In this seminar, we study Reduced Basis Methods (RBMs) for the efficient solution of parametrized PDEs.
The main idea of the RBM is the following: Consider the solution u(p) of the parameter dependent differential equation L(p) u(p) = f(p). If one is interested in u(p) for a particular value of p, one can simply assemble f(p), L(p) and solve the system. In most cases, this will be a high dimensional problem, so the cost for solution maybe high, say O(N). Efficient algorithms for doing have been discussed in the lectures "Numerical methods for elliptic problems, for instance.If one is interested in solving u(p) for many values of p, one can of course call the same solver multiple times. If M evaluations are required, then the overall complexity is O(M*N). If M is very large, this may becom prohibitive.
The goal of the RBM is to drastically reduce the computational complexity of the evaluation of the solution u(p) for many p. This is typicall done in the following way: In a first stage, one computes snapshots ui = u(pi), i=1,...,n. The space spanned by the snapshots can be further reduced by algebraic manipulations. Then the system L(p) u(p) = f(p) is projected to the reduced space Vm spanned by the snapshots or the further reduced basis. These computations might be expensive, say O(n*N + ...) and are tpically done in an offline stage. In the online stage, one only has to deal with the reduced problem which lives in the dimension of the reduced space Vn. The complexity of solving Ln(p) un(p) = fn(p) will be O(n^3) at most, so the evaluation for many parameters p can be reduced to O(m*n^3) at most. In typicall applications, n~10-100, so this can eventually be done very efficiently using highly optimized dense linear algebra routines.
You will find links to further material in these references.