Blatt 08, 03.07.2019¶

Aktivieren Sie wie gewohnt ihre Arbeitsumgebung und starten Sie den Jupyter Notebook server, siehe zB Blatt 1, Aufgabe 0.
Importieren Sie numpy und pymor.basic und machen Sie matplotlib für das Notebook nutzbar.

%matplotlib notebook
import numpy as np
from pymor.basic import *
from matplotlib import pyplot as plt

Wir betrachten das Diffusionsproblem von Blatt 04, Aufgabe 1 mit dem Ausgabe Funktional

$$ s(v) := \frac{1}{|\Omega_\text{out}|} \int_{\Omega_\text{out}} v\; \text{d}x \quad\quad\text{mit } \Omega_\text{out} := (0, 1) \times (0.9, 1) $$

und das folgende Minimierungsproblem: finde das Minimum

$$ \mu* := \underset{\mu \in \mathcal{P}}{\text{argmin}} J(\mu) $$

wobei $J(\mu) := s(u_\mu)$, und $u_\mu$ für $\mu \in \mathcal{P}$ die Lösung des Diffusionsproblems sei; $J$ wird oft Paramter-zu-Ausgabe Abbilding oder input-output map genannt.

Aufgabe 1: der schwache greedy Algorithmus¶

Erstellen Sie ein analytisches Problem unter Verwedung von problem_B4_A1_parametric und fügen Sie dem Problem das Ausgabefunktional mit der id output hinzu (siehe auch API Dokumentation von StationaryProblem).

from problems import problem_B4_A1_parametric

problem = problem_B4_A1_parametric()
problem = problem.with_(functionals={'output': ('l2', ExpressionFunction('(x[..., 1] >= 0.9) * 10.', 2, ())) })

00:00 |WARNING|BitmapFunction: Image R.png not in grayscale mode. Convertig to grayscale.
00:00 |WARNING|BitmapFunction: Image B.png not in grayscale mode. Convertig to grayscale.

Diskretisieren Sie das Problem mit Hilfe des cg discretizers und werten Sie $J$ für den Paramter $\mu = ${'R': 1, 'B': 1} aus.
- extrahieren Sie dazu das Ausgabefunktional s aus dem operators-dict der Diskretisierung
- definieren Sie $J$

d, _ = discretize_stationary_cg(problem)
print(d.operators.keys())

00:00 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:00 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:00 DiffusionOperatorP1: Determine global dofs ...
00:00 DiffusionOperatorP1: Boundary treatment ...
00:00 DiffusionOperatorP1: Assemble system matrix ...
00:00 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:00 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:00 DiffusionOperatorP1: Determine global dofs ...
00:00 DiffusionOperatorP1: Boundary treatment ...
00:00 DiffusionOperatorP1: Assemble system matrix ...
00:00 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:00 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:00 DiffusionOperatorP1: Determine global dofs ...
00:00 DiffusionOperatorP1: Boundary treatment ...
00:00 DiffusionOperatorP1: Assemble system matrix ...
00:00 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:00 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:00 DiffusionOperatorP1: Determine global dofs ...
00:00 DiffusionOperatorP1: Boundary treatment ...
00:00 DiffusionOperatorP1: Assemble system matrix ...
00:00 L2ProductP1: Integrate the products of the shape functions on each element
00:00 L2ProductP1: Determine global dofs ...
00:00 L2ProductP1: Boundary treatment ...
00:00 L2ProductP1: Assemble system matrix ...
00:00 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:01 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:01 DiffusionOperatorP1: Determine global dofs ...
00:01 DiffusionOperatorP1: Boundary treatment ...
00:01 DiffusionOperatorP1: Assemble system matrix ...
00:01 L2ProductP1: Integrate the products of the shape functions on each element
00:01 L2ProductP1: Determine global dofs ...
00:01 L2ProductP1: Boundary treatment ...
00:01 L2ProductP1: Assemble system matrix ...
00:01 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ...
00:01 DiffusionOperatorP1: Calculate all local scalar products beween gradients ...
00:01 DiffusionOperatorP1: Determine global dofs ...
00:01 DiffusionOperatorP1: Boundary treatment ...
00:01 DiffusionOperatorP1: Assemble system matrix ...

dict_keys(['output_functional', 'operator', 'rhs'])

s = d.operators['output_functional']

def J(mu):
    U = d.solve(mu)
    return s.apply(U).data[0][0]

print(J([1, 1]))

00:01 StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1, R: 1} ...

0.003584674108277644

Erstellen Sie einen CoerciveRBReductor mit entsprechenden product und coercivity_estimator

reductor = CoerciveRBReductor(
        d, product=d.h1_0_semi_product,
        coercivity_estimator=ExpressionParameterFunctional('min([1, R, B])', d.parameter_type))

Erstellen Sie eine reduzierte Basis mit Hilfe eines (weak discrete) greedy Algorithmus.
- Machen Sie sich mit dem greedy Algorithmus in pyMOR vertraut. Dieser Nutzt den Residuenbasierten offline/online zerlegbaren Fehlerschätzer (weak) des CoerciveRBReductor, um den Modellreduktionsfehler online-effizient für eine große Menge an Trainingsparametern (discrete) zu schätzen. Es werden also für jedes Element der reduzierten Basis nur eine hochdimensionale Lösung der PDE benötigt.
- Verwenden Sie eine Trainingsmenge von 10000 gleichverteilten Paramtern.
- Stellen Sie sicher, dass der maximale geschätzte Modellreduktionsfehler (über die Trainingsmenge) 1e-3 nicht überschreitet.
- Stellen Sie sicher, dass die reduzierte Basis mit Hilfe eines Gram-Schmidt Verfahrens orthonormalisiert wird, setzen Sie hierzu extension_params={'method': 'gram_schmidt'}.

greedy_data = greedy(
    d, reductor, d.parameter_space.sample_uniformly(100),
    atol=1e-3,
    extension_params={'method': 'gram_schmidt'}
)

00:01 greedy: Started greedy search on 10000 samples
00:01 greedy: Reducing ...
00:01 |   CoerciveRBReductor: RB projection ...
00:01 |   CoerciveRBReductor: Assembling error estimator ...
00:01 |   |   ResidualReductor: Estimating residual range ...
00:01 |   |   |   estimate_image_hierarchical: Estimating image for basis vector -1 ...
00:02 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:02 |   |   ResidualReductor: Projecting residual operator ...
00:02 greedy: Estimating errors ...
00:04 greedy: Maximum error after 0 extensions: 5199.189979573521 (mu = {B: 0.0001, R: 0.0001})
00:04 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.0001} ...
00:04 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.0001} ...
00:04 greedy: Extending basis with solution snapshot ...
      
00:04 greedy: Reducing ...
00:04 |   CoerciveRBReductor: RB projection ...
00:05 |   CoerciveRBReductor: Assembling error estimator ...
00:05 |   |   ResidualReductor: Estimating residual range ...
00:05 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 0 ...
00:05 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:05 |   |   |   |   gram_schmidt: Removing vector 1 of norm 3.713266607296652e-18
00:05 |   |   |   |   gram_schmidt: Removing linear dependent vector 4
00:05 |   |   ResidualReductor: Projecting residual operator ...
00:05 greedy: Estimating errors ...
00:08 greedy: Maximum error after 1 extensions: 3588.8146149577324 (mu = {B: 0.0001, R: 1.0})
00:08 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 1.0} ...
00:08 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 1.0} ...
00:08 greedy: Extending basis with solution snapshot ...
      
00:08 greedy: Reducing ...
00:08 |   CoerciveRBReductor: RB projection ...
00:08 |   CoerciveRBReductor: Assembling error estimator ...
00:08 |   |   ResidualReductor: Estimating residual range ...
00:08 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 1 ...
00:08 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:08 |   |   |   |   gram_schmidt: Removing vector 3 of norm 6.4911410267974024e-18
00:08 |   |   |   |   gram_schmidt: Orthonormalizing vector 4 again
00:08 |   |   |   |   gram_schmidt: Orthonormalizing vector 6 again
00:08 |   |   ResidualReductor: Projecting residual operator ...
00:08 greedy: Estimating errors ...
00:11 greedy: Maximum error after 2 extensions: 2764.7755366045512 (mu = {B: 1.0, R: 0.0001})
00:11 greedy: Computing solution snapshot for mu = {B: 1.0, R: 0.0001} ...
00:11 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1.0, R: 0.0001} ...
00:12 greedy: Extending basis with solution snapshot ...
      
00:12 greedy: Reducing ...
00:12 |   CoerciveRBReductor: RB projection ...
00:12 |   CoerciveRBReductor: Assembling error estimator ...
00:12 |   |   ResidualReductor: Estimating residual range ...
00:12 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 2 ...
00:12 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:12 |   |   |   |   gram_schmidt: Removing vector 6 of norm 2.710973310666844e-17
00:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 8 again
00:12 |   |   ResidualReductor: Projecting residual operator ...
00:12 greedy: Estimating errors ...
00:15 greedy: Maximum error after 3 extensions: 202.732922083436 (mu = {B: 0.0001, R: 0.22229999999999997})
00:15 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.22229999999999997} ...
00:15 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.22229999999999997} ...
00:15 greedy: Extending basis with solution snapshot ...
00:15 |   gram_schmidt: Orthonormalizing vector 3 again
      
00:15 greedy: Reducing ...
00:15 |   CoerciveRBReductor: RB projection ...
00:15 |   CoerciveRBReductor: Assembling error estimator ...
00:15 |   |   ResidualReductor: Estimating residual range ...
00:15 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 3 ...
00:15 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:15 |   |   |   |   gram_schmidt: Removing vector 9 of norm 1.290777571816372e-16
00:15 |   |   |   |   gram_schmidt: Orthonormalizing vector 11 again
00:15 |   |   |   |   gram_schmidt: Orthonormalizing vector 12 again
00:15 |   |   ResidualReductor: Projecting residual operator ...
00:15 greedy: Estimating errors ...
00:19 greedy: Maximum error after 4 extensions: 198.2312470568827 (mu = {B: 0.2728, R: 0.0001})
00:19 greedy: Computing solution snapshot for mu = {B: 0.2728, R: 0.0001} ...
00:19 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.2728, R: 0.0001} ...
00:19 greedy: Extending basis with solution snapshot ...
00:19 |   gram_schmidt: Orthonormalizing vector 4 again
      
00:19 greedy: Reducing ...
00:19 |   CoerciveRBReductor: RB projection ...
00:19 |   CoerciveRBReductor: Assembling error estimator ...
00:19 |   |   ResidualReductor: Estimating residual range ...
00:19 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 4 ...
00:19 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:19 |   |   |   |   gram_schmidt: Removing vector 12 of norm 2.1958436305627265e-16
00:19 |   |   |   |   gram_schmidt: Orthonormalizing vector 15 again
00:19 |   |   ResidualReductor: Projecting residual operator ...
00:19 greedy: Estimating errors ...
00:22 greedy: Maximum error after 5 extensions: 46.28371663457766 (mu = {B: 0.0708, R: 0.0001})
00:22 greedy: Computing solution snapshot for mu = {B: 0.0708, R: 0.0001} ...
00:22 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0708, R: 0.0001} ...
00:22 greedy: Extending basis with solution snapshot ...
00:22 |   gram_schmidt: Orthonormalizing vector 5 again
      
00:22 greedy: Reducing ...
00:22 |   CoerciveRBReductor: RB projection ...
00:22 |   CoerciveRBReductor: Assembling error estimator ...
00:22 |   |   ResidualReductor: Estimating residual range ...
00:22 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 5 ...
00:22 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:22 |   |   |   |   gram_schmidt: Removing vector 15 of norm 4.856513302966699e-16
00:22 |   |   |   |   gram_schmidt: Orthonormalizing vector 17 again
00:22 |   |   |   |   gram_schmidt: Orthonormalizing vector 18 again
00:22 |   |   ResidualReductor: Projecting residual operator ...
00:22 greedy: Estimating errors ...
00:26 greedy: Maximum error after 6 extensions: 41.35728288180628 (mu = {B: 0.0001, R: 0.060700000000000004})
00:26 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.060700000000000004} ...
00:26 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.060700000000000004} ...
00:26 greedy: Extending basis with solution snapshot ...
00:26 |   gram_schmidt: Orthonormalizing vector 6 again
      
00:26 greedy: Reducing ...
00:26 |   CoerciveRBReductor: RB projection ...
00:26 |   CoerciveRBReductor: Assembling error estimator ...
00:26 |   |   ResidualReductor: Estimating residual range ...
00:26 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 6 ...
00:26 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:26 |   |   |   |   gram_schmidt: Removing vector 18 of norm 3.7264021444657053e-16
00:26 |   |   |   |   gram_schmidt: Orthonormalizing vector 21 again
00:26 |   |   ResidualReductor: Projecting residual operator ...
00:26 greedy: Estimating errors ...
00:29 greedy: Maximum error after 7 extensions: 18.96832142909099 (mu = {B: 0.0001, R: 0.596})
00:29 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.596} ...
00:29 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.596} ...
00:29 greedy: Extending basis with solution snapshot ...
00:29 |   gram_schmidt: Orthonormalizing vector 7 again
      
00:29 greedy: Reducing ...
00:29 |   CoerciveRBReductor: RB projection ...
00:29 |   CoerciveRBReductor: Assembling error estimator ...
00:29 |   |   ResidualReductor: Estimating residual range ...
00:29 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 7 ...
00:29 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:29 |   |   |   |   gram_schmidt: Removing vector 21 of norm 1.0456690646094309e-15
00:29 |   |   |   |   gram_schmidt: Orthonormalizing vector 23 again
00:29 |   |   |   |   gram_schmidt: Orthonormalizing vector 24 again
00:29 |   |   ResidualReductor: Projecting residual operator ...
00:29 greedy: Estimating errors ...
00:33 greedy: Maximum error after 8 extensions: 15.670922798571622 (mu = {B: 0.6364, R: 0.0001})
00:33 greedy: Computing solution snapshot for mu = {B: 0.6364, R: 0.0001} ...
00:33 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.6364, R: 0.0001} ...
00:33 greedy: Extending basis with solution snapshot ...
00:33 |   gram_schmidt: Orthonormalizing vector 8 again
      
00:33 greedy: Reducing ...
00:33 |   CoerciveRBReductor: RB projection ...
00:33 |   CoerciveRBReductor: Assembling error estimator ...
00:33 |   |   ResidualReductor: Estimating residual range ...
00:33 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 8 ...
00:33 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:33 |   |   |   |   gram_schmidt: Removing vector 24 of norm 2.3435484491542996e-15
00:33 |   |   |   |   gram_schmidt: Orthonormalizing vector 25 again
00:33 |   |   |   |   gram_schmidt: Orthonormalizing vector 26 again
00:33 |   |   |   |   gram_schmidt: Orthonormalizing vector 27 again
00:33 |   |   ResidualReductor: Projecting residual operator ...
00:33 greedy: Estimating errors ...
00:36 greedy: Maximum error after 9 extensions: 6.281860618858695 (mu = {B: 0.0203, R: 0.0001})
00:36 greedy: Computing solution snapshot for mu = {B: 0.0203, R: 0.0001} ...
00:36 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0203, R: 0.0001} ...
00:37 greedy: Extending basis with solution snapshot ...
00:37 |   gram_schmidt: Orthonormalizing vector 9 again
      
00:37 greedy: Reducing ...
00:37 |   CoerciveRBReductor: RB projection ...
00:37 |   CoerciveRBReductor: Assembling error estimator ...
00:37 |   |   ResidualReductor: Estimating residual range ...
00:37 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 9 ...
00:37 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:37 |   |   |   |   gram_schmidt: Removing vector 27 of norm 2.986575844722279e-15
00:37 |   |   |   |   gram_schmidt: Orthonormalizing vector 29 again
00:37 |   |   |   |   gram_schmidt: Orthonormalizing vector 30 again
00:37 |   |   ResidualReductor: Projecting residual operator ...
00:37 greedy: Estimating errors ...
00:40 greedy: Maximum error after 10 extensions: 4.646201879884575 (mu = {B: 0.0001, R: 0.0203})
00:40 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.0203} ...
00:40 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.0203} ...
00:40 greedy: Extending basis with solution snapshot ...
00:40 |   gram_schmidt: Orthonormalizing vector 10 again
      
00:40 greedy: Reducing ...
00:40 |   CoerciveRBReductor: RB projection ...
00:40 |   CoerciveRBReductor: Assembling error estimator ...
00:40 |   |   ResidualReductor: Estimating residual range ...
00:40 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 10 ...
00:40 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:40 |   |   |   |   gram_schmidt: Removing vector 30 of norm 1.204864401430773e-15
00:40 |   |   |   |   gram_schmidt: Orthonormalizing vector 32 again
00:40 |   |   |   |   gram_schmidt: Orthonormalizing vector 33 again
00:40 |   |   ResidualReductor: Projecting residual operator ...
00:40 greedy: Estimating errors ...
00:43 greedy: Maximum error after 11 extensions: 1.0355654795488056 (mu = {B: 0.010199999999999999, R: 0.0001})
00:43 greedy: Computing solution snapshot for mu = {B: 0.010199999999999999, R: 0.0001} ...
00:43 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.010199999999999999, R: 0.0001} ...
00:43 greedy: Extending basis with solution snapshot ...
00:43 |   gram_schmidt: Orthonormalizing vector 11 again
      
00:43 greedy: Reducing ...
00:43 |   CoerciveRBReductor: RB projection ...
00:44 |   CoerciveRBReductor: Assembling error estimator ...
00:44 |   |   ResidualReductor: Estimating residual range ...
00:44 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 11 ...
00:44 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:44 |   |   |   |   gram_schmidt: Removing vector 33 of norm 5.647768467057802e-15
00:44 |   |   |   |   gram_schmidt: Orthonormalizing vector 35 again
00:44 |   |   |   |   gram_schmidt: Orthonormalizing vector 36 again
00:44 |   |   ResidualReductor: Projecting residual operator ...
00:44 greedy: Estimating errors ...
00:47 greedy: Maximum error after 12 extensions: 0.854208469990706 (mu = {B: 0.0001, R: 0.010199999999999999})
00:47 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.010199999999999999} ...
00:47 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.010199999999999999} ...
00:47 greedy: Extending basis with solution snapshot ...
00:47 |   gram_schmidt: Orthonormalizing vector 12 again
      
00:47 greedy: Reducing ...
00:47 |   CoerciveRBReductor: RB projection ...
00:47 |   CoerciveRBReductor: Assembling error estimator ...
00:47 |   |   ResidualReductor: Estimating residual range ...
00:47 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 12 ...
00:47 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:47 |   |   |   |   gram_schmidt: Removing vector 36 of norm 6.701795057817819e-15
00:47 |   |   |   |   gram_schmidt: Orthonormalizing vector 38 again
00:47 |   |   |   |   gram_schmidt: Orthonormalizing vector 39 again
00:47 |   |   ResidualReductor: Projecting residual operator ...
00:47 greedy: Estimating errors ...
00:51 greedy: Maximum error after 13 extensions: 0.5521868085655179 (mu = {B: 0.0001, R: 0.38389999999999996})
00:51 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.38389999999999996} ...
00:51 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.38389999999999996} ...
00:51 greedy: Extending basis with solution snapshot ...
00:51 |   gram_schmidt: Orthonormalizing vector 13 again
      
00:51 greedy: Reducing ...
00:51 |   CoerciveRBReductor: RB projection ...
00:51 |   CoerciveRBReductor: Assembling error estimator ...
00:51 |   |   ResidualReductor: Estimating residual range ...
00:51 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 13 ...
00:51 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:51 |   |   |   |   gram_schmidt: Removing vector 39 of norm 2.423114965037598e-14
00:51 |   |   |   |   gram_schmidt: Orthonormalizing vector 41 again
00:51 |   |   |   |   gram_schmidt: Orthonormalizing vector 42 again
00:51 |   |   ResidualReductor: Projecting residual operator ...
00:51 greedy: Estimating errors ...
00:55 greedy: Maximum error after 14 extensions: 0.4929653071445868 (mu = {B: 0.15159999999999998, R: 0.0001})
00:55 greedy: Computing solution snapshot for mu = {B: 0.15159999999999998, R: 0.0001} ...
00:55 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.15159999999999998, R: 0.0001} ...
00:55 greedy: Extending basis with solution snapshot ...
00:55 |   gram_schmidt: Orthonormalizing vector 14 again
      
00:55 greedy: Reducing ...
00:55 |   CoerciveRBReductor: RB projection ...
00:55 |   CoerciveRBReductor: Assembling error estimator ...
00:55 |   |   ResidualReductor: Estimating residual range ...
00:55 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 14 ...
00:55 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:55 |   |   |   |   gram_schmidt: Removing vector 42 of norm 2.0323787240387254e-14
00:55 |   |   |   |   gram_schmidt: Orthonormalizing vector 44 again
00:55 |   |   |   |   gram_schmidt: Orthonormalizing vector 45 again
00:55 |   |   ResidualReductor: Projecting residual operator ...
00:55 greedy: Estimating errors ...
00:59 greedy: Maximum error after 15 extensions: 0.2316359088110416 (mu = {B: 0.010199999999999999, R: 1.0})
00:59 greedy: Computing solution snapshot for mu = {B: 0.010199999999999999, R: 1.0} ...
00:59 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.010199999999999999, R: 1.0} ...
00:59 greedy: Extending basis with solution snapshot ...
00:59 |   gram_schmidt: Orthonormalizing vector 15 again
      
00:59 greedy: Reducing ...
00:59 |   CoerciveRBReductor: RB projection ...
00:59 |   CoerciveRBReductor: Assembling error estimator ...
00:59 |   |   ResidualReductor: Estimating residual range ...
00:59 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 15 ...
00:59 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
00:59 |   |   |   |   gram_schmidt: Removing vector 45 of norm 1.7420316366544852e-15
00:59 |   |   |   |   gram_schmidt: Orthonormalizing vector 47 again
00:59 |   |   |   |   gram_schmidt: Orthonormalizing vector 48 again
00:59 |   |   ResidualReductor: Projecting residual operator ...
00:59 greedy: Estimating errors ...
01:03 greedy: Maximum error after 16 extensions: 0.17317119859004293 (mu = {B: 1.0, R: 0.010199999999999999})
01:03 greedy: Computing solution snapshot for mu = {B: 1.0, R: 0.010199999999999999} ...
01:03 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1.0, R: 0.010199999999999999} ...
01:03 greedy: Extending basis with solution snapshot ...
01:03 |   gram_schmidt: Orthonormalizing vector 16 again
      
01:03 greedy: Reducing ...
01:03 |   CoerciveRBReductor: RB projection ...
01:03 |   CoerciveRBReductor: Assembling error estimator ...
01:03 |   |   ResidualReductor: Estimating residual range ...
01:03 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 16 ...
01:03 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:03 |   |   |   |   gram_schmidt: Removing vector 48 of norm 2.406556260774081e-15
01:03 |   |   |   |   gram_schmidt: Orthonormalizing vector 50 again
01:03 |   |   |   |   gram_schmidt: Orthonormalizing vector 51 again
01:03 |   |   ResidualReductor: Projecting residual operator ...
01:03 greedy: Estimating errors ...
01:06 greedy: Maximum error after 17 extensions: 0.13487844931103896 (mu = {B: 0.8484999999999999, R: 0.0001})
01:06 greedy: Computing solution snapshot for mu = {B: 0.8484999999999999, R: 0.0001} ...
01:06 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.8484999999999999, R: 0.0001} ...
01:06 greedy: Extending basis with solution snapshot ...
01:06 |   gram_schmidt: Orthonormalizing vector 17 again
      
01:07 greedy: Reducing ...
01:07 |   CoerciveRBReductor: RB projection ...
01:07 |   CoerciveRBReductor: Assembling error estimator ...
01:07 |   |   ResidualReductor: Estimating residual range ...
01:07 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 17 ...
01:07 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:07 |   |   |   |   gram_schmidt: Orthonormalizing vector 52 again
01:07 |   |   |   |   gram_schmidt: Orthonormalizing vector 53 again
01:07 |   |   |   |   gram_schmidt: Orthonormalizing vector 54 again
01:07 |   |   ResidualReductor: Projecting residual operator ...
01:07 greedy: Estimating errors ...
01:12 greedy: Maximum error after 18 extensions: 0.10537984401529059 (mu = {B: 0.0001, R: 0.1112})
01:12 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.1112} ...
01:12 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.1112} ...
01:12 greedy: Extending basis with solution snapshot ...
01:12 |   gram_schmidt: Orthonormalizing vector 18 again
      
01:12 greedy: Reducing ...
01:12 |   CoerciveRBReductor: RB projection ...
01:12 |   CoerciveRBReductor: Assembling error estimator ...
01:12 |   |   ResidualReductor: Estimating residual range ...
01:12 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 18 ...
01:12 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 57 again
01:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 58 again
01:12 |   |   ResidualReductor: Projecting residual operator ...
01:12 greedy: Estimating errors ...
01:17 greedy: Maximum error after 19 extensions: 0.04232345508114265 (mu = {B: 0.0001, R: 0.8484999999999999})
01:17 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.8484999999999999} ...
01:17 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.8484999999999999} ...
01:17 greedy: Extending basis with solution snapshot ...
01:18 |   gram_schmidt: Orthonormalizing vector 19 again
      
01:18 greedy: Reducing ...
01:18 |   CoerciveRBReductor: RB projection ...
01:18 |   CoerciveRBReductor: Assembling error estimator ...
01:18 |   |   ResidualReductor: Estimating residual range ...
01:18 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 19 ...
01:18 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:18 |   |   |   |   gram_schmidt: Orthonormalizing vector 60 again
01:18 |   |   |   |   gram_schmidt: Orthonormalizing vector 61 again
01:18 |   |   |   |   gram_schmidt: Orthonormalizing vector 62 again
01:18 |   |   ResidualReductor: Projecting residual operator ...
01:18 greedy: Estimating errors ...
01:23 greedy: Maximum error after 20 extensions: 0.036465300659745424 (mu = {B: 0.0405, R: 0.0001})
01:23 greedy: Computing solution snapshot for mu = {B: 0.0405, R: 0.0001} ...
01:23 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0405, R: 0.0001} ...
01:23 greedy: Extending basis with solution snapshot ...
01:23 |   gram_schmidt: Orthonormalizing vector 20 again
      
01:24 greedy: Reducing ...
01:24 |   CoerciveRBReductor: RB projection ...
01:24 |   CoerciveRBReductor: Assembling error estimator ...
01:24 |   |   ResidualReductor: Estimating residual range ...
01:24 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 20 ...
01:24 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:24 |   |   |   |   gram_schmidt: Orthonormalizing vector 65 again
01:24 |   |   |   |   gram_schmidt: Orthonormalizing vector 66 again
01:24 |   |   ResidualReductor: Projecting residual operator ...
01:24 greedy: Estimating errors ...
01:30 greedy: Maximum error after 21 extensions: 0.034676333520369265 (mu = {B: 1.0, R: 0.18189999999999998})
01:30 greedy: Computing solution snapshot for mu = {B: 1.0, R: 0.18189999999999998} ...
01:30 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1.0, R: 0.18189999999999998} ...
01:30 greedy: Extending basis with solution snapshot ...
01:30 |   gram_schmidt: Orthonormalizing vector 21 again
      
01:30 greedy: Reducing ...
01:30 |   CoerciveRBReductor: RB projection ...
01:30 |   CoerciveRBReductor: Assembling error estimator ...
01:30 |   |   ResidualReductor: Estimating residual range ...
01:30 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 21 ...
01:30 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:30 |   |   |   |   gram_schmidt: Orthonormalizing vector 67 again
01:30 |   |   |   |   gram_schmidt: Orthonormalizing vector 68 again
01:30 |   |   |   |   gram_schmidt: Orthonormalizing vector 69 again
01:30 |   |   |   |   gram_schmidt: Orthonormalizing vector 70 again
01:30 |   |   ResidualReductor: Projecting residual operator ...
01:30 greedy: Estimating errors ...
01:36 greedy: Maximum error after 22 extensions: 0.0211620887810872 (mu = {B: 1.0, R: 0.0405})
01:36 greedy: Computing solution snapshot for mu = {B: 1.0, R: 0.0405} ...
01:36 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1.0, R: 0.0405} ...
01:36 greedy: Extending basis with solution snapshot ...
01:36 |   gram_schmidt: Orthonormalizing vector 22 again
      
01:36 greedy: Reducing ...
01:36 |   CoerciveRBReductor: RB projection ...
01:36 |   CoerciveRBReductor: Assembling error estimator ...
01:36 |   |   ResidualReductor: Estimating residual range ...
01:36 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 22 ...
01:36 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:36 |   |   |   |   gram_schmidt: Removing vector 71 of norm 5.832003254848112e-14
01:36 |   |   |   |   gram_schmidt: Orthonormalizing vector 73 again
01:36 |   |   |   |   gram_schmidt: Orthonormalizing vector 74 again
01:36 |   |   ResidualReductor: Projecting residual operator ...
01:36 greedy: Estimating errors ...
01:42 greedy: Maximum error after 23 extensions: 0.014536586870023361 (mu = {B: 0.1011, R: 1.0})
01:42 greedy: Computing solution snapshot for mu = {B: 0.1011, R: 1.0} ...
01:42 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.1011, R: 1.0} ...
01:42 greedy: Extending basis with solution snapshot ...
01:42 |   gram_schmidt: Orthonormalizing vector 23 again
      
01:42 greedy: Reducing ...
01:42 |   CoerciveRBReductor: RB projection ...
01:42 |   CoerciveRBReductor: Assembling error estimator ...
01:42 |   |   ResidualReductor: Estimating residual range ...
01:42 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 23 ...
01:42 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:42 |   |   |   |   gram_schmidt: Orthonormalizing vector 74 again
01:42 |   |   |   |   gram_schmidt: Orthonormalizing vector 76 again
01:42 |   |   |   |   gram_schmidt: Orthonormalizing vector 77 again
01:42 |   |   ResidualReductor: Projecting residual operator ...
01:42 greedy: Estimating errors ...
01:48 greedy: Maximum error after 24 extensions: 0.013329432239524012 (mu = {B: 0.2829, R: 0.010199999999999999})
01:48 greedy: Computing solution snapshot for mu = {B: 0.2829, R: 0.010199999999999999} ...
01:48 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.2829, R: 0.010199999999999999} ...
01:48 greedy: Extending basis with solution snapshot ...
01:48 |   gram_schmidt: Orthonormalizing vector 24 again
      
01:48 greedy: Reducing ...
01:48 |   CoerciveRBReductor: RB projection ...
01:48 |   CoerciveRBReductor: Assembling error estimator ...
01:48 |   |   ResidualReductor: Estimating residual range ...
01:48 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 24 ...
01:48 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:48 |   |   |   |   gram_schmidt: Removing vector 78 of norm 8.229241952026786e-14
01:48 |   |   |   |   gram_schmidt: Orthonormalizing vector 81 again
01:48 |   |   ResidualReductor: Projecting residual operator ...
01:48 greedy: Estimating errors ...
01:53 greedy: Maximum error after 25 extensions: 0.008296328595144847 (mu = {B: 0.0001, R: 0.0304})
01:53 greedy: Computing solution snapshot for mu = {B: 0.0001, R: 0.0304} ...
01:53 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0001, R: 0.0304} ...
01:53 greedy: Extending basis with solution snapshot ...
01:53 |   gram_schmidt: Orthonormalizing vector 25 again
      
01:53 greedy: Reducing ...
01:53 |   CoerciveRBReductor: RB projection ...
01:54 |   CoerciveRBReductor: Assembling error estimator ...
01:54 |   |   ResidualReductor: Estimating residual range ...
01:54 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 25 ...
01:54 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
01:54 |   |   |   |   gram_schmidt: Orthonormalizing vector 83 again
01:54 |   |   |   |   gram_schmidt: Orthonormalizing vector 84 again
01:54 |   |   ResidualReductor: Projecting residual operator ...
01:54 greedy: Estimating errors ...
01:59 greedy: Maximum error after 26 extensions: 0.008246202641230882 (mu = {B: 0.42429999999999995, R: 0.0001})
01:59 greedy: Computing solution snapshot for mu = {B: 0.42429999999999995, R: 0.0001} ...
01:59 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.42429999999999995, R: 0.0001} ...
01:59 greedy: Extending basis with solution snapshot ...
01:59 |   gram_schmidt: Orthonormalizing vector 26 again
      
01:59 greedy: Reducing ...
01:59 |   CoerciveRBReductor: RB projection ...
01:59 |   CoerciveRBReductor: Assembling error estimator ...
01:59 |   |   ResidualReductor: Estimating residual range ...
01:59 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 26 ...
01:59 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:00 |   |   |   |   gram_schmidt: Orthonormalizing vector 87 again
02:00 |   |   |   |   gram_schmidt: Orthonormalizing vector 88 again
02:00 |   |   ResidualReductor: Projecting residual operator ...
02:00 greedy: Estimating errors ...
02:05 greedy: Maximum error after 27 extensions: 0.0058066040836533585 (mu = {B: 0.0203, R: 0.3132})
02:05 greedy: Computing solution snapshot for mu = {B: 0.0203, R: 0.3132} ...
02:05 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0203, R: 0.3132} ...
02:05 greedy: Extending basis with solution snapshot ...
02:05 |   gram_schmidt: Orthonormalizing vector 27 again
      
02:05 greedy: Reducing ...
02:05 |   CoerciveRBReductor: RB projection ...
02:05 |   CoerciveRBReductor: Assembling error estimator ...
02:05 |   |   ResidualReductor: Estimating residual range ...
02:05 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 27 ...
02:05 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:05 |   |   |   |   gram_schmidt: Orthonormalizing vector 89 again
02:05 |   |   |   |   gram_schmidt: Orthonormalizing vector 92 again
02:06 |   |   ResidualReductor: Projecting residual operator ...
02:06 greedy: Estimating errors ...
02:12 greedy: Maximum error after 28 extensions: 0.0016575362490125064 (mu = {B: 0.6162, R: 0.010199999999999999})
02:12 greedy: Computing solution snapshot for mu = {B: 0.6162, R: 0.010199999999999999} ...
02:12 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.6162, R: 0.010199999999999999} ...
02:12 greedy: Extending basis with solution snapshot ...
02:12 |   gram_schmidt: Orthonormalizing vector 28 again
      
02:12 greedy: Reducing ...
02:12 |   CoerciveRBReductor: RB projection ...
02:12 |   CoerciveRBReductor: Assembling error estimator ...
02:12 |   |   ResidualReductor: Estimating residual range ...
02:12 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 28 ...
02:12 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 93 again
02:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 94 again
02:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 95 again
02:12 |   |   |   |   gram_schmidt: Orthonormalizing vector 96 again
02:12 |   |   ResidualReductor: Projecting residual operator ...
02:12 greedy: Estimating errors ...
02:20 greedy: Maximum error after 29 extensions: 0.0015271039860315994 (mu = {B: 1.0, R: 0.8282999999999999})
02:20 greedy: Computing solution snapshot for mu = {B: 1.0, R: 0.8282999999999999} ...
02:20 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 1.0, R: 0.8282999999999999} ...
02:20 greedy: Extending basis with solution snapshot ...
02:20 |   gram_schmidt: Orthonormalizing vector 29 again
      
02:20 greedy: Reducing ...
02:20 |   CoerciveRBReductor: RB projection ...
02:20 |   CoerciveRBReductor: Assembling error estimator ...
02:20 |   |   ResidualReductor: Estimating residual range ...
02:20 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 29 ...
02:20 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:20 |   |   |   |   gram_schmidt: Orthonormalizing vector 97 again
02:20 |   |   |   |   gram_schmidt: Orthonormalizing vector 98 again
02:21 |   |   |   |   gram_schmidt: Orthonormalizing vector 99 again
02:21 |   |   |   |   gram_schmidt: Orthonormalizing vector 100 again
02:21 |   |   ResidualReductor: Projecting residual operator ...
02:21 greedy: Estimating errors ...
02:27 greedy: Maximum error after 30 extensions: 0.001579610532117633 (mu = {B: 0.0304, R: 1.0})
02:27 greedy: Computing solution snapshot for mu = {B: 0.0304, R: 1.0} ...
02:27 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0304, R: 1.0} ...
02:28 greedy: Extending basis with solution snapshot ...
02:28 |   gram_schmidt: Orthonormalizing vector 30 again
      
02:28 greedy: Reducing ...
02:28 |   CoerciveRBReductor: RB projection ...
02:28 |   CoerciveRBReductor: Assembling error estimator ...
02:28 |   |   ResidualReductor: Estimating residual range ...
02:28 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 30 ...
02:28 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:28 |   |   |   |   gram_schmidt: Orthonormalizing vector 101 again
02:28 |   |   |   |   gram_schmidt: Orthonormalizing vector 103 again
02:28 |   |   |   |   gram_schmidt: Orthonormalizing vector 104 again
02:28 |   |   ResidualReductor: Projecting residual operator ...
02:28 greedy: Estimating errors ...
02:35 greedy: Maximum error after 31 extensions: 0.0013002145648843286 (mu = {B: 0.0304, R: 0.0001})
02:35 greedy: Computing solution snapshot for mu = {B: 0.0304, R: 0.0001} ...
02:35 |   StationaryDiscretization: Solving Blatt4_Aufgabe_1_parametric_CG for {B: 0.0304, R: 0.0001} ...
02:35 greedy: Extending basis with solution snapshot ...
02:35 |   gram_schmidt: Orthonormalizing vector 31 again
      
02:35 greedy: Reducing ...
02:35 |   CoerciveRBReductor: RB projection ...
02:35 |   CoerciveRBReductor: Assembling error estimator ...
02:35 |   |   ResidualReductor: Estimating residual range ...
02:35 |   |   |   estimate_image_hierarchical: Estimating image for basis vector 31 ...
02:35 |   |   |   estimate_image_hierarchical: Orthonormalizing ...
02:35 |   |   |   |   gram_schmidt: Orthonormalizing vector 105 again
02:35 |   |   |   |   gram_schmidt: Orthonormalizing vector 107 again
02:35 |   |   |   |   gram_schmidt: Orthonormalizing vector 108 again
02:36 |   |   ResidualReductor: Projecting residual operator ...
02:36 greedy: Estimating errors ...
02:43 greedy: Maximum error after 32 extensions: 0.0008611643216361232 (mu = {B: 0.0001, R: 0.16169999999999998})
02:43 greedy: Absolute error tolerance (0.001) reached! Stoping extension loop.
02:43 greedy: Greedy search took 161.71575474739075 seconds

Stellen Sie den Fehlerabfall während der Basisgenerierung mit Hilfe der vom greedy Algorithmus zurück gegebenen Daten grafisch dar.

max_errs = greedy_data['max_errs']
plt.figure()
plt.semilogy(max_errs)
plt.xlabel('RB size')
plt.ylabel('max estimated error')
plt.title('Error decay (over training set) during greedy basis generation')

Text(0.5, 1.0, 'Error decay (over training set) during greedy basis generation')

Aufgabe 2: PDE-constrained minimization¶

Definieren Sie eine Funktione
```
def parameter_to_output(R, B):
    ...
```
welche mit Hilfe der reduzierten Diskretisierung und des reduzierten Ausgabefunktionals eine RB-Approximation von $J$ berechnet.
- Extrahieren Sie dazu die reduzierte Diskretisierung aus den vom greedy Algorithmus übergebenen Daten.

rd = greedy_data['rd']
J = rd.operators['output_functional']

def parameter_to_output(R, B):
    u = rd.solve({'R': R, 'B': B})
    return J.apply(u).data[0][0]

def plot_3d(x, y, f, xlabel='X', ylabel='Y'):
    # compute_value_matrix
    f_of_x = np.zeros((len(x), len(y)))
    import time
    t = time.time()
    for ii in range(len(x)):
        for jj in range(len(y)):
            f_of_x[jj][ii] = f(x[ii], y[jj]) # ii and jj are switched on purpose!
    x, y = np.meshgrid(x, y)
    print('{} evaluations took {}s'.format(len(x)*len(y), time.time() - t))
    # plot
    from mpl_toolkits.mplot3d import Axes3D # required for 3d plots
    from matplotlib import cm # required for colormaps
    fig = plt.figure()
    ax = fig.gca(projection='3d')
    ax.plot_surface(x, y, f_of_x, cmap=cm.viridis)
    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
    ax.invert_xaxis()
    return ax

def plot_point(ax, point, color):
    x, y = point[0], point[1]
    ax.plot([x, x], [y, y], ax.get_zlim(), color, zorder=10)

Erstellen Sie einen 3d Plot der RB-Approximation von $J$ mit Hilfe von plot_3d.
- Geben Sie als x-Werte 100 gleichverteilte Werte im Definitionsbereich der Parameterkomponente R an, entsprechend als y-Werte B. Nutzen Sie dazu den Parameterraum der reduzierten Diskretisierung.

ax = plot_3d(np.linspace(*rd.parameter_space.ranges['R'], 100),
             np.linspace(*rd.parameter_space.ranges['B'], 100),
             parameter_to_output, xlabel='R', ylabel='B')
xlocs, xlabels = plt.xticks()
ylocs, ylabels = plt.yticks()

10000 evaluations took 3.9125216007232666s

Machen Sie sich mit der minimize Funktion in scipy vertraut und vervollständigen Sie die Definition von find_minimum, um das Minimierungsproblem zu lösen.
- wählen Sie als Minimierer den BFGS Algorithmus für "Bound-Constrained minimization"
- lassen Sie den Gradienten von $J$ mit Hilfe von Finiten Differenzen Approximieren
- nutzen Sie den Paramterraum der reduzierten Diskretisierung um das "Bound-Constrained" zu definieren
- übergeben Sie options={'ftol': 1e-15}

def find_minimum(initial_guess):
    from scipy.optimize import minimize
    return minimize(lambda mu: parameter_to_output(mu[0], mu[1]),
                    initial_guess,
                    method='L-BFGS-B', jac=False,
                    bounds=(rd.parameter_space.ranges['R'], rd.parameter_space.ranges['B']),
                    options={'ftol': 1e-15})

def print_report(result):
    if result.success:
        print('succeded after {} iterations (required {} PDE solves)'.format(result.nit, result.nfev))
        print('J({}) = {}'.format(result.x, result.fun))
    else:
        print('failed!')

Lassen Sie die Minimierung für vier verschiedene Anfangswerte mit Hilfe des folgenden Codes druchlaufen. Was können Sie beobachten?

for mu, color in zip(rd.parameter_space.sample_uniformly(2), ('red', 'yellow', 'orange', 'purple')):
    initial_guess = [mu['R'], mu['B']]    
    print('minimizing from {}...'.format(initial_guess))
    result = find_minimum(initial_guess)
    print_report(result)
    plot_point(ax, result.x, color)
    print('')

minimizing from [array(0.0001), array(0.0001)]...
succeded after 7 iterations (required 24 PDE solves)
J([0.0001     0.07097346]) = 0.002849831289926982

minimizing from [array(1.), array(0.0001)]...
succeded after 9 iterations (required 39 PDE solves)
J([0.0001     0.07093573]) = 0.0028498314120881576

minimizing from [array(0.0001), array(1.)]...
succeded after 3 iterations (required 51 PDE solves)
J([0.0001     0.07163161]) = 0.0028498319014712424

minimizing from [array(1.), array(1.)]...
succeded after 10 iterations (required 69 PDE solves)
J([0.4280417  0.01167548]) = 0.0031462490223433167

Erstellen Sie einen 3d plot der RB-Approximation von $J$ wie oben, wobei die $x$ und $y$ Punkte nicht gleichverteilt, sonder mit Hilfe des Logarithmus zur Basis 10 verteilt sind.

ax = plot_3d(np.log10(np.linspace(*rd.parameter_space.ranges['R'], 100)),
             np.log10(np.linspace(*rd.parameter_space.ranges['B'], 100)),
             lambda x, y: parameter_to_output(10**x, 10**y), xlabel='R', ylabel='B')
xlocs, _ = plt.xticks()
ylocs, _ = plt.yticks()
plt.xticks(np.linspace(min(xlocs), max(xlocs), len(xlabels)), xlabels)
plt.yticks(np.linspace(min(ylocs), max(ylocs), len(ylabels)), ylabels)

10000 evaluations took 3.5173699855804443s

([<matplotlib.axis.XTick at 0x7f68492632b0>,
  <matplotlib.axis.XTick at 0x7f6849463a90>,
  <matplotlib.axis.XTick at 0x7f6849328da0>,
  <matplotlib.axis.XTick at 0x7f68496cd978>,
  <matplotlib.axis.XTick at 0x7f684926a2b0>,
  <matplotlib.axis.XTick at 0x7f684926a7b8>,
  <matplotlib.axis.XTick at 0x7f684926acc0>,
  <matplotlib.axis.XTick at 0x7f6843833208>],
 <a list of 8 Text yticklabel objects>)

Wiederholen Sie auch die Minimierung und grafische Darstellung der gefundenen Minima. Passen Sie letzteren entsprechend an.

for mu, color in zip(rd.parameter_space.sample_uniformly(2), ('red', 'yellow', 'orange', 'purple')):
    initial_guess = [mu['R'], mu['B']]    
    print('minimizing from {}...'.format(initial_guess))
    result = find_minimum(initial_guess)
    print_report(result)
    plot_point(ax, np.log10(result.x), color)
    print('')

minimizing from [array(0.0001), array(0.0001)]...
succeded after 7 iterations (required 24 PDE solves)
J([0.0001     0.07097346]) = 0.002849831289926982

minimizing from [array(1.), array(0.0001)]...
succeded after 9 iterations (required 39 PDE solves)
J([0.0001     0.07093573]) = 0.0028498314120881576

minimizing from [array(0.0001), array(1.)]...
succeded after 3 iterations (required 51 PDE solves)
J([0.0001     0.07163161]) = 0.0028498319014712424

minimizing from [array(1.), array(1.)]...
succeded after 10 iterations (required 69 PDE solves)
J([0.4280417  0.01167548]) = 0.0031462490223433167