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approximant_taylor_rb.py
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approximant_taylor_rb.py
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# Copyright (C) 2018 by the RROMPy authors
#
# This file is part of RROMPy.
#
# RROMPy is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# RROMPy is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with RROMPy. If not, see <http://www.gnu.org/licenses/>.
#
from copy import copy
import warnings
import numpy as np
import scipy as sp
from rrompy.reduction_methods.taylor.generic_approximant_taylor import GenericApproximantTaylor
from rrompy.utilities.types import Np1D, ListAny, DictAny, HFEng, HSEng
from rrompy.utilities import purgeDict
__all__ = ['ApproximantTaylorRB']
class ApproximantTaylorRB(GenericApproximantTaylor):
"""
ROM single-point fast RB approximant computation for parametric problems
with polynomial dependence up to degree 2.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: sparse matrix (in CSC format) associated to
w-weighted H10 norm;
- problemData: list of HF problem data (excluding mu);
- setProblemData: set HF problem data and mu to prescribed values;
- getLSBlocks: get blocks of HF linear system as sparse matrices in
CSC format;
- liftDirichletData: perform lifting of Dirichlet BC as numpy
complex vector;
- setupDerivativeComputation: setup HF problem data to compute j-th
solution derivative at mu0;
- solve: get HF solution as complex numpy vector.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
mu0(optional): Default parameter. Defaults to 0.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'R': rank for Galerkin projection; defaults to E + 1;
- 'E': total number of derivatives current approximant relies upon;
defaults to Emax;
- 'Emax': total number of derivatives of solution map to be
computed; defaults to E;
- 'sampleType': label of sampling type; available values are:
- 'ARNOLDI': orthogonalization of solution derivatives through
Arnoldi algorithm;
- 'KRYLOV': standard computation of solution derivatives.
Defaults to 'KRYLOV'.
Defaults to empty dict.
plotDer(optional): What to plot of derivatives of the Helmholtz
solution map. See plot method in HSEngine. Defaults to [].
plotDerSpecs(optional): How to plot derivatives of the Helmholtz
solution map. See plot method in HSEngine. Defaults to {}.
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
mu0: Default parameter.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are in parameterList.
parameterList: Recognized keys of approximant parameters:
- 'POD': whether to compute POD of snapshots;
- 'R': rank for Galerkin projection;
- 'E': total number of derivatives current approximant relies upon;
- 'Emax': total number of derivatives of solution map to be
computed;
- 'sampleType': label of sampling type.
solDerivatives: Array whose columns are FE dofs of solution
derivatives.
RArnoldi: Right factor of derivatives QR factorization. If no Arnoldi,
set to None.
HArnoldi: Hessenberg factor of derivatives Arnoldi algorithm. If no
Arnoldi, set to None.
POD: Whether to compute QR factorization of derivatives.
R: Rank for Galerkin projection.
E: Number of solution derivatives over which current approximant is
based upon.
Emax: Total number of solution derivatives to be computed.
sampleType: Label of sampling type, i.e. 'KRYLOV'.
plotDer: What to plot of derivatives of the Helmholtz solution map.
plotDerSpecs: How to plot derivatives of the Helmholtz solution map.
energyNormMatrix: Sparse matrix (in CSC format) associated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
solLifting: Numpy complex vector with lifting of real part of Dirichlet
boundary datum.
projMat: Numpy matrix representing projection onto RB space.
projMat: Numpy matrix representing projection onto RB space.
As: List of sparse matrices (in CSC format) representing coefficients
of linear system matrix wrt mu.
bs: List of numpy vectors representing coefficients of linear system
RHS wrt mu.
ARBs: List of sparse matrices (in CSC format) representing RB
coefficients of linear system matrix wrt mu.
bRBs: List of numpy vectors representing RB coefficients of linear
system RHS wrt mu.
"""
def __init__(self, HFEngine:HFEng, HSEngine:HSEng, mu0 : complex = 0,
approxParameters : DictAny = {}, plotDer : ListAny = [],
plotDerSpecs : DictAny = {}):
if not hasattr(self, "parameterList"):
self.parameterList = []
self.parameterList += ["R"]
super().__init__(HFEngine = HFEngine, HSEngine = HSEngine, mu0 = mu0,
approxParameters = approxParameters,
plotDer = plotDer, plotDerSpecs = plotDerSpecs)
def resetSamples(self):
"""Reset samples."""
super().resetSamples()
self.projMat = None
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change M, N and S.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
approxParameters = purgeDict(approxParams, self.parameterList,
dictname = self.name() + ".approxParameters")
approxParametersCopy = purgeDict(approxParameters, ["R"],
True, True)
GenericApproximantTaylor.approxParameters.fset(self,
approxParametersCopy)
keyList = list(approxParameters.keys())
if "R" in keyList:
self.R = approxParameters["R"]
elif hasattr(self, "R"):
self.R = self.R
else:
self.R = self.E + 1
@property
def POD(self):
"""Value of POD."""
return self._POD
@POD.setter
def POD(self, POD):
GenericApproximantTaylor.POD.fset(self, POD)
if (hasattr(self, "sampleType") and self.sampleType == "ARNOLDI"
and not self.POD):
warnings.warn(("Arnoldi sampling implicitly forces POD-type "
"derivative management."), stacklevel = 2)
@property
def sampleType(self):
"""Value of sampleType."""
return self._sampleType
@sampleType.setter
def sampleType(self, sampleType):
GenericApproximantTaylor.sampleType.fset(self, sampleType)
if (hasattr(self, "POD") and not self.POD
and self.sampleType == "ARNOLDI"):
warnings.warn(("Arnoldi sampling implicitly forces POD-type "
"derivative management."), stacklevel = 2)
@property
def R(self):
"""Value of R. Its assignment may change S."""
return self._R
@R.setter
def R(self, R):
if R < 0: raise ArithmeticError("R must be non-negative.")
self._R = R
self._approxParameters["R"] = self.R
if hasattr(self, "E") and self.E + 1 < self.R:
warnings.warn("Prescribed E is too small. Updating E to R - 1.",
stacklevel = 2)
self.E = self.R - 1
def setupApprox(self):
"""
Setup RB system. For usage of correlation matrix in POD see Section
6.3.1 in 'Reduced Basis Methods for PDEs. An introduction', A.
Quarteroni, A. Manzoni, F. Negri, Springer, 2016.
"""
if not self.checkComputedApprox():
self.computeDerivatives()
if self.POD and not self.sampleType == "ARNOLDI":
A = copy(self.solDerivatives)
M = self.energyNormMatrix
V = np.zeros_like(A, dtype = np.complex)
Q = np.zeros_like(A, dtype = np.complex)
R = np.zeros((A.shape[1], A.shape[1]), dtype = np.complex)
for k in range(A.shape[1]):
Q[:, k] = (np.random.randn(Q.shape[0])
+ 1.j * np.random.randn(Q.shape[0]))
if k > 0:
for times in range(2): #twice is enough!
Q[:, k] = Q[:, k] - Q[:, :k].dot(
(Q[:, k].conj().dot(M.dot(Q[:, :k]))).conj())
Q[:, k] = Q[:, k]/(Q[:, k].conj().dot(M.dot(Q[:, k])))**.5
R[k, k] = np.abs(A[:, k].conj().dot(M.dot(A[:, k]))) ** .5
alpha = Q[:, k].conj().dot(M.dot(A[:, k]))
if np.isclose(np.abs(alpha), 0.): s = 1.
else: s = - alpha / np.abs(alpha)
Q[:, k] = s * Q[:, k]
v = R[k, k] * Q[:, k] - A[:, k]
for times in range(2): #twice is enough!
v = v - Q[:, :k].dot((v.conj().dot(M.dot(Q[:, :k]))
).conj())
sigma = np.abs(v.conj().dot(M.dot(v))) ** .5
if np.isclose(sigma, 0.): v = Q[:, k]
else: v = v / sigma
V[:, k] = v
J = np.arange(k + 1, A.shape[1])
vtQ = v.conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - 2 * np.outer(v, vtQ)
R[k, J] = Q[:, k].conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - np.outer(Q[:, k], R[k, J])
for k in range(A.shape[1] - 1, -1, -1):
v = V[:, k]
J = np.arange(k, A.shape[1])
vtQ = v.conj().dot(M.dot(Q[:, J]))
Q[:, J] = Q[:, J] - 2 * np.outer(v, vtQ)
self.projMatQ = Q
self.projMatR = R
if self.POD and not self.sampleType == "ARNOLDI":
U, _, _ = np.linalg.svd(self.projMatR[: self.R, : self.R])
self.projMat = self.projMatQ[:, : self.R].dot(U)
else:
self.projMat = self.solDerivatives[:, : self.R]
self.assembleReducedSystem()
self.lastApproxParameters = copy(self.approxParameters)
def assembleReducedSystem(self):
"""Build affine blocks of RB linear system through projections."""
projMatH = self.projMat.T.conjugate()
self.ARBs = [None] * len(self.As)
self.bRBs = [None] * len(self.bs)
for j in range(len(self.As)):
self.ARBs[j] = projMatH.dot(self.As[j].dot(self.projMat))
for j in range(len(self.bs)):
self.bRBs[j] = projMatH.dot(self.bs[j])
def solveReducedSystem(self, mu:complex) -> Np1D:
"""
Solve RB linear system.
Args:
mu: Target parameter.
Returns:
Solution of RB linear system.
"""
self.setupApprox()
ARBmu = self.ARBs[0][: self.R, : self.R]
bRBmu = self.bRBs[0][: self.R]
for j in range(1, len(self.ARBs)):
ARBmu = ARBmu + np.power(mu, j) * self.ARBs[j][:self.R, :self.R]
for j in range(1, len(self.bRBs)):
bRBmu = bRBmu + np.power(mu, j) * self.bRBs[j][:self.R]
return self.projMat[:, :self.R].dot(np.linalg.solve(ARBmu, bRBmu))
def evalApprox(self, mu:complex):
"""
Evaluate RB approximant at arbitrary wavenumber.
Args:
mu: Target parameter.
"""
self.setupApprox()
self.uApp = self.solveReducedSystem(mu)
def getPoles(self) -> Np1D:
"""
Obtain approximant poles.
Returns:
Numpy complex vector of poles.
"""
self.setupApprox()
A = np.eye(self.ARBs[0].shape[0] * (len(self.ARBs) - 1),
dtype = np.complex)
B = np.zeros_like(A)
A[: self.ARBs[0].shape[0], : self.ARBs[0].shape[0]] = - self.ARBs[0]
for j in range(len(self.ARBs) - 1):
Aj = self.ARBs[j + 1]
B[: Aj.shape[0], j * Aj.shape[0] : (j + 1) * Aj.shape[0]] = Aj
II = np.arange(self.ARBs[0].shape[0],
self.ARBs[0].shape[0] * (len(self.ARBs) - 1))
B[II, II - self.ARBs[0].shape[0]] = 1.
try:
return sp.linalg.eigvals(A, B)
except:
warnings.warn("Generalized eig algorithm did not converge.",
stacklevel = 2)
x = np.empty(A.shape[0])
x[:] = np.NaN
return x

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