### Extrap: added connection to Krylov methods

parent ea01f34d
 ... ... @@ -531,4 +531,35 @@ Anderson mixing solves $F_{n,k} \Delta \vec{u} = \fxi{n+k}$ in a least-squares s \end{equation*} For $\beta=0$ this is exactly the multisecant equations. \section{Connection to Krylov methods} Recall that the extrapolation methods may be expressed as \begin{equation*} \begin{bmatrix} 1^\top \\ B^\top F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \hat{\vec{x}} = X_{n,k} \vec{u} \end{equation*} where $B = F_{n,k-1}$ for MPE, $B = F_{n+1,k} - F_{n,k-1} = F_{n,k} \Delta$ for RRE, and $B = \begin{bmatrix} \vec{q}_1 & \dots & \vec{q}_k \end{bmatrix}$ for MMPE. TEA cannot be expressed in this form in general. If the columns of $F_{n,k}$ form a Krylov subspace, that is $\fxi{n+1} = A \fxi{n}$ for all $n$, then these methods are Krylov methods. For example, MPE now requires that $F_{n,k} \vec{u}$ is orthogonal to all $\fxi{n+i}$ for $i < k$. Therefore, the residual $F_{n,k} \vec{u}$ is orthogonal to the Krylov subspace $\mathcal{K}_{k-1}(A,\fxi{n})$. This is equivalent then to GMRES (see Sidi and Walter \& Ni for details). TEA may now be expressed as the form above with $B = \begin{bmatrix} \vec{q} & A^\top \vec{q} & \dots & (A^\top)^{k-1} \vec{q} \end{bmatrix}$. Table \ref{tab:KrylovExtrap} gives the orthogonalization conditions and corresponding Krylov methods for these four extrapolation methods. \begin{table} \begin{tabular}{c | c | c} Extrapolation method & Residual is orthogonal to... & Associated Krylov method \\ \hline MPE & $\mathcal{K}_{k-1} (A,\fxi{n})$ (Arnoldi) & GMRES \\ RRE & $\mathcal{K}_{k-1} (A-I,\fxi{n})$ & generalized conjugate residual \\ MMPE & $\mathcal{K}_{k-1} (G,\vec{q}_0)$ & n/a \\ TEA & $\mathcal{K}_{k-1} (A^\top, \vec{q})$ (Lanczos) & BiCG \end{tabular} \caption{Connections between extrapolation methods and Krylov methods.} \label{tab:KrylovExtrap} \end{table} The methods become Krylov methods most readily when the sequence to be accelerated may be expressed as $\vec{x}_{n+1} = A \vec{x}_n + \vec{b}$. \end{document} \ No newline at end of file
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