Have a look here for an application up to [$]d = 64[$]: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical financeSo, the advantages only kick in when [$]d \geq 4[$]?

- FaridMoussaoui
**Posts:**401**Joined:****Location:**Genève, Genf, Ginevra, Geneva

Have a look here for an application up to [$]d = 64[$]: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical financeSo, the advantages only kick in when [$]d \geq 4[$]?

- Cuchulainn
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And for [$]d \le 3[$], do we FDM or meshless?Have a look here for an application up to [$]d = 64[$]: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical financeSo, the advantages only kick in when [$]d \geq 4[$]?

You can find it online hereI am going to read Villani's book: Optimal Transport. Fortunately, as a Geneva resident, I can borrow the book from the university mathematics library.

wrong typing sorry !And for [$]d \le 3[$], do we FDM or meshless?So, the advantages only kick in when [$]d \geq 4[$]?

Last edited by JohnLeM on February 11th, 2019, 5:55 pm, edited 1 time in total.

We had this conversation some years before: these algorithms converges at rate [$]\frac{1}{N^2}[$] for call/puts, ( [$]\frac{1}{N}[$] for digitals). The overall complexity is [$]N^3[$]. In other words : to reach an error of [$]\epsilon[$], one needs to perform [$]O(\frac{1}{\epsilon^{3/2}})[$] operations ([$]O(\frac{1}{\epsilon^{3}})[$] for digitals). To compare, a straight FDM method needs [$]O(\frac{1}{\epsilon^{D/2}})[$] operations ([$]O(\frac{1}{\epsilon^{D}})[$] for digitals): FDM methods performs better for two dimensions (D=2) (even if I think that I could fix this, specializing the algorithms), and are worse starting at dimensions three.So, the advantages only kick in when [$]d \geq 4[$]?

Last edited by JohnLeM on February 11th, 2019, 6:25 pm, edited 4 times in total.

- FaridMoussaoui
**Posts:**401**Joined:****Location:**Genève, Genf, Ginevra, Geneva

Thank you. I already have the book as a pdf but I spend too much time in front of my screens. I am going to read the paper book only .You can find it online hereI am going to read Villani's book: Optimal Transport. Fortunately, as a Geneva resident, I can borrow the book from the university mathematics library.

@Cuchullain, I wrote in 2008 a paper regarding the one-dimensional case. Since I never published it, I recompiled it and put it on my drive (as you could see, I already was excited by the curse of dimensionality ...). Note, the one-dimensional scheme is still interpreted there as a FDM one (indeed, FDM methods can be interpreted as meshfree methods, as are IA - DNN ones).I am studing Jean-Marc's paper (CRAS) to implement the methodology

Does the article flesh out the algorithm to aid implementation? Or is a PhD in FEM needed? I would personally have difficulty in mapping the high-level maths to an implementation. (I did meshfree but ages ago.. AFAIR it was not as good as FDM)

I would even reduce the scope even more by taking a 1-factor problem and working it out in excruciating detail, I prefer this approach to a top-down one.

Last edited by JohnLeM on February 11th, 2019, 7:13 pm, edited 1 time in total.

Yes, these are quite old results now. For instance we tested up to D=784 : indeed, since we can integrate any Neural Networks (NN) in this PDE framework, we tested some of these NN that are publicly available for the MNIST academic problem, that is a D=784 problem.So, the advantages only kick in when [$]d \geq 4[$]?

I implemented these methods using C++. But it is quite a challenge (> 100k lines, 12 years ...). I think that we should include these methods for instance in TensorFlow (or another framework) to give others access to it. But that's a huge work, I only have two hands and not enough ressources to hire people :/I am studing Jean-Marc's paper (CRAS) to implement the methodology

Does the article flesh out the algorithm to aid implementation? Or is a PhD in FEM needed? I would personally have difficulty in mapping the high-level maths to an implementation. (I did meshfree but ages ago.. AFAIR it was not as good as FDM)

I would even reduce the scope even more by taking a 1-factor problem and working it out in excruciating detail, I prefer this approach to a top-down one.

Well, you know, a CRAS is limited to 7 pages, thats few space...Indeed, I just adapted Meshfree methods to mathematical Finance equations: the scheme for the Kolmogorov equations is fully described there (see eq 10 and 11). I agree, the scheme to solve Fokker-Planck equations (non linear hyperbolic / parabolic equations) is not described, but one can just use S instead of [$]P\circ S[$] in (10) and reverse time.It is said that the PDE is solved by a meshfree method (Fasshauer notes) but it is not explicited which one is used.

PS1: The goal of a meshfree method and quantization is to solve the curse of dimentionality.....

PS2: Have a look to Jameson or Löhner papers on meshfree methods. They are more accurate.

- Cuchulainn
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The problem with many articles is that the reader cannot reproduce or check the results.And too many lacunae.

Well, technically the CRAS is self-contained: there are two algorithms to implement. You can reproduce both. And we can proove theoretically the errors bounds.

Last edited by JohnLeM on February 12th, 2019, 2:19 am, edited 1 time in total.

- Cuchulainn
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OK. Hoiw many manhours would it take to implement this algorithm, days, weeks or months?Well, technically the CRAS is self-contained: there is two algorithms to implement. You can reproduce all of it. And we can proove theoretically the errors bounds.

What are the two algorithms called?

Yes i agree :/ we definitely should let it more toyable.

No name. I think maybe lagrangian meshfree methods or something like that but more sexy !

No name. I think maybe lagrangian meshfree methods or something like that but more sexy !

- Cuchulainn
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I think there's a typo on page 3; you say [$]\nabla^2[$] is the Jacobian but it is the Laplacian operator (or Hessian which become singular/negative definite).

BTW is reference [6] available yet?

I don't get what [$] A \circ S[$] does/is: it seem to be used a number of times without a sharp definition.

// Haven't got my head yet around article [1] and how to pull-back a measure..It uses upper case # while the article uses lower case #. Are they the same #?

BTW is reference [6] available yet?

I don't get what [$] A \circ S[$] does/is: it seem to be used a number of times without a sharp definition.

// Haven't got my head yet around article [1] and how to pull-back a measure..It uses upper case # while the article uses lower case #. Are they the same #?

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