I have a radiation-transfer matrix F that had been computed using some raytracing Monte Carlo method. From physics it is clear that it must be row-stochastic :$\sum_{j=1}^{N}F_{ij}=1$ because energy cannot be lost. It is also clear that it must be symmetrizable by a diagonalmatrix S which contains the areas of the emitting surfaces: $SF=(SF)^{T}$. Matrix F, however, does not comply with these requirements when it comes out of the raytracing process. Deviations from symmetry and row-stochasticity cannot be ignored. Due to the statistical nature of the raytracing process it can be assumed that the matrixelements scatter around their actual value with a known variance proportional $\sqrt{F_{ij}}$. I assume that this problem can best (or only) be solved by some nonlinear optimisation method. Is there a numerical method by which I can improve F while at the same time respecting the constraint posed by the variance?
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I am a bit lost regarding symmetrizability. How you derive $S$? Is it predefined matrix or it is some function of $F$? Does it also have errors? – Alexander Vigodner Sep 12 '14 at 20:39
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And also variance or std deviation is proportional to $\sqrt F_{ij}$ – Alexander Vigodner Sep 12 '14 at 20:44