Divergence for higher orders using Pseudo-spectral projection

[hamzabuw]

Hi Jonathan,

I am trying to use projection method to quantify uncertainty in mass output (as mass loss rate) due to uncertainty in activation energy (E) using Arrhenius equation:

Up till order 4 of PCE, I get a reasonable result as below:

The shaded region in blue is the standard deviation.

But from order 5 on, the solution diverges as below:

I used truncated normal distribution to generate quadrature as below:

E_1_exact = 188E+03
E_variation = 0.02
E_alpha = chaospy.Normal(E_1_exact, E_variation*E_1_exact)
E_alpha = chaospy.Trunc(E_alpha, lower= E_1_exact -2*E_variation*E_1_exact,upper= E_1_exact+ 2*E_variation*E_1_exact)
gauss_quads = chaospy.generate_quadrature(order, joint, rule="gaussian")

Can you guide me why does the solution diverge from order 5 on? I am experiencing similar issue with other variables even without using the truncation.

[jonathf]

Could you try using chaospy.TruncNormal instead of composing with chaospy.Normal and chaospy.Trunc? I suspect numerical instabilities in the truncation approximation.

[hamzabuw]

Chaospy.TruncNormal creates undefined weights for me using the code below:
E_1_exact = 188E+03
E_variation = 0.02
E_alpha = chaospy.TruncNormal(lower= E_1_exact -2*E_variation*E_1_exact, upper= E_1_exact+ 2*E_variation*E_1_exact, mu=E_1_exact, sigma=E_variation*E_1_exact)
gauss_quads = chaospy.generate_quadrature(order, joint)

But even without truncation and just using normal distribution, I am unable to reach higher order even though the weights appear to be well defined.

[jonathf]

Then that is not it. I don't have all information available about your problem, so I can only suggest ideas of what is wrong.

I have two other hypothesis hat can be tested.

1. You order is too low. Try to increase it.

2. The equation hints at a latent log-normal probability distribution. Log-normal is one of those distributions that behaves really bad with polynomial chaos expansion. In that case, you are out of luck, as the problem is inherently not stable.