Helmholtz FMM inaccurate when Re(zk)<0

[bobbielf2]

Hi folks,

I was in MATLAB testing hfmm3d with a negative wavenumber zk<0 (see code below), but keep getting only ~2 digits when using a moderate tolerance (e.g. 1e-6). (The original goal was to do fast adjoint mat-vec.)

Here is some more info:

1. Tests would fail when Re(zk)<0
2. However, can get full 15 digits once the number of sources ns is small enough (e.g. ns=1000)
3. Can also get 15 digits once tolerance tol is small enough (e.g. tol=1e-12)

So I am getting either 2 digits or 15 digits. Could you confirm if this is a bug or am I doing something wrong here?

Thanks, Bobbie

ns = 4000;
srcinfo.sources = rand(3,ns);
srcinfo.charges = rand(1,ns)+1i*rand(1,ns);
stmp = srcinfo.sources;
pg = 1;

zk = complex(-1.1);
tol = 1e-6;

fprintf('Re(zk) = %.1f\ntol = %.1e\n',real(zk),tol)

U1 = hfmm3d(tol,zk,srcinfo,pg);
U2 = h3ddir(zk,srcinfo,stmp,pg);
U2.pot = U2.pottarg;
err = norm(U1.pot-U2.pot)/norm(U2.pot);
if err<tol, fprintf('Test successful, '), 
else, fprintf('Test failed, '); end
fprintf('err = %.2e\n',err);

[mrachh]

Hi, Thanks for bringing this to our notice. Currently, the Helmholtz wave number zk should be in the first quadrant. In case you do want to run the calculation for zk in the second quadrant, you can conjugate the density on input and the result on output. We will update our documentation soon to reflect this restriction.

Regards, Manas

[bobbielf2]

Thanks, Manas! I will use the double conjugation you said.

On Wed, Nov 18, 2020, 8:05 AM mrachh notifications@github.com wrote:

Hi, Thanks for bringing this to our notice. Currently, the Helmholtz wave number zk should be in the first quadrant. In case you do want to run the calculation for zk in the second quadrant, you can conjugate the density on input and the result on output. We will update our documentation soon to reflect this restriction.

Regards, Manas

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[ahbarnett]

Hello - just wanted to chime in here that 2nd or 3rd quadrant are negative Re part of k, which is modified Helmholtz (decaying) rather than Helmholtz (oscillating). I'm sure you can't access that via conjugation (which would only get you to 4th quadrant). A different FMM is needed. (Unless your kernel scale is << system scale, when no FMM is needed, just a NN search.) Best, Alex

On Wed, Nov 18, 2020 at 10:27 AM bobbielf2 notifications@github.com wrote:

Thanks, Manas! I will use the double conjugation you said.

On Wed, Nov 18, 2020, 8:05 AM mrachh notifications@github.com wrote:

Hi, Thanks for bringing this to our notice. Currently, the Helmholtz wave number zk should be in the first quadrant. In case you do want to run the calculation for zk in the second quadrant, you can conjugate the density on input and the result on output. We will update our documentation soon to reflect this restriction.

Regards, Manas

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.

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[lgreengard]

Hi - negative real part is actually the other side of the Fourier transform and annoying to incorporate in the code since it implies a different propagating direction, a different radiation condition, etc. Double conjugation gets around that. (Modified Helmholtz corresponds to pure imaginary k.)

• Leslie

On 11/18/20 2:58 PM, Alex Barnett wrote:

Hello - just wanted to chime in here that 2nd or 3rd quadrant are negative Re part of k, which is modified Helmholtz (decaying) rather than Helmholtz (oscillating). I'm sure you can't access that via conjugation (which would only get you to 4th quadrant). A different FMM is needed. (Unless your kernel scale is << system scale, when no FMM is needed, just a NN search.) Best, Alex

On Wed, Nov 18, 2020 at 10:27 AM bobbielf2 notifications@github.com wrote:

Thanks, Manas! I will use the double conjugation you said.

On Wed, Nov 18, 2020, 8:05 AM mrachh notifications@github.com wrote:

Hi, Thanks for bringing this to our notice. Currently, the Helmholtz wave number zk should be in the first quadrant. In case you do want to run the calculation for zk in the second quadrant, you can conjugate the density on input and the result on output. We will update our documentation soon to reflect this restriction.

Regards, Manas

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[bobbielf2]

Howdy Alex! I am not familiar with modified Helmholtz. Are you talking about the Re(k^2)<0 case (i.e. screened Coulomb/Yukawa potential)? That would indeed require different FMM. To be more specific, what I was trying to do (conjugate transpose mat-vec) is still the unmodified Helmholtz. E.g. SLPmatrix'*x = conj(SLPmatrix*conj(x)) (the case with quadrature weights is analogous). Thanks for joining in (and thanks for your input on the other thread)! Best, Bobbie

[bobbielf2]

Hi Leslie, Thanks for the explanation. (I see that hfmm3d works for pure imaginary k.) Best, Bobbie

[ahbarnett]

My bad - was thinking k^2 (Helmholtz parameter), not k. Best, Alex

On Wed, Nov 18, 2020 at 3:44 PM Leslie Greengard notifications@github.com wrote:

Hi - negative real part is actually the other side of the Fourier transform and annoying to incorporate in the code since it implies a different propagating direction, a different radiation condition, etc. Double conjugation gets around that. (Modified Helmholtz corresponds to pure imaginary k.)

• Leslie

On 11/18/20 2:58 PM, Alex Barnett wrote:

Hello - just wanted to chime in here that 2nd or 3rd quadrant are negative Re part of k, which is modified Helmholtz (decaying) rather than Helmholtz (oscillating). I'm sure you can't access that via conjugation (which would only get you to 4th quadrant). A different FMM is needed. (Unless your kernel scale is << system scale, when no FMM is needed, just a NN search.) Best, Alex

On Wed, Nov 18, 2020 at 10:27 AM bobbielf2 notifications@github.com wrote:

Thanks, Manas! I will use the double conjugation you said.

On Wed, Nov 18, 2020, 8:05 AM mrachh notifications@github.com wrote:

Hi, Thanks for bringing this to our notice. Currently, the Helmholtz wave number zk should be in the first quadrant. In case you do want to run the calculation for zk in the second quadrant, you can conjugate the density on input and the result on output. We will update our documentation soon to reflect this restriction.

Regards, Manas

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