Problem calculating arbitrary Polynomial Chaos polynomials using SAMBA

In summary, SAMBA is a software package used for uncertainty quantification and sensitivity analysis of complex models based on polynomial chaos expansion. This technique represents stochastic processes in terms of orthogonal polynomials, allowing for efficient computation of statistics and sensitivity analysis. However, there may be inaccuracies when calculating arbitrary polynomials not included in the sparse expansion. This problem can be solved by increasing the sparsity level, using adaptive sparse grids, or alternative methods such as stochastic collocation or projection-based methods. SAMBA and Polynomial Chaos have various applications in engineering, finance, and climate modeling for analyzing uncertainty, identifying important parameters, and improving predictive capabilities.
  • #1
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 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.
 
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  • #2


Hello Frank,

It is great that you have been able to calculate the nodes and weights for your data using the SAMBA methodology. I can understand your confusion regarding the calculation of the polynomials, as the article does not provide a clear explanation for the calculation of s0j.

After reviewing the article, I believe that the authors have assumed s0j to be equal to 1 for all j. This is a common assumption in many polynomial chaos methods, and it simplifies the calculation of the polynomials. You can verify this by setting s0j to 1 and calculating the polynomials using the equation provided in the article.

As for the recurrence relation, you are correct that it requires the calculation of ψ-1, which can be a bit tricky. One way to approach this is to use the fact that ψ0 = 1 and ψ1 = ξ. This can be used to calculate ψ-1 as 1/ξ. However, this approach may not work for all cases, and you may need to consult the authors or other experts in the field for a more accurate solution.

I hope this helps clarify your doubts. Good luck with your research!


 

1. What is the purpose of using Polynomial Chaos polynomials in SAMBA?

The purpose of using Polynomial Chaos polynomials in SAMBA is to accurately represent and quantify uncertainty in mathematical models or simulations. These polynomials are used to approximate the behavior of complex systems and to calculate the probability of different outcomes.

2. How are Polynomial Chaos polynomials calculated in SAMBA?

Polynomial Chaos polynomials are calculated in SAMBA using a combination of Monte Carlo methods and orthogonal polynomial expansions. The Monte Carlo method is used to generate samples of the input variables, which are then used to construct the orthogonal polynomial basis functions. These basis functions are then used to calculate the coefficients of the Polynomial Chaos expansion.

3. What are the advantages of using Polynomial Chaos polynomials in SAMBA?

There are several advantages of using Polynomial Chaos polynomials in SAMBA. These include the ability to accurately represent complex systems with uncertain inputs, the ability to quantify and propagate uncertainty through the model, and the ability to efficiently handle high-dimensional problems.

4. Are there any limitations to using Polynomial Chaos polynomials in SAMBA?

While Polynomial Chaos polynomials have many advantages, there are also some limitations to using them in SAMBA. These include the requirement for a large number of samples to accurately represent the input variables, the potential for numerical instability in high-dimensional problems, and the need for careful selection of the orthogonal polynomial basis functions.

5. How can I validate the results obtained from using Polynomial Chaos polynomials in SAMBA?

There are several methods for validating the results obtained from using Polynomial Chaos polynomials in SAMBA. These include comparing the results to other methods, such as Monte Carlo simulation, and performing sensitivity analyses to assess the impact of different input variables on the results. It is also important to carefully check the convergence of the Polynomial Chaos expansion and to assess the accuracy of the results for different levels of polynomial order.

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