical computing with matrices is the occurrence of ill-conditioned matrices, which can result in significant amplification of error. This in turn can lead to poor or even invalid results. Ill-conditioning presents some challenging problems for code developers:

\(1\) Ill-conditioning can occur for systems that are correctly formulated. In addition, a particular formulation of the equations may be well conditioned under certain circumstances, and ill-conditioned in other circumstances. How can ill-conditioning be efficiently detected?

\(2\) Ill-conditioning can arise from the solution algorithm employed to solve the equations. How can this be avoided?

\(3\) When a matrix is determined to be ill-conditioned, how does one determine the cause of ill-conditioning? Once the cause is determined, how does one go about correcting the problem?

While the equations solved in a matrix may be formulated in various ways, ill-conditioning is fundamentally a geometric problem that can be understood with geometric concepts associated with matrices and vectors.  Ill-conditioning arises from equations or system variables that are not adequately orthogonal. In other words, the equations or system variables convey information that is too similar. The simplest measure of similarity is the angle between vectors in a matrix which provides a direct measure of the orthogonality or linear independence of two system equations (i.e., matrix row vectors) or two system variables (i.e., matrix column vectors). The angle between vectors provides a simple measure of the cause of ill-conditioning and directs the developer to the location where changes to the formulation of the system are needed to improve conditioning.

However, the angle between vectors is an insufficient measure of linear independence of a system. Another measure of the linear independence of a system of equations is the determinant, where a determinant of zero indicates a system that is not linearly independent and a determinant near zero indicates ill-conditioning. A small modification to the determinant formulation is needed to properly detect a “near zero” measure, which indicates ill-conditioning of the system as a whole. Between the techniques of measuring the angle between vectors and using the determinant as a measure of the conditioning of the system as a whole are techniques for measuring the linear independence of subsets of vectors to diagnose the primary causes of ill-conditioning in the system.

A geometric understanding of ill-conditioning helps provide a developer with the tools necessary to identify the cause of ill-conditioning, and thus provides the information needed to know where to focus efforts on fixing the formulation. A geometric understanding of ill-conditioning also aids in developing dynamic preconditioners to improve matrix conditioning and ultimately improve the accuracy of solutions.


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

Division / Sponsors: 
Power Division & Nuclear Engineering Division (NED)

Power Conference
Paper Proceedings: https://asmedigitalcollection.asme.org/POWER/POWER2020/volume/83747

International Conference on Nuclear Engineering
Paper Proceedings: https://asmedigitalcollection.asme.org/ICONE/ICONE2020/volume/83761

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