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TL: added notebook with iteration generator
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Lines changed: 243 additions & 7 deletions

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‎blockops/blockIterationGenerator.py‎

Lines changed: 5 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -81,8 +81,8 @@ def checkResults(self, u):
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def generateSymbols(self, nBlocks, nLevels, nPreSmooth=1, nPostSmooth=0):
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save_symbols = {}
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u_0 = sy.Symbol(r'u_0', commutative=False)
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phi = [sy.Symbol(f'\phi_{i}', commutative=False) for i in range(nLevels)]
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chi = [sy.Symbol(f'\chi_{i}', commutative=False) for i in range(nLevels)]
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phi = [sy.Symbol(rf'\phi_{i}', commutative=False) for i in range(nLevels)]
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chi = [sy.Symbol(rf'\chi_{i}', commutative=False) for i in range(nLevels)]
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T_c_to_f = [sy.Symbol(f'T_{i + 1}^{i}', commutative=False) for i in range(nLevels)]
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T_f_to_c = [sy.Symbol(f'T_{i}^{i + 1}', commutative=False) for i in range(nLevels)]
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A = [sy.Matrix(np.eye(nBlocks, dtype=int) * phi[i]) + sy.Matrix(np.eye(nBlocks, k=-1, dtype=int) * -chi[i]) for i in
@@ -237,10 +237,10 @@ def check(self, expr: sy.Expr, n: int):
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Current block
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"""
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expr_str = f'{expr}'
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unknowns = list(set(re.findall(re.compile('u\^\d+_\d+'), expr_str)))
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unknowns = list(set(re.findall(re.compile(r'u\^\d+_\d+'), expr_str)))
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tmpWildcard = {}
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for i in range(len(unknowns)):
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tmp_split = re.split('_|\^', unknowns[i])
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tmp_split = re.split(r'_|\^', unknowns[i])
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iteration = int(tmp_split[1])
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block = int(tmp_split[2])
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tmp_block = 'n' if n - int(block) == 0 else f'n-{n - int(block)}'
@@ -291,4 +291,4 @@ def u(n, k):
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if __name__ == "__main__":
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# PararealGenerator(n=4)
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MultilevelGenerator(nBlocks=7, nLevels=3, nPreSmooth=1, nPostSmooth=0)
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MultilevelGenerator(nBlocks=7, nLevels=4, nPreSmooth=1, nPostSmooth=0)

‎notebook/08_blockGenerator.ipynb‎

Lines changed: 236 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,236 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "5a905a68",
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"metadata": {},
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"source": [
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"# BlockIteration generator\n",
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"\n",
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"Using the formalism of block iterations, one can generate different algorithms using symbolic computations. In particular, we can use the time-multigrid formalism to determine the block iteration for Parareal and many of its extension.\n",
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"\n",
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"For instance, Parareal consist on a 2-level time-multigrid approach with 1 pre-smoothing step, and the block update formula can be obtained like this :"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "0d19156e",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Found rule: \n",
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"\n",
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"u_{n+1}^{k+1}=\n",
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" F*u_{n}^{k}\n",
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" +G*u_{n}^{k+1}\n",
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" -G*u_{n}^{k}\n",
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"\n"
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]
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}
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],
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"source": [
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"from blockops.blockIterationGenerator import MultilevelGenerator\n",
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"\n",
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"MultilevelGenerator(nBlocks=7, nLevels=2, nPreSmooth=1, nPostSmooth=0);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d58e00bc",
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"metadata": {},
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"source": [
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"Now, if we consider 2 pre-smoothing, then we get Parareal with overlap (or MGRIT with FCF relaxation) :"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "ea75d86f",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Found rule: \n",
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"\n",
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"u_{n+1}^{k+1}=\n",
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" G*u_{n}^{k+1}\n",
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" +F**2*u_{n-1}^{k}\n",
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" -G*F*u_{n-1}^{k}\n",
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"\n"
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]
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}
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],
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"source": [
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"MultilevelGenerator(nBlocks=7, nLevels=2, nPreSmooth=2, nPostSmooth=0);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4891764a",
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"metadata": {},
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"source": [
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"and this can be continued to get FCFCF[...] relaxation :"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "8ba5dd3b",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Found rule: \n",
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"\n",
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"u_{n+1}^{k+1}=\n",
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" G*u_{n}^{k+1}\n",
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" +F**3*u_{n-2}^{k}\n",
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" -G*F**2*u_{n-2}^{k}\n",
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"\n"
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]
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}
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],
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"source": [
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"MultilevelGenerator(nBlocks=7, nLevels=2, nPreSmooth=3, nPostSmooth=0);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ceb74720",
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"metadata": {},
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"source": [
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"Now, more interesting could be to build a block formula for the three-level Parareal approach :"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "ddcd2b14",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Found rule: \n",
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"\n",
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"u_{n+1}^{k+1}=\n",
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" F*u_{n}^{k}\n",
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" +H*u_{n}^{k+1}\n",
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" -G*u_{n}^{k}\n",
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" +G*F*u_{n-1}^{k}\n",
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" -H*F*u_{n-1}^{k}\n",
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"\n"
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]
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}
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],
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"source": [
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"MultilevelGenerator(nBlocks=7, nLevels=3, nPreSmooth=1, nPostSmooth=0);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "12c5ce1d",
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"metadata": {},
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"source": [
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"... and even go to 4 levels eventually :"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "4c45b9d5",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Found rule: \n",
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"\n",
159+
"u_{n+1}^{k+1}=\n",
160+
" F*u_{n}^{k}\n",
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" +K*u_{n}^{k+1}\n",
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" -G*u_{n}^{k}\n",
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" +G*F*u_{n-1}^{k}\n",
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" -K*F*u_{n-1}^{k}\n",
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" -T_1^0*T_2^1*\\phi_2**(-1)*\\chi_2*T_1^2*\\phi_1**(-1)*\\chi_1*T_0^1*u_{n-1}^{k}\n",
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" +T_1^0*T_2^1*\\phi_2**(-1)*\\chi_2*T_1^2*\\phi_1**(-1)*\\chi_1*T_0^1*F*u_{n-2}^{k}\n",
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" +T_1^0*T_2^1*T_3^2*\\phi_3**(-1)*\\chi_3*T_2^3*T_1^2*\\phi_1**(-1)*\\chi_1*T_0^1*u_{n-1}^{k}\n",
168+
" -T_1^0*T_2^1*T_3^2*\\phi_3**(-1)*\\chi_3*T_2^3*T_1^2*\\phi_1**(-1)*\\chi_1*T_0^1*F*u_{n-2}^{k}\n",
169+
"\n"
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]
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}
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],
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"source": [
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"MultilevelGenerator(nBlocks=7, nLevels=4, nPreSmooth=1, nPostSmooth=0);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f3671960",
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"metadata": {},
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"source": [
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"Note that until now, the formula were automatically simplified using those notations :\n",
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"\n",
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"$$\n",
185+
"\\begin{align}\n",
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"F &= \\phi_0^{-1}\\chi_0 \\\\\n",
187+
"G &= T_0^1 \\phi_1^{-1}\\chi_1 T_1^0 \\\\\n",
188+
"H &= T_0^1 T_1^2 \\phi_2^{-1}\\chi_2 T_2^1 T_1^0 \\\\\n",
189+
"K &= T_0^1 T_1^2 T_2^3 \\phi_3^{-1}\\chi_3 T_3^2 T_2^1 T_1^0 \\\\\n",
190+
"...&\n",
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"\\end{align}\n",
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"$$\n",
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"\n",
194+
"where $0$ is the index used for the finest level, and the increases with the coarsening. $T_{i}^{j}$ is then the transfer operator to go from level $i$ to level $j$.\n",
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"\n",
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"Considering the hypothesis that if $j<i$, $T_{i}^{j} T_{j}^{i} = I$ (which means that going from a coarse level to the fine level above then back to the coarse level again does not change the solution), we can simplify the formula for 4-level Parareal above as :\n",
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"\n",
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"```latex\n",
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"u_{n+1}^{k+1}=\n",
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" F*u_{n}^{k}\n",
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" +K*u_{n}^{k+1}\n",
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" -G*u_{n}^{k}\n",
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" +G*F*u_{n-1}^{k}\n",
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" -K*F*u_{n-1}^{k}\n",
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" -H*G*u_{n-1}^{k}\n",
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" +H*G*F*u_{n-2}^{k}\n",
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" +K*G*u_{n-1}^{k}\n",
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" -K*G*F*u_{n-2}^{k}\n",
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"```\n",
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"\n",
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"Now, don't hesitate to play and generate all kind of Parareal-like formula !"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "base",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.13.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

‎pyproject.toml‎

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -3,13 +3,13 @@ requires = ["hatchling"]
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build-backend = "hatchling.build"
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[tool.hatch.build.targets.wheel]
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packages = ["time4apint"]
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packages = ["blockops"]
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[tool.hatch.metadata]
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allow-direct-references = true
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[project]
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name = "time4apint"
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name = "blockops"
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version = "0.0.2"
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description = "Analysis tool for iterative Parallel-in-Time algorithms"
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dependencies = [

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