Generalized minimal residual method

Denote the Euclidean norm of any vector v by ${\displaystyle \|v\|}$. Denote the (square) system of linear equations to be solved by

$${\displaystyle Ax=b.}$$

The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., that ${\displaystyle \|b\|=1}$.

The n-th Krylov subspace for this problem is

$${\displaystyle K_{n}=K_{n}(A,r_{0})=\operatorname {span} \,\{r_{0},Ar_{0},A^{2}r_{0},\ldots ,A^{n-1}r_{0}\}.\,}$$

where ${\displaystyle r_{0}=b-Ax_{0}}$ is the initial error given an initial guess ${\displaystyle x_{0}\neq 0}$. Clearly ${\displaystyle r_{0}=b}$ if ${\displaystyle x_{0}=0}$.

GMRES approximates the exact solution of ${\displaystyle Ax=b}$ by the vector ${\displaystyle x_{n}\in K_{n}}$ that minimizes the Euclidean norm of the residual ${\displaystyle r_{n}=b-Ax_{n}}$.

The vectors ${\displaystyle r_{0},Ar_{0},\ldots A^{n-1}r_{0}}$ might be close to linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors ${\displaystyle q_{1},q_{2},\ldots ,q_{n}\,}$ which form a basis for ${\displaystyle K_{n}}$. In particular, ${\displaystyle q_{1}=\|r_{0}\|_{2}^{-1}r_{0}}$.

Therefore, the vector ${\displaystyle x_{n}\in K_{n}}$ can be written as ${\displaystyle x_{n}=x_{0}+Q_{n}y_{n}}$ with ${\displaystyle y_{n}\in \mathbb {R} ^{n}}$, where ${\displaystyle Q_{n}}$ is the m-by-n matrix formed by ${\displaystyle q_{1},\ldots ,q_{n}}$. In other words, finding the n-th approximation of the solution (i.e., ${\displaystyle x_{n}}$) is reduced to finding the vector ${\displaystyle y_{n}}$, which is determined via minimizing the residue as described below.

The Arnoldi process also constructs ${\displaystyle {\tilde {H}}_{n}}$, an (${\displaystyle n+1}$)-by-${\displaystyle n}$ upper Hessenberg matrix which satisfies

$${\displaystyle AQ_{n}=Q_{n+1}{\tilde {H}}_{n}\,}$$

an equality which is used to simplify the calculation of ${\displaystyle y_{n}}$ (see below). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method.

Because columns of ${\displaystyle Q_{n}}$ are orthonormal, we have

$${\displaystyle {\begin{aligned}\|r_{n}\|&=\|b-Ax_{n}\|\\&=\|b-A(x_{0}+Q_{n}y_{n})\|\\&=\|r_{0}-AQ_{n}y_{n}\|\\&=\|\beta q_{1}-AQ_{n}y_{n}\|\\&=\|\beta q_{1}-Q_{n+1}{\tilde {H}}_{n}y_{n}\|\\&=\|Q_{n+1}(\beta e_{1}-{\tilde {H}}_{n}y_{n})\|\\&=\|\beta e_{1}-{\tilde {H}}_{n}y_{n}\|\end{aligned}}}$$

where

$${\displaystyle e_{1}=(1,0,0,\ldots ,0)^{T}\,}$$

is the first vector in the standard basis of ${\displaystyle \mathbb {R} ^{n+1}}$, and

$${\displaystyle \beta =\|r_{0}\|\,,}$$

${\displaystyle r_{0}}$ being the first trial vector (usually zero). Hence, ${\displaystyle x_{n}}$ can be found by minimizing the Euclidean norm of the residual

$${\displaystyle r_{n}={\tilde {H}}_{n}y_{n}-\beta e_{1}.}$$

This is a linear least squares problem of size n.

This yields the GMRES method. On the ${\displaystyle n}$-th iteration:

1. calculate ${\displaystyle q_{n}}$ with the Arnoldi method;
2. find the ${\displaystyle y_{n}}$ which minimizes ${\displaystyle \|r_{n}\|}$;
3. compute ${\displaystyle x_{n}=x_{0}+Q_{n}y_{n}}$;
4. repeat if the residual is not yet small enough.

At every iteration, a matrix-vector product ${\displaystyle Aq_{n}}$ must be computed. This costs about ${\displaystyle 2m^{2}}$ floating-point operations for general dense matrices of size ${\displaystyle m}$, but the cost can decrease to ${\displaystyle O(m)}$ for sparse matrices. In addition to the matrix-vector product, ${\displaystyle O(nm)}$ floating-point operations must be computed at the n -th iteration.

The nth iterate minimizes the residual in the Krylov subspace ${\displaystyle K_{n}}$. Since every subspace is contained in the next subspace, the residual does not increase. After m iterations, where m is the size of the matrix A, the Krylov space K_m is the whole of R^m and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to m), the vector x_n is already a good approximation to the exact solution.

This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence a₁, ..., a_m−1, a_m = 0, one can find a matrix A such that the ‖r_n‖ = a_n for all n, where r_n is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for m − 1 iterations, and only drops to zero at the last iteration.

In practice, though, GMRES often performs well. This can be proven in specific situations. If the symmetric part of A, that is ${\displaystyle (A^{T}+A)/2}$, is positive definite, then

$${\displaystyle \|r_{n}\|\leq \left(1-{\frac {\lambda _{\min }^{2}(1/2(A^{T}+A))}{\lambda _{\max }(A^{T}A)}}\right)^{n/2}\|r_{0}\|,}$$

where ${\displaystyle \lambda _{\mathrm {min} }(M)}$ and ${\displaystyle \lambda _{\mathrm {max} }(M)}$ denote the smallest and largest eigenvalue of the matrix ${\displaystyle M}$, respectively.[4]

If A is symmetric and positive definite, then we even have

$${\displaystyle \|r_{n}\|\leq \left({\frac {\kappa _{2}(A)^{2}-1}{\kappa _{2}(A)^{2}}}\right)^{n/2}\|r_{0}\|.}$$

where ${\displaystyle \kappa _{2}(A)}$ denotes the condition number of A in the Euclidean norm.

In the general case, where A is not positive definite, we have

$${\displaystyle {\frac {\|r_{n}\|}{\|b\|}}\leq \inf _{p\in P_{n}}\|p(A)\|\leq \kappa _{2}(V)\inf _{p\in P_{n}}\max _{\lambda \in \sigma (A)}|p(\lambda )|,\,}$$

where P_n denotes the set of polynomials of degree at most n with p(0) = 1, V is the matrix appearing in the spectral decomposition of A, and σ(A) is the spectrum of A. Roughly speaking, this says that fast convergence occurs when the eigenvalues of A are clustered away from the origin and A is not too far from normality.[5]

All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate x_n and the exact solution.

Like other iterative methods, GMRES is usually combined with a preconditioning method in order to speed up convergence.

The cost of the iterations grow as O(n²), where n is the iteration number. Therefore, the method is sometimes restarted after a number, say k, of iterations, with x_k as initial guess. The resulting method is called GMRES(k) or Restarted GMRES. For non-positive definite matrices, this method may suffer from stagnation in convergence as the restarted subspace is often close to the earlier subspace.

The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR.[6] Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.[7]

The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.

Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed.

The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynomials of the iteration sequence suitably.

None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.

One part of the GMRES method is to find the vector ${\displaystyle y_{n}}$ which minimizes

$${\displaystyle \left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|.}$$

Note that ${\displaystyle {\tilde {H}}_{n}}$ is an (n + 1)-by-n matrix, hence it gives an over-constrained linear system of n+1 equations for n unknowns.

The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix Ω_n and an (n + 1)-by-n upper triangular matrix ${\displaystyle {\tilde {R}}_{n}}$ such that

$${\displaystyle \Omega _{n}{\tilde {H}}_{n}={\tilde {R}}_{n}.}$$

The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as

$${\displaystyle {\tilde {R}}_{n}={\begin{bmatrix}R_{n}\\0\end{bmatrix}},}$$

where ${\displaystyle R_{n}}$ is an n-by-n (thus square) triangular matrix.

The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:

$${\displaystyle {\tilde {H}}_{n+1}={\begin{bmatrix}{\tilde {H}}_{n}&h_{n+1}\\0&h_{n+2,n+1}\end{bmatrix}},}$$

where h_n+1 = (h_1,n+1, …, h_n+1,n+1)^T. This implies that premultiplying the Hessenberg matrix with Ω_n, augmented with zeroes and a row with multiplicative identity, yields almost a triangular matrix:

$${\displaystyle {\begin{bmatrix}\Omega _{n}&0\\0&1\end{bmatrix}}{\tilde {H}}_{n+1}={\begin{bmatrix}R_{n}&r_{n+1}\\0&\rho \\0&\sigma \end{bmatrix}}}$$

This would be triangular if σ is zero. To remedy this, one needs the Givens rotation

$${\displaystyle G_{n}={\begin{bmatrix}I_{n}&0&0\\0&c_{n}&s_{n}\\0&-s_{n}&c_{n}\end{bmatrix}}}$$

where

$${\displaystyle c_{n}={\frac {\rho }{\sqrt {\rho ^{2}+\sigma ^{2}}}}\quad {\text{and}}\quad s_{n}={\frac {\sigma }{\sqrt {\rho ^{2}+\sigma ^{2}}}}.}$$

With this Givens rotation, we form

$${\displaystyle \Omega _{n+1}=G_{n}{\begin{bmatrix}\Omega _{n}&0\\0&1\end{bmatrix}}.}$$

Indeed,

$${\displaystyle \Omega _{n+1}{\tilde {H}}_{n+1}={\begin{bmatrix}R_{n}&r_{n+1}\\0&r_{n+1,n+1}\\0&0\end{bmatrix}}}$$

is a triangular matrix with ${\displaystyle r_{n+1,n+1}={\sqrt {\rho ^{2}+\sigma ^{2}}}}$.

Given the QR decomposition, the minimization problem is easily solved by noting that

$${\displaystyle {\begin{aligned}\left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|&=\left\|\Omega _{n}({\tilde {H}}_{n}y_{n}-\beta e_{1})\right\|\\&=\left\|{\tilde {R}}_{n}y_{n}-\beta \Omega _{n}e_{1}\right\|.\end{aligned}}}$$

Denoting the vector ${\displaystyle \beta \Omega _{n}e_{1}}$ by

$${\displaystyle {\tilde {g}}_{n}={\begin{bmatrix}g_{n}\\\gamma _{n}\end{bmatrix}}}$$

with g_n ∈ Rⁿ and γ_n ∈ R, this is

$${\displaystyle {\begin{aligned}\left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|&=\left\|{\tilde {R}}_{n}y_{n}-\beta \Omega _{n}e_{1}\right\|\\&=\left\|{\begin{bmatrix}R_{n}\\0\end{bmatrix}}y_{n}-{\begin{bmatrix}g_{n}\\\gamma _{n}\end{bmatrix}}\right\|.\end{aligned}}}$$

The vector y that minimizes this expression is given by

$${\displaystyle y_{n}=R_{n}^{-1}g_{n}.}$$

Again, the vectors ${\displaystyle g_{n}}$ are easy to update.[8]

• Biconjugate gradient method

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• Amritkar, Amit; de Sturler, Eric; Świrydowicz, Katarzyna; Tafti, Danesh; Ahuja, Kapil (2015). "Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver". Journal of Computational Physics 303: 222. doi:10.1016/j.jcp.2015.09.040
• Imankulov T, Lebedev D, Matkerim B, Daribayev B, Kassymbek N. Numerical Simulation of Multiphase Multicomponent Flow in Porous Media: Efficiency Analysis of Newton-Based Method. Fluids. 2021; 6(10):355. https://doi.org/10.3390/fluids6100355