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updating documentation and adding python file
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mrachh committed Feb 18, 2024

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172 changes: 157 additions & 15 deletions docs/math.rst
Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
Definitions
===========
Let $x_{j} \in \mathbb{R}^{2}$, $i=1,2,\ldots N$, denote a collection
of source locations and let $t_{i} \in \mathbb{R}^{2}$ denote a collection
Let $x_{j} = (x_{j,1}, x_{j,2}) \in \mathbb{R}^{2}$, $i=1,2,\ldots N$, denote a collection
of source locations and let $t_{i} = (t_{i,1}, t_{i,2}) \in \mathbb{R}^{2}$ denote a collection
of target locations.


@@ -13,12 +13,12 @@ The Laplace FMM comes in three varieties
gradients, and hessians are all double precision
* lfmm2d: All of the above quantities are double complex
* cfmm2d: Same as lfmm2d except that there are no dipole orientation vectors,
and the dipole kernel is given by $1/(zt_{i} - zx_{j})$ where $zt_{i}, zx_{j}$
and the dipole kernel is given by $1/(z_{i} - \xi_{j})$ where $z_{i}, \xi_{j}$
are the complex versions of the source and target locations given by
$zt_{i} = t_{i}(1) + i*t_{i}(2)$, and $zx_{j} = x_{j}(1) + i*y_{j}(2)$.
$z_{i} = t_{i,1} + i\cdot t_{i,2}$, and $\xi_{j} = x_{j,1} + i \cdot x_{j,2}$.
Moreover, the gradients, and hessians
in this case are derivatives with respect to $z$, and in a slight abuse of
notation, we set $d/dz log(|z|) = 1/z$
notation, we set $\frac{\mathrm{d}}{\mathrm{d} z} \log(\|z\|) = 1/z$


Laplace FMM (rfmm2d)
@@ -31,14 +31,14 @@ denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$,
$j=1,2,\ldots N$, denote the corresponding dipole orientation vectors.

The Laplace FMM (rfmm2d) computes
the potential $u(x) \in \mathbb{R}$ and the its gradient $\nabla u(x) \in
the potential $u(x) \in \mathbb{R}$ and its gradient $\nabla u(x) \in
\mathbb{R}^{2}$
given by

.. math::
:label: rlap_nbody
u(x) = \sum_{j=1}^{N} c_{j} log(\|x-x_{j}\|) - v_{j} d_{j}\cdot \nabla log(\|x-x_{j}\|) \, ,
u(x) = \sum_{j=1}^{N} c_{j} \log{(\|x-x_{j}\|)} - v_{j} d_{j}\cdot \nabla \log{(\|x-x_{j}\|)} \, ,
at the source and target locations. When $x=x_{j}$, the term
corresponding to $x_{j}$ is dropped from the sum.
@@ -54,14 +54,14 @@ denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$,
$j=1,2,\ldots N$, denote the corresponding dipole orientation vectors.

The Laplace FMM (rfmm2d) computes
the potential $u(x) \in \mathbb{C}$ and the its gradient $\nabla u(x) \in
the potential $u(x) \in \mathbb{C}$ and its gradient $\nabla u(x) \in
\mathbb{C}^{2}$
given by

.. math::
:label: llap_nbody
u(x) = \sum_{j=1}^{N} c_{j} log(\|x-x_{j}\|) - v_{j} d_{j}\cdot \nabla log(\|x-x_{j}\|) \, ,
u(x) = \sum_{j=1}^{N} c_{j} \log{(\|x-x_{j}\|)} - v_{j} d_{j}\cdot \nabla \log{(\|x-x_{j}\|)} \, ,
at the source and target locations. When $x=x_{j}$, the term
corresponding to $x_{j}$ is dropped from the sum.
@@ -76,18 +76,18 @@ $j=1,2,\ldots N$,
denote a collection of dipole strengths.

The Laplace FMM (cfmm2d) computes
the potential $u(z) \in \mathbb{C}$ and the its gradient $d/dz u(z) \in
the potential $u(z) \in \mathbb{C}$ and its gradient $\frac{\mathrm{d}}{\mathrm{d}z} u(z) \in
\mathbb{C}$
given by

.. math::
:label: clap_nbody
u(z) = \sum_{j=1}^{N} c_{j} log(\|z-zx_{j}\|) - \frac{v_{j}}{z-zx_{j}} \, ,
u(z) = \sum_{j=1}^{N} c_{j} \log{(\|z-\xi_{j}\|)} - \frac{v_{j}}{z-\xi_{j}} \, ,
at the source and target locations. When $z=zx_{j}$, the term
corresponding to $zx_{j}$ is dropped from the sum. Here
$z = x(1) + i x(2)$, and $zx_{j} = x_{j}(1) + ix_{j}(2)$, are the complex
at the source and target locations. When $z=\xi_{j}$, the term
corresponding to $\xi_{j}$ is dropped from the sum. Here
$z = x(1) + i x(2)$, and $\xi_{j} = x_{j}(1) + ix_{j}(2)$, are the complex
numbers corresponding to $x$ and $x_{j}$ respectively.


@@ -103,7 +103,7 @@ Let $k\in\mathbb{C}$ denote the wave number or the Helmholtz
parameter.

The Helmholtz FMM computes
the potential $u(x)$ and the its gradient $\nabla u(x)$
the potential $u(x)$ and its gradient $\nabla u(x)$
given by

.. math::
@@ -118,13 +118,155 @@ corresponding to $x_{j}$ is dropped from the sum.

Biharmonic FMM
***************
Let $c_{j} = (c_{j,1}, c_{j,2})\in \mathbb{C}^2$,
$j=1,2,\ldots N$,
denote a collection of charge strengths, and
$v_{j} = (v_{j,1}, v_{j,2}, v_{j,3}) \in \mathbb{C}^{3}$,
$j=1,2,\ldots N$,
denote a collection of dipole strengths.

The Biharmonic FMM computes
the potential $u(x)$ and its `gradient` = $(P_{z} \frac{\mathrm{d}}{\mathrm{d}z}, P_{\overline{z}} \frac{\mathrm{d}}{\mathrm{d}z}, \frac{\mathrm{d}}{\mathrm{d}\overline{z}})$
given by

.. math::
:label: biharm_nbody
u(z) &= \sum_{j=1}^{N} c_{j,1} \log{\|z - \xi_{j}\|} +
c_{j,2} \frac{z-\xi_{j}}{\overline{z-\xi_{j}}} + \frac{v_{j,1}}{z - \xi_{j}} +
\frac{v_{j,3}}{\overline{z-\xi_{j}}} +
v_{j,2} \frac{z - \xi_{j}}{(\overline{z-\xi_{j}})^2} \, , \\
P_{z} \frac{\mathrm{d}}{\mathrm{d} z}u(z) &= \sum_{j=1}^{N} \frac{c_{j,1}}{z-\xi_{j}} -
\frac{v_{j,1}}{(z-\xi_{j})^2} \, \\
P_{\overline{z}} \frac{\mathrm{d}}{\mathrm{d} z} u(z) &=
\sum_{j=1}^{N} \frac{c_{j,2}}{\overline{z-\xi_{j}}} +
\frac{v_{j,2}}{(\overline{z-\xi_{j}})^2} \,
,\\
\frac{\mathrm{d}}{\mathrm{d}\overline{z}} u(z) &=
\sum_{j=1}^{N} \frac{c_{j,1}}{\overline{z-\xi_{j}}} -
c_{j,2} \frac{z-\xi_{j}}{(\overline{z-\xi_{j}})^2} -
\frac{v_{j,3}}{(\overline{z-\xi_{j}})^2} -
2v_{j,2} \frac{z - \xi_{j}}{(\overline{z-\xi_{j}})^3} \, , \\
at the source and target locations. When $z=\xi_{j}$, the term
corresponding to $\xi_{j}$ is dropped from the sum. Here
$z = x(1) + i x(2)$, and $\xi_{j} = x_{j}(1) + ix_{j}(2)$, are the complex
numbers corresponding to $x$ and $x_{j}$ respectively.
When $x=x_{j}$, the term
corresponding to $x_{j}$ is dropped from the sum.


Modified Biharmonic FMM
************************

Let $G^{\textrm{mbh}}(x,y)$ denote the modified biharmonic
Green's function given by

.. math::
G^{\textrm{mbh}}(x,y) = \frac{1}{2\pi \beta^2}\left(K_{0}(\beta \|x-y \|) - \log{\|x-y\|}\right)
where $K_{0}$ is the modified Bessel function of order $0$, and $\beta$ is the
Modified Biharmonic wavenumber.


Let $c_{j} \in \mathbb{C}$,
denote a collection of charge strengths,
$v_{j} \in \mathbb{C}$,
denote a collection of dipole strengths,
$d_{j} = (d_{j,1}, d_{j,2})$ denote a collection
of dipole vectors,
$q_{j} \in \mathbb{C}$ denote a collection of
quadrupole strengths,
$w_{j} = (w_{j,1}, w_{j,2}, w_{j,3}) \in \mathbb{C}^{3}$,
denote a collection of quadrupole orientation vectors,
$o_{j}$ denote a collection of octopole strengths, and
$p_{j} = (p_{j,1}, p_{j,2}, p_{j,3}, p_{j,4}) \in \mathbb{C}^{4}$,
denote a collceetion of octopole strengths. For all the vectors,
$j=1,2,\ldots N$.

The Modified Biharmonic FMM computes
the potential $u(x)$ and its gradients, and hessians,
given by

.. math::
:label: modbiharm_nbody
u(x) = \sum_{j=1}^{N} &c_{j} G^{\textrm{mbh}}(x,x_{j}) +
v_{j} d_{j} \cdot \nabla_{y} G^{\textrm{mbh}}(x,x_{j}) + \\
&q_{j} \left(w_{j,1} \partial_{y_{1},y_{1}} G^{\textrm{mbh}}(x,x_{j}) + w_{j,2}
\partial_{y_{1},y_{2}} G^{\textrm{mbh}}(x,x_{j}) + w_{j,3}
\partial_{y_{2},y_{2}} G^{\textrm{mbh}}(x,x_{j}) \right) + \\
&o_{j} \big( p_{j,1} \partial_{y_{1},y_{1},y_{1}} G^{\textrm{mbh}}(x,x_{j}) +
p_{j,2} \partial_{y_{1},y_{1},y_{2}} G^{\textrm{mbh}}(x,x_{j}) + \\
&p_{j,3} \partial_{y_{1},y_{2},y_{2}} G^{\textrm{mbh}}(x,x_{j}) +
p_{j,4} \partial_{y_{2},y_{2},y_{2}} G^{\textrm{mbh}}(x,x_{j}) \big) \, ,
at the source and target locations. When $x=x_{j}$, the term
corresponding to $x_{j}$ is dropped from the sum.

Stokes FMM
************

Let $G^{\textrm{stok}}(x,y)$ denote the Stokeslet given by

.. math::
G^{\textrm{stok}}(x,y) = \frac{1}{2}
\begin{bmatrix}
-\log{\|x-y \|} + \frac{(x_{1}-y_{1})^2}{\|x-y\|^2} & \frac{(x_{1}-y_{1})
(x_{2}-y_{2})}{\|x-y \|^2} \\
\frac{(x_{1}-y_{1})(x_{2}-y_{2})}{\|x-y \|^2} &
-\log{\|x-y \|} + \frac{(x_{2}-y_{2})^2}{\|x-y \|^2}
\end{bmatrix} \, ,
$P^{\textrm{stok}}(x,y)$ denote the associated pressure tensor

.. math::
P(x,y) = \frac{1}{\|x-y \|^2}\begin{bmatrix}
(x_{1}-y_{1}) &
(x_{2}-y_{2})
\end{bmatrix} \, ,
$T^{\textrm{stok}}(x,y)$ denote the Stresslet whose action of a vector $v$
is given by

.. math::
v\cdot T^{\textrm{stok}}(x,y) = -\frac{2 v \cdot (x-y)}{\|x-y \|^4}
\begin{bmatrix}
(x_{1} - y_{1})^2 & (x_{1}-y_{1})(x_{2}-y_{2}) \\
(x_{1}-y_{1})(x_{2}-y_{2}) & (x_{2} - y_{2})^2
\end{bmatrix} \, ,
and finally let $\Pi^{\textrm{stok}} (x,y)$ denote its assocaited pressure tensor given by

.. math::
v\cdot \Pi(x,y)^{textrm{stok}} = -\frac{v}{\|x-y \|^2} + \frac{2 v \cdot(x-y)}{\|x-y \|^4}
\begin{bmatrix}
(x_{1}-y_{1}) \\
(x_{2}-y_{2})
\end{bmatrix}
Let $c_{j} \in \mathbb{R}^2$,
$j=1,2,\ldots N$,
denote a collection of stokeslet strengths, $v_{j} \in \mathbb{R}^2$,
$j=1,2,\ldots N$,
denote a collection of stresslet strengths, and $d_{j} \in \mathbb{R}^{2}$,
$j=1,2,\ldots N$, denote the corresponding stresslet orientation vectors.

The Stokes FMM computes
the potential $u(x) \in \mathbb{R}^2$, its gradient $\nabla u(x) \in
\mathbb{R}^{2\times 2}$, and the pressure $p$ given by

.. math::
:label: stok_nbody
u(x) &= \sum_{j=1}^{N} G^{\textrm{stok}}(x,x_{j}) c_{j} + d_{j} \cdot
T^{\textrm{stok}}(x,x_{j}) \cdot v_{j} \, , \\
p(x) &= \sum_{j=1}^{N} c_{j} P^{\textrm{stok}}(x,x_{j}) + d_{j} \cdot
\Pi^{\textrm{stok}}(x,x_{j}) \cdot v_{j}^{T}
at the source and target locations. When $x=x_{j}$, the term
corresponding to $x_{j}$ is dropped from the sum.

Vectorized versions
*******************
The vectorized versions of the Helmholtz FMM,
34 changes: 34 additions & 0 deletions python/bhfmmexample.py
Original file line number Diff line number Diff line change
@@ -0,0 +1,34 @@
#!/usr/bin/env python

import fmm2dpy as fmm
import numpy as np


#
# This is a sample code to demonstrate how to use
# the fmm libraries
#

# sample with one density, sources to sources,
# charge interactions, and potential only
#
n = 100000
nd = 1
sources = np.random.uniform(0,1,(2,n))
eps = 10**(-5)

charges = np.random.uniform(0,1,(2,n)) + 1j*np.random.uniform(0,1,(2,n))
dipoles = np.random.uniform(0,1,(3,n)) + 1j*np.random.uniform(0,1,(3,n))
out = fmm.bhfmm2d(eps=eps,sources=sources,charges=charges,dipoles=dipoles,pg=1)


# sample with a vector of densities, sources to
# sources and targets, dipole interactions,
# potential and gradietns

nd = 3
nt = 1870
targ = np.random.uniform(0,1,(2,nt))
dipoles = np.random.uniform(0,1,(nd,3,n)) + 1j*np.random.uniform(0,1,(nd,3,n))
out2 = fmm.bhfmm2d(eps=eps,sources=sources,dipoles=dipoles,\
targets=targ,nd=nd,pg=2,pgt=2)
6 changes: 3 additions & 3 deletions src/biharmonic/bhfmm2d.f
Original file line number Diff line number Diff line change
@@ -51,7 +51,7 @@ subroutine bhfmm2d(nd,eps,ns,sources,ifcharge,charge,
c and velocity are interchangably used
c
c
c vel(z) = \sum charge1_j *\log|z-z_j| +
c vel(z) = \sum 2*charge1_j *\log|z-z_j| +
c (charge2_j) (z-z_j)/(z-z_j)_bar+
c dip1_j/(z-z_j) + dip2_j/(z-z_j)_bar-
c dip3_j (z-z_j)/(z-z_j)^2_bar
@@ -65,8 +65,8 @@ subroutine bhfmm2d(nd,eps,ns,sources,ifcharge,charge,
c \phi(z) = \sum charge_j log(z-z_j)+
c dip1/(z-z_j)
c
c \psi(z) = \sum dippar2_bar/(z-z_j)+
c z_j_bar dippar1/(z-z_j)^2 +
c \psi(z) = \sum dip2_bar/(z-z_j)+
c z_j_bar dip1/(z-z_j)^2 +
c
c In all of the sums above, the terms for which |z-z_{j}| <= 2^-51|b|
c where |b| is the size of the root box will be ignored.
6 changes: 3 additions & 3 deletions src/modified-biharmonic/mbhfmm2d.f
Original file line number Diff line number Diff line change
@@ -36,10 +36,10 @@ subroutine mbhfmm2d(nd,eps,beta,ns,sources,ifcharge,charge,
c u(x) = sum_j charge(j)*G(x,y(j))
c + dipstr(j)*(G_{y1}(x,y(j))*dipvec(1,j) + G_{y2}*dipvec(2,j))
c + quadstr(j)*(G_{y1y1}(x,y(j))*quadvec(1,j)
c + G_{y1y2}*quadvec(2,j) + G_{y2y2}*quadvec(2,j))
c + G_{y1y2}*quadvec(2,j) + G_{y2y2}*quadvec(3,j))
c + octstr(j)*(G_{y1y1y1}(x,y(j))*octvec(1,j)
c + G_{y1y1y2}*octvec(2,j) + G_{y1y2y2}*octvec(2,j)
c + G_{y1y2y2}*octvec(2,j))
c + G_{y1y1y2}*octvec(2,j) + G_{y1y2y2}*octvec(3,j)
c + G_{y2y2y2}*octvec(4,j))
c
c INPUT PARAMETERS:
c
14 changes: 7 additions & 7 deletions src/stokes/stfmm2d.f
Original file line number Diff line number Diff line change
@@ -11,16 +11,16 @@
c The Stokeslet, G_{ij}, and its associated pressure tensor, P_j,
c (without the 1/2pi scaling) are
c
c UPDATE formulae
c
c G_{ij}(x,y) = (r_i r_j)/(2r^3) + delta_{ij}/(2r)
c P_j(x,y) = r_j/r^3
c G_{ij}(x,y) = (r_i r_j)/(2r^2) - delta_{ij}log(r)/(2)
c P_j(x,y) = r_j/r^2
c
c The (Type I) stresslet, T_{ijk}, and its associated pressure
c tensor, PI_{jk}, (without the 1/4pi scaling) are
c tensor, PI_{jk}, (without the 1/2pi scaling) are
c
c T_{ijk}(x,y) = -3 r_i r_j r_k/ r^5
c PI_{jk} = -2 delta_{jk} + 6 r_j r_k/r^5
c T_{ijk}(x,y) = -2 r_i r_j r_k/ r^4
c PI_{jk} = - delta_{jk}/r^2 + 2 r_j r_k/r^4
c
c
c


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