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Frank Einstein

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- TL;DR Summary
- I have an article in which the polynomials of the aPC expansion of a stochstic process are calculated. However, i am unable to follow the calculations presented in said article.

Hello everyone. I have recently read the following article (which title is

In the proposed methodology, to find out the polynomials, one has to calculate hankel's matrix of moments and then perform the Cholesky descomposition over it, obtaining a triangular matrix. Then, one has to calculate the inverse of said triangular matrix obtaining another triangular matrix which first row is composed of s11 s12 s13 and so on, the second line of the matrix is 0 s22 s23 and so on and the third line is composed of 0 0 s33 and so on. I have been able to calculate these too. the problem comes with the definition of the polynomials as:

ψj(ξ) = s0j ξ^0 + s1j ξ^1 + s2j ξ^2 + ...

As you can see, the definition of these polynomials include the term s0jξ^0. As you can see, the elements of the matrix in the article are defined starting at s11, so, i don't know what to do with s00, s01 and so on. Has someone worked with this article and can tell me how to calculate s0j?

Another way to calculate said polynomials provided by the article is the recurrence relation:

ξψj−1 (ξ ) = bj−1ψj−2 (ξ ) + ajψj−1 (ξ ) + b jψj (ξ )

Being both a and b known coefficients. However, i run into a simmilar problem. i know that by deffinition ψ0=1, but I have no idea on how to calculate ψ1, since I would need ψ-1 in adition to,ψ0. If someone could explain to me how to do this it would be as useful as understanding the method explained in earlier paragraphs.

Any help is appreciated.

Regards.

Frank.

**SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos**) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate the nodes and weights, but now, I want to calculate the polynomials.In the proposed methodology, to find out the polynomials, one has to calculate hankel's matrix of moments and then perform the Cholesky descomposition over it, obtaining a triangular matrix. Then, one has to calculate the inverse of said triangular matrix obtaining another triangular matrix which first row is composed of s11 s12 s13 and so on, the second line of the matrix is 0 s22 s23 and so on and the third line is composed of 0 0 s33 and so on. I have been able to calculate these too. the problem comes with the definition of the polynomials as:

ψj(ξ) = s0j ξ^0 + s1j ξ^1 + s2j ξ^2 + ...

As you can see, the definition of these polynomials include the term s0jξ^0. As you can see, the elements of the matrix in the article are defined starting at s11, so, i don't know what to do with s00, s01 and so on. Has someone worked with this article and can tell me how to calculate s0j?

Another way to calculate said polynomials provided by the article is the recurrence relation:

ξψj−1 (ξ ) = bj−1ψj−2 (ξ ) + ajψj−1 (ξ ) + b jψj (ξ )

Being both a and b known coefficients. However, i run into a simmilar problem. i know that by deffinition ψ0=1, but I have no idea on how to calculate ψ1, since I would need ψ-1 in adition to,ψ0. If someone could explain to me how to do this it would be as useful as understanding the method explained in earlier paragraphs.

Any help is appreciated.

Regards.

Frank.