Matrix Relative Threshold

Matrix Relative Threshold = <float>

Description / Usage

This card is only available with the Trilinos library. The effect of this card is to impose a relative lower bound to either a diagonal value or a singular value. The legal values for <float> are:

<float>

r, a floating point number ( r ≥ 0.0 ) that specifies a relative threshold.

If this card is omitted, the default is 0.0.

Examples

A sample input card follows:

Matrix Relative Threshold = 1.e-4

Technical Discussion

This card, along with the Matrix Absolute Threshold card, allow the user to modify the linear system prior to calculation of the preconditioner. Note that the modification is only to change the “initial condition” of the preconditioner–it does not actually change the linear system.

Let t be the value specified with the Matrix Absolute Threshold card. For a scalar-based preconditioner (ilut, ilu, rilu, icc), each value on the diagonal undergoes the following substitution:

\[d_{\mathrm{new}} = r * d_{\mathrm{old}} + \mathrm{sgn} \left( d_{\mathrm{old}} \right) * t\]

For the bilu preconditioner, each singular value of the diagonal block preconditioner is compared to:

\[\sigma_{\mathrm{min}} = r * \sigma_1 + t\]

where σ1 is the largest singular value of the diagonal block under consideration. All \(\sigma_k\) are modified (if necessary) to be at least as large as \(\sigma_{\mathrm{min}}\).

The appropriate values for the threshold can vary over many orders of magnitude depending on the situation. Refer to Schunk, et. al., 2002 for information and for further guidance.

References

SAND2001-3512J: Iterative Solvers and Preconditioners for Fully-coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems, P. R. Schunk, M. A. Heroux, R. R. Rao, T. A. Baer, S. R. Subia and A. C. Sun, March 2002.