Tutorial: Building a Reduced Basis¶

In this tutorial we will learn more about VectorArrays and how to construct a reduced basis using pyMOR.

A reduced basis spans a low dimensional subspace of a Model’s solution_space, in which the solutions of the Model can be well approximated for all parameter values. In this context, time is treated as an additional parameter. So for time-dependent problems, the reduced space (the span of the reduced basis) should approximate the solution for all parameter values and time instances.

An upper bound for the possible quality of a reduced space is given by the so-called Kolmogorov $N$ -width $d_{N}$ given as

\begin{array}{r} d_{N} := inf_{\begin{array}{c} V_{N} \subseteq V \\ \dim (V_{N}) \leq N \end{array}} sup_{μ \in P} inf_{v \in V_{N}} ‖ u (μ) - v ‖ . \end{array}

In this formula $V$ denotes the solution_space, $P \subset R^{p}$ denotes the set of all parameter values we are interested in, and $u (μ)$ denotes the solution of the Model for the given parameter values $μ$ . In pyMOR the set $P$ is called the ParameterSpace.

How to read this formula? For each candidate reduced space $V_{N}$ we look at all possible parameter values $μ$ and compute the best-approximation error in $V_{N}$ (the second infimum). The supremum over the infimum is thus the worst-case best-approximation error over all parameter values of interest. Now we take the infimum of the worst-case best-approximation errors over all possible reduced spaces of dimension at most $N$ , and this is $d_{N}$ .

So whatever reduced space of dimension $N$ we pick, we will always find a $μ$ for which the best-approximation error in our space is at least $d_{N}$ . Reduced basis methods aim at constructing spaces $V_{N}$ for which the worst-case best-approximation error is as close to $d_{N}$ as possible.

However, we will only find a good $V_{N}$ of small dimension $N$ if the values $d_{N}$ decrease quickly for growing $N$ . It can be shown that this is the case as soon as $u (μ)$ analytically depends on $μ$ , which is true for many problems of interest. More precisely, it can be shown [BCD+11], [RD13] that there are constants $C, c > 0$ such that

d_{N} \leq C \cdot e^{- N^{c}} .

In this tutorial we will construct reduced spaces $V_{N}$ for a concrete problem with pyMOR and study their error decay.

Model setup¶

First we need to define a Model and a ParameterSpace for which we want to build a reduced basis. We choose here the standard thermal block benchmark problem shipped with pyMOR (see Getting started). However, any pyMOR Model can be used, except for Section Weak Greedy Algorithm where some more assumptions have to be made on the Model.

First we import everything we need:

import numpy as np
from pymor.basic import *

Then we build a 3-by-3 thermalblock problem that we discretize using pyMOR’s builtin discretizers (see Tutorial: Using pyMOR’s discretization toolkit for an introduction to pyMOR’s discretization toolkit).

problem = thermal_block_problem((3,3))
fom, _ = discretize_stationary_cg(problem, diameter=1/100)

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