Commit 759a13ce authored by Martin Reinecke's avatar Martin Reinecke
Browse files

tweak docs

parent a4c2996b
Pipeline #24371 passed with stage
in 6 minutes and 18 seconds
......@@ -82,11 +82,11 @@
"\n",
"- One-dimensional signal with power spectrum: $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n",
"with $P_0 = 0.2, k_0 = 5, \\gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$.\n",
"- $N = 0.5 \\cdot \\text{id}$.\n",
"- Number data points $N_{pix} = 512$.\n",
"- $N = 0.05 \\cdot \\text{id}$.\n",
"- Number of data points $N_{pix} = 512$.\n",
"- reconstruction in harmonic space.\n",
"- Response operator:\n",
"$$R_x=\\begin{pmatrix} \\delta(x-0)\\\\\\delta(x-1)\\\\\\ldots\\\\ \\delta(x-511) \\end{pmatrix}.$$\n",
"However, the signal space is also discrete on the computer and $R = \\text{id}$."
"$$R = FFT(\\text{harmonic} \\rightarrow \\text{position})$$\n"
]
},
{
......@@ -167,7 +167,7 @@
"outputs": [],
"source": [
"def PropagatorOperator(R, N, Sh):\n",
" IC = ift.GradientNormController(name=\"inverter\", iteration_limit=50000,\n",
" IC = ift.GradientNormController(iteration_limit=50000,\n",
" tol_abs_gradnorm=0.1)\n",
" inverter = ift.ConjugateGradient(controller=IC)\n",
" D = (R.adjoint*N.inverse*R + Sh.inverse).inverse\n",
......@@ -186,9 +186,10 @@
"### Conjugate Gradient Preconditioning\n",
"\n",
"- $D$ is defined via:\n",
"$$D^{-1} = \\mathcal F^\\dagger S_h^{-1}\\mathcal F + R^\\dagger N^{-1} R.$$\n",
"$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",
"In the end, we want to apply $D$ to $j$, i.e. we need the inverse action of $D^{-1}$. This is done numerically (algorithm: *Conjugate Gradient*). \n",
"\n",
"<!--\n",
"- One can define the *condition number* of a non-singular and normal matrix $A$:\n",
"$$\\kappa (A) := \\frac{|\\lambda_{\\text{max}}|}{|\\lambda_{\\text{min}}|},$$\n",
"where $\\lambda_{\\text{max}}$ and $\\lambda_{\\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.\n",
......@@ -201,7 +202,8 @@
"\n",
"- In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose\n",
"\n",
"$$T = \\mathcal F^\\dagger S_h^{-1} \\mathcal F.$$"
"$$T = \\mathcal F^\\dagger S_h^{-1} \\mathcal F.$$\n",
"-->"
]
},
{
......
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