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I am running into some numerical issues when trying to generalize the LGM (linear Gauss-Markov) model to multiple factors and associate a "high" mean reversion value to one of the factors. See here https://ssrn.com/abstract=4317184. Is there a simple way out that I am missing oder is the model formulation just not fit for high reversion values?

I believe it's a valid concern and in line with my own observations - the model can explode. In practice you would only use a high kappa for a short-dated factor, thus your sigma would not be 1% for 30y. 

Thanks. And yes, the example in the note is simplified, in reality I use e.g. sigma = 0.0030 and kappa = 0.50 for one of the factors, but the instability remains. 

In the meantime I think the issue is quite obvious. When you develop the multi factor Hull-White model from the HJM equation you naturally arrive at a point where you are tempted to use the state from the LGM formulation because of its simple SDE. But then you _want_ to switch to a mean reverting state exactly because of the issue described above. The Piterbarg / Andersen book is quite explicit on this, see Proposition 12.1.1, Equation (12.3) and the remark at the beginning of section 12.1.1.1.