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pyMOR
pymor
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a040d4b1
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a040d4b1
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Dec 08, 2020
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Tim Keil
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[docs/tutorials] last comments from stephan
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@@ -183,7 +183,7 @@ to resolve the data structure in the diffusion. This suggests to use an
even finer mesh. However, for enabling a faster runtime for this
tutorial, we stick with this mesh and remark that refining the mesh does
not change the interpretation of the methods that are discussed below.
It rather fur
h
ter improves the speedups achieved by model reduction.
It rather furt
h
er improves the speedups achieved by model reduction.
Before we discuss the first optimization method, we define helpful
functions for visualizations.
...
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@@ -360,7 +360,7 @@ a perfect surrogate model in the sense that a low error tolerance for
the :func:`~pymor.algorithms.greedy.rb_greedy` already suffices to converge
to the same minimum.
In our case we choose ``atol=1e-2`` and yield a very low dimensional space.
In general, however, it is not a priori
ly
clear how to choose ``atol``
In general, however, it is not a priori clear how to choose ``atol``
in order to arrive at a minimum which is close enough to the true
optimum.
...
...
@@ -484,7 +484,7 @@ equation, i.e.
.. math::
r_\mu^{\text{pr}(u)[v] := l_\mu(v) - a_\mu(u, v) &&\text{for all }v \in V
r_\mu^{\text{pr}
}
(u)[v] := l_\mu(v) - a_\mu(u, v) &&\text{for all }v \in V
A major issue of this approach is that the computation of the
full gradient requires :math:`P` solutions of :math:`\eqref{sens}`.
...
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@@ -603,7 +603,7 @@ output functional.
report(opt_rom_result, opt_rom_minimization_data, reference_mu)
The online phase is even slightly faster than before but the offline
phase is
obviously
still the same as before. We also conclude that the
phase is still the same as before. We also conclude that the
ROM model eventually gives less speedup by using a better optimization
method for the FOM and ROM.
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@@ -697,8 +697,9 @@ for the gradients since we compute the dual solutions with the ROM.
Adaptively enriching along the path
-----------------------------------
This gives rise to another idea where we only enrich if it is
necessary. For example it could be the case that the model is already good at
In order to further speedup the above algorithm, we enhance it
by only adaptive enrichments of the model.
For instance it may happen that the model is already good at
the next iteration, which we can easily check by evaluating the standard
error estimator which is also used in the greedy algorithm. In the next
example we will implement this adaptive way of enriching and set a
...
...
@@ -821,21 +822,17 @@ the traditional offline/online splitting by only enriching the model along
the path of optimization or (even better) only enrich
the model if the standard error estimator goes above a certain tolerance.
A main drawback of the content in this tutorial was that the choice of
the tolerance ``atol`` that has been used to build the RB spaces
cannot be known a priorily. This shows the need for
certified and robust reduced methods.
For some standard literature for faster and robust optimization tools we refer to
`CGT00 <https://epubs.siam.org/doi/book/10.1137/1.9780898719857?mobileUi=0>`__ and
In this tutorial we have only covered a few basic approaches to combine model
reduction with optimization.
For faster and more robust optimization algorithms we refer to the textbooks
`CGT00 <https://epubs.siam.org/doi/book/10.1137/1.9780898719857>`__ and
`NW06 <https://link.springer.com/book/10.1007/978-0-387-40065-5>`__.
For recent research on using trust-region methods for MOR of PDE-constrained
optimization problems, we refer to
For recent research on combining trust-region methods with model reduction for
PDE-constrained optimization problems we refer to
`YM13 <https://epubs.siam.org/doi/abs/10.1137/120869171>`__,
`QGVW17 <https://epubs.siam.org/doi/abs/10.1137/16M1081981>`__ and
`KMSOV <https://arxiv.org/abs/2006.09297>`__ where for the latter
,
pyMOR
has been used for the numerical experiments
.
`KMSOV
20
<https://arxiv.org/abs/2006.09297>`__ where for the latter
a
pyMOR
implementation is available as supplementary material
.
Download the code:
:jupyter-download:script:`tutorial_optimization`
...
...
René Fritze
@r_milk01
mentioned in commit
fdc00a9d
·
Dec 08, 2020
mentioned in commit
fdc00a9d
mentioned in commit fdc00a9d3592c808795d12802de5c92ce20d40bb
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