From bf05bd27ffabafd2150eba37eeef8ff720c23640 Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 23 Apr 2009 16:39:28 +0000 Subject: [PATCH] Explain the mean value filter in a bit more detail. git-svn-id: https://svn.dealii.org/trunk@18706 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-34/step-34.cc | 64 +++++++++++++++++++++++------ 1 file changed, 51 insertions(+), 13 deletions(-) diff --git a/deal.II/examples/step-34/step-34.cc b/deal.II/examples/step-34/step-34.cc index 5a81e76a48..0c7920b56b 100644 --- a/deal.II/examples/step-34/step-34.cc +++ b/deal.II/examples/step-34/step-34.cc @@ -1328,19 +1328,57 @@ void BEMProblem::assemble_system() // address this issue, we use two new // instruments of the library: the // MeanValueFilter class, and the - // ProductMatrix class. The - // MeanValueFilter has the interface - // of a matrix (i.e. it has a - // function MeanValueFilter::vmult), - // with the effect that the output - // vector equals the input vector - // minus its mean value. We cascade - // this operator with the system - // matrix, and we obtain a matrix - // whose result is renormalized to a - // zero mean value Vector. In other - // words, vectors that are multiplied - // have mean value zero and therefore + // ProductMatrix class. + // + // In essence, the idea is this: all + // Krylov subspace solvers construct + // an approximation the solution in + // the space $\text{span} + // \{b,Ab,A^2b,A^3b,\ldots,A^{n-1}b\}$ + // in the $n$-th iteration. We would + // like the vectors in this space to + // have mean value zero. To guarantee + // this sort of thing, we should + // instead consider the problem + // $FAx=Fb$ where $F=I-\frac 1n + // \mathbf{e}\mathbf{e}^T$ (with + // $\mathbf e$ a vector of length $n$ + // with all entries equal to + // one). $F$ is the matrix that given + // a vector filters out its mean + // value. The Krylov subspace that + // GMRES constructs from this is + // $\text{span} + // \{Fb,FAb,FA^2b,FA^3b,\ldots,FA^{n-1}b\}$ + // (note here that $(FA)^k=FA^k$ + // because $A$ maps any vector $t$ to + // exactly the same result as it + // would map $Ft$ - that's the + // definition of its kernel!). So + // each of the elements of Krylov + // subspace has mean value zero, and + // as a consequence so does the + // approximation $x^{(n)}$ + // constructed in the $n$-th + // iteration. + // + // To implement this, we need a class + // that represents the action of the + // filter $F$. Sure enough, deal.II + // has one of these: the + // MeanValueFilter class has the + // interface of a matrix (i.e. it has + // a function + // MeanValueFilter::vmult), with the + // effect that the output vector + // equals the input vector minus its + // mean value. We cascade this + // operator with the system matrix, + // and we obtain a matrix $FA$ whose + // result is renormalized to a zero + // mean value vector. In other words, + // vectors that are multiplied have + // mean value zero and therefore // never feel the fact that the // system matrix has a kernel for // these. The combined matrix object -- 2.39.5