JuliaApproximation/ApproxFun.jl

Sheehan Olver

@dlfivefifty

It’s pretty simple: just differentiate wrt u but treating d/dx and d/dy as constants.

To get the Jacobian

Mitko Georgiev

@mitkoge

excuse if my question is too naive but i try this

x= Fun(0..1); sx=space(x);
a=Fun(0..pi); sa=space(a);
fa= DefiniteIntegral(x*a,sx)
# i expect fa= a here

but i get MethodError. Are integrals of two arguments possible?

Mitko Georgiev

@mitkoge

or may be like this

x= Fun(0..1); sx=space(x)
a=Fun(0..pi); sa=space(a)
fm= x*a; sm= space(fm)
fI=  DefiniteIntegral(sx)
(fI*fm)(0.2)
#expect 0.2

Sheehan Olver

@dlfivefifty

Are you expecting it to be a 2D function? And do you actually expect the result to be 0.1 (that is, a/2 since you integrate out x)?

That's possible via:

x,a = Fun((0..1) × (0..π))
Q = DefiniteIntegral(0..1) ⊗ I
(Q*(x*a))(0.0,0.2)

Mitko Georgiev

@mitkoge

Wow exciting how elegant is this! Thank you!
As you can see i am exploring it the dummy way so please bear with me.

Mitko Georgiev

@mitkoge

let Q2= Q (x a) . Q2 is still 2D Fun((x,a)-> smth). Is there a way to curry it to 1D function Q1 that is Fun(a-> smth)?

Sheehan Olver

@dlfivefifty

Not an easy way. What we actually want is DefiniteIntegral(S, [1,0]) that wraps the kronecker product. It’s not too hard to add this but I don’t have time right now

Mitko Georgiev

@mitkoge

to practice your guiding i wrote

function Hankel(fx::Fun, spx,spa::Space)
    spxa= spx ⊗ spa
    fx2= Fun((x,a)-> fx(x), spxa)
    fj= Fun((x,a) -> besselj0(x*a), spxa)
    fa2= (DefiniteIntegral(dmx1) ⊗ I)*(fx2*fj*x)
    fa= Fun(a->fa2(0,a), spa)
end

from the definition. It seems to work. Thank you!
Is this the style ApproxFun is expected to be used?

Sheehan Olver

@dlfivefifty

Not an easy way.

I take that back. I think you can always do ApproxFun.SpaceOperator(op, newdomainspace, newrangespace) change the spaces of an operator. So just write Q with the "right" spaces

Is this the style ApproxFun is expected to be used?

In Julia, capital letters are only used when it returns a special type of the same name. So this would be better as lower case. Also, there is support for Green's functions in https://github.com/JuliaApproximation/SingularIntegralEquations.jl including Helmholtz / hankel kernels

Mitko Georgiev

@mitkoge

I see. I will check SpaceOperator and will have to learn what rangespace is.
And thank you for the link. It is probably some more efficient implementation? Will check it.
I tried hankel as an learning example.
The kind of high level style the ApproxFun works is fascinating me.
Like a poetry for the crowd. I do not necessarily understand but like it. :-)

Mitko Georgiev

@mitkoge

here is the revised

function hankel(fx::Fun, spx,spa::Space)
    spxa= spx ⊗ spa
    fx2= Fun((x,a)-> fx(x), spxa)
    fjx= Fun((x,a)-> x*besselj0(x*a), spxa)
    fa2= (DefiniteIntegral(spx.domain) ⊗ I)*(fx2*fjx)
    fa= Fun(a->fa2(0,a), spa)
end

Mitko Georgiev

@mitkoge

when i blind try

spx= Space(0..1); spa= Space(0..π); spxa= spx ⊗ spa
Q2= DefiniteIntegral(spx.domain) ⊗ I
Q1= ApproxFun.SpaceOperator(Q2, spxa, spa ) 

x=Fun((x,a)->x,spxa); a=Fun((x,a)->a,spxa)
(Q2*(x*a))(0,1)  #=0.5
(Q1*(x*a))(1)      #=0.5*pi/2

it seems to work up to scaling cofficient

Mitko Georgiev

@mitkoge

i gues it is because the default domain for I is -1..1

Mitko Georgiev

@mitkoge

may i ask if a least quare fit by bivariate Fun to data is possible?
Given data is Array{Float64,2}.
I liked your nice 1D example from the docs and wonder how to extend to 2D.

Sheehan Olver

@dlfivefifty

Yes an example is in the FAQ

Mitko Georgiev

@mitkoge

Thank you! I found it and will have a look.
Is the example for standard grid ponts also extendable for 2D?
Or it is not preferable because data points in 2D are mostly large ammont?

Art Gower

@arturgower

I'm quite sure Fun used to work on intervals in the complex domain. But now

f(x) = cos(x)
Fun(f, Interval(1.0+1.0im, 2.0+2.0im))

throws the error

ERROR: MethodError: no method matching isless(::Complex{Float64}, ::Complex{Float64})
Closest candidates are:
  isless(::Missing, ::Any) at missing.jl:66
  isless(::InfiniteArrays.OrientedInfinity{Bool}, ::Number) at /.julia/packages/InfiniteArrays/Z4yap/src/Infinity.jl:145
  isless(::Number, ::InfiniteArrays.OrientedInfinity{Bool}) at /.julia/packages/InfiniteArrays/Z4yap/src/Infinity.jl:144
  ...
Stacktrace:
 [1] <(::Complex{Float64}, ::Complex{Float64}) at ./operators.jl:260
 [2] >(::Complex{Float64}, ::Complex{Float64}) at ./operators.jl:286
 [3] isempty(::Interval{:closed,:closed,Complex{Float64}}) at /.julia/packages/IntervalSets/xr34V/src/IntervalSets.jl:153

Sheehan Olver

@dlfivefifty

Use Segment(a,b) for line segments in the complex plane. (The previous support for intervals in the complex plane violated the definition of an interval.)

Art Gower

@arturgower

A ha! Yes I see. Thanks so much

Shi Pengcheng

@shipengcheng1230

Hello, I was playing around with the poisson equation example and wonder if I could replace the RHS $f$ with something like $\delta (x) \delta (y)$ . I tried to construct the RHS like this:

fx = KroneckerDelta()
fy = KroneckerDelta()
f = Fun((x,y) -> fx(x) * fy(y))

But I got the error that ERROR: MethodError: no method matching isless(::Int64, ::Nothing). What is the proper way for me to do that? Thanks in advance!

Sheehan Olver

@dlfivefifty

Essentially you want to calculate the Greens function? That’s a tough question which we are looking at right now.

Shi Pengcheng

@shipengcheng1230

Thanks for the reply. I guess I have to wait for now :)

Quentin C.

@robocop

Hello ! I would like to consider ApproxFun for the following problem: I have a N-L operator Phi: L_s -> L_s, where L_s is the set of continuous functions that goes exponentially fast to zero at speed s > 0, that is: x belongs to Ls if \sup{t \geq 0} |x(t)| e^{s t} < +oo. Moreover given x in L_s, Phi(x) is the solution of a Volterra integral equation of the form Phi(x)(t) = K_0(x)(t) + \int_0^t{K(x) (t, u)Phi(x) (u) du}, where the Kernels K_0(x) and K(x) are known explicitly in term of x. I would like to compute the spectrum of the Frechet differential of Phi at the point x = 0. Do you think it is possible to do that?

Let me know if the question is unclear... I do not have a background in numerical simulations. Thanks :)

Quentin C.

@robocop

I think I have started to understand how to do that with ApproxFun. But, I have got few questions. How to encode the domain

\mathbb{R}_+

and how to encode

\{ (x, y) \in mathbb{R}_+, x \geq y \}

Quentin C.

@robocop

More generally the fact that I want to work on

\mathbb{R}_+

and not on a segment seems to be a problem. Is that correct?

Sheehan Olver

@dlfivefifty

Sorry forgot to reply. I have a student working on Volterra integral equations so once that’s up and running your problem should be easy. We’d love to know the application so we can mention it in the paper

Quentin C.

@robocop

Ok great! Looking forward to see that. The Volterra equation I am looking at appears in a neuroscience problem (see https://arxiv.org/abs/1810.08562) The dynamic of the mean-field network can be reduce to a N-L Volterra equation (see equations (7) and (8) of the paper). Btw the same problem can be solved numerically by looking at the associated PDE (see equation (3)) - but I am curious to know if more efficient numerically methods can be develop by specifically exploit this Volterra equation.

Art Gower

@arturgower

Hello! For the space of Jacobi polynomials, what package does ApproxFun use? Or is it internally implemented? I struggled to find it in the code... =(

Sheehan Olver

@dlfivefifty

Internal: see src/Spaces/Jacobi

Art Gower

@arturgower

Cheers! What method do you use: is it Clenshaw for all PolynomialSpaces? In which case, where do you specify the recurrence relation?

Sheehan Olver

@dlfivefifty

Yes. There’s a bunch of recα , etc., overrides

Clenshaw is implemented in PolynomialSpace.jl

Art Gower

@arturgower

thanks, I get it now

Mitko Georgiev

@mitkoge

May i ask may be not directly related to ApproxFun
I found FastAsyTransforms.m on github and wondered if such functionality is already available as julia code?
I looked at FastTransforms.jl but could not find something like a HankelTransform. Would you kindly direct?
May be SingularIntegralEquations.jl? But no idea how to start with it? Or other place?

Sheehan Olver

@dlfivefifty

@MikaelSlevinsky should be able to help, but I'm not aware of anything.

Christoph Ortner

@cortner

I’m trying to implement something like this FAQ example,

S = Chebyshev(1..2);
p = points(S,20); # the default grid
v = exp.(p);      # values at the default grid
f = Fun(S,ApproxFun.transform(S,v));

but multi-variate (2D tensor Chebyshev will do). The canonical thing,

S = Chebyshev((1..2)^2)
p = points(S, 20)

errors. Of course I could just construct the points via tensor products, but then I’m unsure how to use the ApproxFun.transform(S,v) correctly. Is this documented somewhere? Is there an example I can look at?

Basically, I just want to freeze the polynomial degree, rather than prescribe a solver tolerance.

Sheehan Olver

@dlfivefifty

You want Chebyshev(1..2)^2. It actually uses Padua points so if you say ` Fun(f, S, div(n*(n+1),2)) it should give the degree n interpolant.

Christoph Ortner

@cortner

Interesting - thanks. are tensor product grids a special case of these?

I didn’t know about this family of points, readig up on it now, very very interesting...

Sheehan Olver

@dlfivefifty

No, but anything other than Chebyshev^2 will use a tensor grid. Padua points are nice because you don’t oversample, and the transform is a single one dimensional DCT (though as implemented we form a tensor product by filling with zeros and use the 2D DCT, which due to aliasing we can recover the coefficients)

There’s a Chebfun example describing it

Christoph Ortner

@cortner

So this is very interesting, but for teaching purposes I would have still preferred the standard tensor Chebyshev grid. But this is not as straightforward?

Sheehan Olver

@dlfivefifty

If you comment out the lines after ## Multivariate in https://github.com/JuliaApproximation/ApproxFun.jl/blob/master/src/Spaces/Chebyshev/Chebyshev.jl it will go back to the default tensor version, probably a keyword would be appropriate here to allow switching

Christoph Ortner

@cortner

ok - I’ll try

thanks for the suggestions

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