# coding: utf-8

# # Skeletonization using Proxies

# In[1]:

import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

import scipy.linalg.interpolative as sli

eps = 1e-7


# In[2]:

sources = np.random.rand(2, 200)
targets = np.random.rand(2, 200) + 3

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")


# In[3]:

def interaction_mat(t, s):
    all_distvecs = s.reshape(2, 1, -1) - t.reshape(2, -1, 1)
    dists = np.sqrt(np.sum(all_distvecs**2, axis=0))
    return np.log(dists)


# In[4]:

def numerical_rank(A, eps):
    _, sigma, _ = la.svd(A)
    return np.sum(sigma >= eps)


# Check the interaction rank:

# In[5]:

numerical_rank(interaction_mat(targets, sources), eps)


# **Idea:**
# 
# * Don't want to build whole matrix to find the few rows/columns that actually matter.
# * Introduces "proxies" that stand in for
#     * *all sources outside the targets* or
#     * *all targets outside these sources*

# ## Target Skeletonization

# In[38]:

nproxies = 25

angles = np.linspace(0, 2*np.pi, nproxies)
target_proxies = 3.5 + 1.5 * np.array([np.cos(angles), np.sin(angles)])


# In[39]:

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(target_proxies[0], target_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")


# Construct the interaction matrix *from* the target proxies *to* the targets as `target_proxy_mat`.
# 
# **A note on terminology:** The `target_proxies` are *near* the targets but *stand in* for far-away sources.

# In[61]:

target_proxy_mat = interaction_mat(targets, target_proxies)


# Check its numerical rank and shape:

# In[62]:

numerical_rank(target_proxy_mat, eps)


# In[42]:

target_proxy_mat.shape


# Now compute an ID (row or column?):

# In[43]:

idx, proj = sli.interp_decomp(target_proxy_mat.T, nproxies)


# Find the target skeleton as `target_skeleton`, i.e. the indices of the targets from which the remaining values can be recovered:

# In[44]:

target_skeleton = idx[:nproxies]


# Check that the ID does what is promises:

# In[45]:

P = np.hstack([np.eye(nproxies), proj])[:,np.argsort(idx)]
tpm_approx = P.T.dot(target_proxy_mat[target_skeleton])

la.norm(tpm_approx - target_proxy_mat, 2)


# Plot the chosen proxies:

# In[46]:

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro", alpha=0.05)
pt.plot(targets[0, target_skeleton], targets[1, target_skeleton], "ro")
pt.plot(target_proxies[0], target_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")


# What does this mean?
# 
# * We have now got a moral equivalent to a local expansion: The point values at the target skeleton points.
# * Is it a coincidence that the skeleton points sit at the boundary of the target region?
# * How many target proxies should we choose?
# * Can cheaply recompute potential at any target from those few points.
# * Have thus reduce LA-based evaluation cost to same as expansion-based cost.
# 
# Can we come up with an equivalent of a multipole expansion?

# ----------------
# 
# Check that this works for 'our' sources:

# In[49]:

imat_error = (
    P.T.dot(interaction_mat(targets[:, target_skeleton], sources))
    -
    interaction_mat(targets, sources))

la.norm(imat_error, 2)


# ## Source Skeletonization

# In[54]:

nproxies = 25

angles = np.linspace(0, 2*np.pi, nproxies)
source_proxies = 0.5 + 1.5 * np.array([np.cos(angles), np.sin(angles)])


# In[56]:

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(source_proxies[0], source_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")


# Construct the interaction matrix *from* the sources *to* the source proxies as `source_proxy_mat`:
# 
# **A note on terminology:** The `source_proxies` are *near* the sources but *stand in* for far-away targets.

# In[73]:

source_proxy_mat = interaction_mat(source_proxies, sources)


# In[74]:

source_proxy_mat.shape


# Now compute an ID (row or column?):

# In[75]:

idx, proj = sli.interp_decomp(source_proxy_mat, nproxies)


# In[76]:

source_skeleton = idx[:nproxies]


# In[78]:

P = np.hstack([np.eye(nproxies), proj])[:,np.argsort(idx)]
tsm_approx = source_proxy_mat[:, source_skeleton].dot(P)

la.norm(tsm_approx - source_proxy_mat, 2)


# Plot the chosen proxies:

# In[79]:

pt.plot(sources[0], sources[1], "go", alpha=0.05)
pt.plot(targets[0], targets[1], "ro")
pt.plot(sources[0, source_skeleton], sources[1, source_skeleton], "go")
pt.plot(source_proxies[0], source_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")


# Check that it works for 'our' targets:

# In[81]:

imat_error = (
    interaction_mat(targets, sources[:, source_skeleton]).dot(P)
    -
    interaction_mat(targets, sources))

la.norm(imat_error, 2)


# * Sensibly, this is just the transpose of the target skeletonization process.
#     * For a given point cluster, the same skeleton can serve for target and source skeletonization!
# * Computationally, starting from your original charges $x$, you accumulate 'new' charges $Px$ at the skeleton points and then *only* compute the interaction from the source skeleton to the targets.