# Reduced Basis Method

**General Philosophy **

It is worth to spend high computational costs to get ''good approximation'' subspaces. These subspaces are built hierarchically in a greedy manner until we satisfy some tolerance. At the end we have an approximation subspace,

where are called snapshots and solves (2) for . The are chosen using the Greedy Algorithm. If we want the solution to (2) for any we employ a Galerkin Projection onto ,

**Three Key Ingredients**

- Training Set
- Greedy Algorithm
- A posteriori error bound

**Training Set**

In our project we always start off by picking a training set , consisting of points from parameter space . We run the greedy algorithm on this set. The training set needs to be easy to compute without too many useless samples in order to avoid unnecessary computation but on the other hand it must be sufficient to capture most representative snapshots.

**Greedy Algorithm**

Suppose we are given a training set , a sample set and a reduced basis space . We seek to pick and build nested reduced bases spaces in a greedy manner by solving the following optimization problem: For find,

where is the a posteriori error bound (see below). We then add to the sample to get and augment our basis space to get . We finally orthonormalize the 's in , where .

**A posteriori error bound**

Let be the true error. Then we have the following bound,

where is the lower bound for the coercivity constant and is given by the formula,

where,

and and are such that and , for all , and are the orthonormalised snapshots.