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by JohnLeM
October 7th, 2020, 6:17 am
Forum: Numerical Methods Forum
Topic: Machine Learning: Frequency asked Questions
Replies: 61
Views: 8448

Re: Machine Learning: Frequency asked Questions

BTW, the heatmap is sensitivity analysis of the output (i.e. the probability distribution of the classes) to the activations of the last convolutional layer. In simple words, it shows where the model sees those strange things - indeed, in my cat! Strangely, lots of objects are classified as a dishw...
by JohnLeM
October 6th, 2020, 8:56 pm
Forum: Numerical Methods Forum
Topic: Machine Learning: Frequency asked Questions
Replies: 61
Views: 8448

Re: Machine Learning: Frequency asked Questions

Who needs maths anyways; I remember this infamous post here My deeper point was that  Cauchy  sequences may be an approximation for some things in the real world (as long one does not look too far ahead in time or too closely at tiny epsilons) but  Cauchy  sequences don't actually occur in the real...
by JohnLeM
October 6th, 2020, 7:57 pm
Forum: Numerical Methods Forum
Topic: Machine Learning: Frequency asked Questions
Replies: 61
Views: 8448

Re: Machine Learning: Frequency asked Questions

BTW, the heatmap is sensitivity analysis of the output (i.e. the probability distribution of the classes) to the activations of the last convolutional layer. In simple words, it shows where the model sees those strange things - indeed, in my cat! Strangely, lots of objects are classified as a dishw...
by JohnLeM
October 6th, 2020, 1:53 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

Looks like overkill. What's the story about Markov?? There are several methods >> CN around. Overkill is the only way at my disposal to tackle efficiently high dimensional problems. For Markov: you can interpret a scheme as a Markov Chain provided u^{n+1}=Au^n, A stochastic matrix (a monotone schem...
by JohnLeM
October 5th, 2020, 7:19 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

That paper was from 1998, so it was corrected in my later books. I am a trainer so I expect you to read what I post :-) Especially stuff I invented. One way to get monotonicity without fitting is to use use upwinding in convection. Another way is to split the equation : move the grid according to t...
by JohnLeM
October 5th, 2020, 4:34 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

That paper was from 1998, so it was corrected in my later books. I am a trainer so I expect you to read what I post :-) Especially stuff I invented. One way to get monotonicity without fitting is to use use upwinding in convection. Another way is to split the equation : move the grid according to t...
by JohnLeM
October 5th, 2020, 12:14 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

You're guessing! please do a bit of research. Exponential fitting A-Z is described here, RTFM in bocca al lupo! I read quickly. You define the fitting factor in (31), the full scheme is at (33). (34) shows that the scheme is consistent with the equation [$]\sigma_h u'' + \mu u' = 0[$] (forget the o...
by JohnLeM
October 5th, 2020, 10:15 am
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

Not exactly. We use a modified diffusion term. Is it the diffusion term is denoted [$]D^+D^-[$] in your paper ? Isn't it a standard discretization of second order derivative on non uniform grid ? You're guessing! please do a bit of research. Exponential fitting A-Z is described here, RTFM in bocca ...
by JohnLeM
October 5th, 2020, 9:05 am
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

@Cuchulain I was rereading your exponential fitting  paper. As far as I understand, you produce a monotone scheme using a pertinent change of variable of the mesh itself. This change of variable ensures that the discretization of the advection terms are dominated by the diffusion ones. Technically,...
by JohnLeM
October 5th, 2020, 8:29 am
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4820

Re: Monotone Schemes: what are they and why are they good?

@Cuchulain I was rereading your exponential fitting  paper. As far as I understand, you produce a monotone scheme using a pertinent change of variable of the mesh itself. This change of variable ensures that the discretization of the advection terms are dominated by the diffusion ones. Technically, ...
by JohnLeM
October 5th, 2020, 7:38 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2796

Re: What are the boundary conditions for the Forward contract PDE?

The intent of pointing out the delta 1 thing was to suggest that the boundary conditions for the forward should be quite similar to the that of the spot. BTW are there any embedded deliver options?  The pit might be here: it seems that this is true only at maturity for people using Schwartz modelin...
by JohnLeM
October 4th, 2020, 2:50 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2796

Re: What are the boundary conditions for the Forward contract PDE?

I think that the owner of this thread is asking for educational advises here. We are trying to do our best to help him. "Screwing around with PDEs" is an interesting turn of phrase. Given that the exact solution to the valuation problem is given by equation (2), I think any further efforts at numer...
by JohnLeM
October 4th, 2020, 8:01 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2796

Re: What are the boundary conditions for the Forward contract PDE?

Any desire to further screw around with PDEs would presumably just be for educational purposes, which is fine. 
I think that the owner of this thread is asking for educational advises here. We are trying to do our best to help him.
by JohnLeM
October 4th, 2020, 7:42 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2796

Re: What are the boundary conditions for the Forward contract PDE?

A trick that I was using for this kind of problem: try using the boundary conditions [$]u^{N} = 2 u^{N-1} - u^{N-2} [$],  [$]u^{0} = 2 u^{1} - u^{2}[$] modelling [$]\partial_{xx}u=0[$] (linear behavior at infinity). AFAIR, this saved me to compute complex, payoff dependent, Dirichlet or Neuman boun...
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