On Optimal Algebraic Multigrid Methods

TL;DR

This paper introduces a new spectral characterization approach to derive optimal interpolation operators for algebraic multigrid methods applied to Hermitian positive definite systems, improving efficiency and robustness.

Contribution

It presents a spectral-based method for obtaining optimal interpolation operators, offering a simpler and more general approach compared to previous $A$-norm based techniques.

Findings

Optimal interpolation operators are derived using spectral characterization.

Operators are optimal with respect to $A$-norm, spectral radius, and condition number.

The method applies to both symmetric and non-symmetric two-grid methods.

Abstract

In this note we present an alternative way to obtain optimal interpolation operators for two-grid methods applied to Hermitian positive definite linear systems. Falgout and Vassilevski in [SIAM J. Numer. Anal, 42 (2004), pp. 1669-1693] and Zikatanov [Numer. Linear Algebra Appl., 15 (2008), pp. 439-454] have characterized the $A$-norm of the error propagation operator of algebraic multigrid methods. These results have been recently used by Xu and Zikatanov [Acta Numer., 26 (2017), pp. 591-721] and Brannick, Cao et al. [SIAM J. Sci. Comp, 40 (2018), pp. 591-721] to determine optimal interpolation operators. Here we use a characterization not of the $A$-norm but of the spectrum of the error propagation operator of two-grid methods, which was proved by Garc\'ia Ramos, Nabben and Kehl and holds for arbitrary matrices. For Hermitian positive definite systems this result leads to optimal interpolation operators with respect to the $A$-norm in a short way, moreover, it also leads to optimal interpolation operators with respect to the spectral radius. For the symmetric two-grid method (with pre- and post-smoothing) the optimal interpolation operators are the same. But for a two-grid method with only post-smoothing the optimal interpolations (and hence the optimal algebraic multigrid methods) can be different. Moreover, using the characterization of the spectrum, we can show that the found optimal interpolation operators are also optimal with respect to the condition number of the multigrid preconditioned system.

Equations

$$Ax=b,$$

$$A_C:=RAP\in {\mathbb{C}}^{r\times r}.$$

$$E_M=(I-M_2^{-1}A{)}^{{\nu }_2}(I-P{A}_C^{-1}RA)(I-M_1^{-1}A{)}^{{\nu }_1},$$

$$\rho (E_M)\le \Vert E_M\Vert .$$

$$(I-X^{-1}A)=(I-M_1^{-1}A{)}^{{\nu }_1}(I-M_2^{-1}A{)}^{{\nu }_2},$$

$$E_M=I-BA,$$

$$\sigma (BA)=\{1\}\cup \sigma ({\tilde{P}}^H{X}^{-1}\tilde{R}({\tilde{P}}^H{A}^{-1}\tilde{R}{)}^{-1}).$$

$$\Vert v{\Vert }_A^2=(v,v{)}_A=\Vert A^{\frac{1}{2}}v{\Vert }_2^2,$$

$$\Vert S{\Vert }_A=\Vert A^{\frac{1}{2}}S{A}^{-\frac{1}{2}}{\Vert }_2.$$

$$E_{TG}=(I-M^{-H}A)(I-P{A}_C^{-1}{P}^{H}A)$$

$$E_{STG}=(I-M^{-H}A)(I-P{A}_C^{-1}{P}^{H}A)(I-M^{-1}A).$$

$$\Vert I-M^{-1}A{\Vert }_A<1,$$

$$M+M^H-A\text{mbox}ispositivedefinite,$$

$$\Vert E_{STG}{\Vert }_A=\Vert E_{TG}{\Vert }_A^2,$$

$$\Vert E_{TG}{\Vert }_A^2=1-\frac{1}{K(V_c)},$$

$$K(V_c)=\underset{v\in {\mathbb{C}}^n}{\sup }\frac{\Vert (I-Q)v{\Vert }_{\tilde{M}}^2}{\Vert v{\Vert }_A^2}.$$

$$\sigma (BA)=\{1\}\cup \sigma ({\tilde{U}}^H{X}^{-1}\tilde{U}({\tilde{U}}^H{A}^{-1}\tilde{U}{)}^{-1}).$$

$${\tilde{U}}^{\star }\in \underset{\tilde{U}\in {\mathbb{C}}^{n\times n-r},{\tilde{U}}^H\tilde{U}=I}{\mathrm{argmax}}{\lambda }_{\min }({\tilde{U}}^H{X}^{-1}\tilde{U}({\tilde{U}}^H{A}^{-1}\tilde{U}{)}^{-1}),$$

$$X^{-1}w=\mu A^{-1}w,$$

$$0<{\mu }_1\le {\mu }_2\le \dots \le {\mu }_n.$$

$$\underset{\tilde{U}\in {\mathbb{C}}^{n\times n-r},{\tilde{U}}^H\tilde{U}=I}{\max }{\lambda }_{\min }({\tilde{U}}^H{X}^{-1}\tilde{U}({\tilde{U}}^H{A}^{-1}\tilde{U}{)}^{-1})={\mu }_{r+1}$$

$$\tilde{W}=[{\tilde{w}}_{r+1},\dots ,{\tilde{w}}_n],\in {\mathbb{C}}^{n-r}$$

$${\lambda }_{\min }({\tilde{U}}^H{X}^{-1}\tilde{U}({\tilde{U}}^H{A}^{-1}\tilde{U}{)}^{-1})$$

$$=\underset{\begin{array}{c}z\in \mathcal{R}(\tilde{U})\\ z\ne 0\end{array}}{\min }\frac{z^H{X}^{-1}z}{z^H{A}^{-1}z},$$

$$\underset{\tilde{U}\in {\mathbb{C}}^{n\times n-r},{\tilde{U}}^H\tilde{U}=I}{\max }{\lambda }_{\min }({\tilde{U}}^H{X}^{-1}\tilde{U}({\tilde{U}}^H{A}^{-1}\tilde{U}{)}^{-1})=\underset{\tilde{\mathcal{U}}\in \mathbf{V}}{\max }\underset{\begin{array}{c}z\in \tilde{\mathcal{U}}\\ z\ne 0\end{array}}{\min }\frac{z^H{X}^{-1}z}{z^H{A}^{-1}z}={\mu }_{r+1},$$

$$\underset{\begin{array}{c}P\in {\mathbb{C}}^{n\times r}\\ \text{operator@font}rank(P)=r\end{array}}{\min }\rho (E_M)=1-\underset{\begin{array}{c}P\in {\mathbb{C}}^{n\times r}\\ \text{operator@font}rank(P)=r\end{array}}{\min }{\lambda }_{\min }(BA)=1-{\lambda }_{r+1}.$$

$$P_{\mathrm{opt}}=[u_1,\dots ,u_r].$$

$$\rho (E_M)=1-{\lambda }_{\min }(BA).$$

$$E_{STG}$$

$$E_{TG}$$

$$\Vert E_{STG}{\Vert }_A=\Vert I-B_{STG}A{\Vert }_A=\rho (I-B_{STG}A).$$

Full text

1 Introduction

Typical multigrid methods to solve the linear system

[TABLE]

where $A$ is an $n\times n$ matrix, consist of two ingredients, the smoothing and the coarse grid correction. The smoothing is typically done by a few steps of a basic stationary iterative method, like the Jacobi or Gauss-Seidel method. For the coarse grid correction, a prolongation or interpolation operator $P\in\mathbb{C}^{n\times r}$ and a restriction operator $R\in\mathbb{C}^{r\times n}$ are needed. The coarse grid matrix is then defined as

[TABLE]

Here we always assume that $A$ and $A_{C}$ are non-singular. The multigrid or algebraic multigrid (AMG) error propagation matrix is then given by

[TABLE]

where $M_{1}^{-1}\in\mathbb{C}^{n\times n}$ and $M_{2}^{-1}\in\mathbb{C}^{n\times n}$ are smoothers, $\nu_{1}$ and $\nu_{2}$ are the number of pre- and post-smoothing steps respectively, and $PA_{C}^{-1}R$ is the coarse grid correction matrix. The multigrid method is convergent if and only if the spectral radius of the error propagation matrix $\rho(E_{m})$ is less than one. Alternatively, the norm of the error propagation matrix $\|E_{M}\|$ can be considered, where $\|\cdot\|$ is a consistent matrix norm, and in this case one has

[TABLE]

The aim of algebraic multigrid methods is to balance the interplay between smoothing and coarse grid correction steps. However, most of the existing AMG methods first fix a smoother and then optimize a certain quantity to choose the interpolation $P$ and restriction $R$.

To simplify the analysis, we assume that there exists a non-singular matrix $X$ such that

[TABLE]

it can be shown that such a non-singular matrix $X$ exists if the spectral radius of $(I-M_{1}^{-1}A)^{\nu_{1}}(I-M_{2}^{-1}A)^{\nu_{2}}$ is less than one, see e.g. [2]. Note that the matrix $E_{M}$ can be written as

[TABLE]

where the matrix $B$ is known as the multigrid preconditioner, i.e., $B$ is an approximation of $A^{-1}$. Therefore, eigenvalue estimates of $BA$ are of interest and they lead to estimates for the eigenvalues of $E_{M}$.

The following theorem, proved by García Ramos, Kehl and Nabben in [6], gives a characterization of the spectrum of $BA$, denoted by $\sigma(BA)$, and hence a characterization of the spectrum of the general error propagation matrix $E_{M}$.

Theorem 1**.**

Let $A\in\mathbb{C}^{n\times n}$ be non-singular, and let $P\in\mathbb{C}^{n\times r}$ and $R\in\mathbb{C}^{r\times n}$ such that $RAP$ is non-singular. Moreover, let $M_{1}\in\mathbb{C}^{n\times n}$ and $M_{2}\in\mathbb{C}^{n\times n}$ be such that that the matrices $X$ in (1.3) and $RXP$ are non-singular. Then the following statements hold:

1. (a)

The multigrid preconditioner $B$ in (1.4) is non-singular. 2. (b)

If $\tilde{P},\tilde{R}\in\mathbb{C}^{n\times n-r}$ are matrices such that the columns of $\tilde{P}$ and $\tilde{R}$ form orthonormal bases of $({\cal R}(P))^{\perp}$ and $({\cal R}(R^{H}))^{\perp}$ (the orthogonal complements of ${\cal R}(P)$ and ${\cal R}(R^{H})$ in the Euclidean inner product) respectively, then the matrices $\tilde{P}^{H}A^{-1}\tilde{R}$ and $\tilde{P}^{H}X^{-1}\tilde{R}$ are non-singular and the spectrum of $BA$ is given by

[TABLE]

We will apply this theorem to Hermitian positive definite (HPD) matrices to determine the optimal interpolation operators of AMG methods with respect to the spectral radius of the error propagation matrix. For HPD matrices, optimal interpolation operators with respect to the $A$-norm have been obtained recently in [9, 3]. We will show that the optimal interpolation operators with respect to the spectral radius for the symmetric/symmetrized multigrid method (with pre- and post-smoothing) and the optimal interpolation operator with respect to the $A$-norm are the same. But for multigrid with only a post-smoothing step the optimal interpolation operators with respect to the spectral radius and $A$-norm (and hence the optimal algebraic multigrid methods) can be different. Using Theorem 1 we can also show that the interpolation operators with respect to the spectral radius are also optimal with respect to the condition number of the multigrid preconditioned system.

2 Optimal interpolation for Hermitian positive definite matrices

Let $A\in\mathbb{C}^{n\times n}$ be HPD and recall that the norm induced by $A$ (or $A$-norm) is defined for $v\in\mathbb{C}^{n}$ and $S\in\mathbb{C}^{n\times n}$ by

[TABLE]

and

[TABLE]

We will study the following two-grid methods given by the error propagation operators

[TABLE]

and the symmetrized version

[TABLE]

Thus we are using $R=P^{H}$. The range of $P$, i.e. ${\cal R}(P)$, is called the coarse space $V_{c}$. We assume that the smoother $M^{-1}$ is fixed and let $E_{TG}$ and $E_{STG}$ vary with respect to the choice of the interpolation operator $P$. In addition, we assume that the smoother $M^{-1}$ satisfies

[TABLE]

which is equivalent to the condition

[TABLE]

see, e.g., [8]. Given a fixed smoother $M^{-1}$ such that $\|I-M^{-1}A\|_{A}<1$, many AMG methods are designed to minimize $\|E_{TG}\|_{A}$ or a related quantity. We say an interpolation operator $P^{\star}$ is optimal if it minimizes $\|E_{TG}\|_{A}$. In view of the equality

[TABLE]

proved by Falgout and Vassilevski in [4], we can conclude that an optimal interpolation operator $P^{\star}$ also minimizes $\|E_{STG}\|_{A}$. Zikatanov proved in [10, Lemma 2.3] (see also [5, Theorem 4.1]) that

[TABLE]

where $K(V_{c})$ is a quantity depending on the coarse space, defined by

[TABLE]

Here $\tilde{M}^{-1}=M^{-1}+M^{-H}-M^{-1}AM^{-H}$ is the symmetrized smoother and $Q=P(P^{T}\tilde{M}P)^{-1}\tilde{M}$. Although this equality has been known for a long time, only recently it was used to determine optimal prolongation operators formulated in terms of eigenvectors, which lead to a minimal value of $\|E_{TG}\|_{A}$ for a given smoother (see [9, 3]). We will give an alternative proof of this result using the characterization of the eigenvalues of the multigrid iteration operator given in Theorem 1.

We consider first the more general error propagation matrix $E_{M}$ in (1.2) with $R=P^{H}$ and $E_{M}=I-BA$. Let $\mathcal{U}=\mathcal{R}(P)$ be the range of the interpolation operator $P\in\mathbb{C}^{n\times r}$, and $\tilde{U}\in\mathbb{C}^{n\times n-r}$ be a matrix with orthonormal columns that span $\mathcal{U}^{\perp}$ (the orthogonal complement of $\mathcal{U}$ with respect to the Euclidean inner product). Then Theorem 1 leads to

[TABLE]

In what follows, given a matrix $C\in\mathbb{C}^{n\times n}$ with real eigenvalues we will denote by $\lambda_{\max}(C)$ and $\lambda_{\min}(C)$ the maximum and minimum eigenvalues of $C$ respectively.

Assuming that $X$ is Hermitian positive definite and that $\lambda_{\max}(BA)$ is at most one, we have $\rho(E_{M})=1-\lambda_{\min}(BA)$. In order to find an optimal interpolation operator for the error propagation matrix, we need to first find

[TABLE]

and then find an interpolation operator $P^{\star}\in\mathbb{C}^{n\times r}$ such that ${\cal R}(P^{\star})={\cal R}(\tilde{U}^{\star})^{\perp}$. The following lemma solves the first problem.

Lemma 2**.**

Let $A,X\in\mathbb{C}^{n\times n}$ be Hermitian positive definite and let $\{(\mu_{i},w_{i})\}_{i=1}^{n}$ be the eigenpairs of the generalized eigenvalue problem

[TABLE]

where

[TABLE]

Then

[TABLE]

which is achieved by

[TABLE]

where the columns of $\tilde{W}$ are orthogonal in the Euclidean inner product and satisfy $\operatorname*{span}\{\tilde{w}_{i}\}_{i=1}^{n}=\operatorname*{span}\{w_{i}\}_{i=1}^{n}$.

Proof.

Let $\tilde{U}\in\mathbb{C}^{n\times n-r}$ with $\tilde{U}^{H}\tilde{U}=I$. By the Courant-Fischer theorem we obtain

[TABLE]

Thus, if $\mathbf{V}$ is the set of subspaces of $\mathbb{C}^{n}$ of dimension $n-r$, we have

[TABLE]

and the maximum is attained by choosing a matrix $\tilde{W}=[\tilde{w}_{r+1},\ldots,\tilde{w}_{n}]$ such that the columns of $\tilde{W}$ are orthogonal in the Euclidean inner product and satisfy $\operatorname*{span}\{\tilde{w}_{i}\}_{i=1}^{n}=\operatorname*{span}\{w_{i}\}_{i=1}^{n}$. ∎

The previous lemma is the main tool to obtain the optimal interpolation operators.

Theorem 3**.**

Let $A\in\mathbb{C}^{n\times n}$ and $X\in\mathbb{C}^{n\times n}$ as in (1.3) be Hermitian positive definite. Let $\{(\lambda_{i},u_{i})\}_{i=1}^{n}$ be the eigenpairs of $X^{-1}A$, where $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$, and suppose that $\lambda_{\max}(BA)\leq 1$. Then

[TABLE]

An optimal interpolation operator is given by

[TABLE]

Proof.

Since $\lambda_{\max}(BA)\leq 1$, we have that

[TABLE]

Note that the eigenvalues $\lambda_{i}$ are the same as the $\mu_{i}$ in Lemma 2. According to Lemma 2, we need to find vectors which are orthogonal to the eigenvectors $w_{r+1},\ldots,w_{n}$ of the generalized eigenvalue problem $X^{-1}w=\mu A^{-1}w$. Now, consider the vectors $\{u_{i}\}_{i=1}^{r}$. The $u_{i}$ are also eigenvectors of the generalized eigenvalue problem $Au=\lambda Xu$. Moreover, the vectors $Xu_{i}=w_{i}$ are eigenvectors of the generalized eigenvalue problem $X^{-1}w=\mu A^{-1}w$. But the $w_{i}$ are $X^{-1}$-orthogonal (the $X^{-\frac{1}{2}}w_{i}$ are eigenvectors of the Hermitian matrix $X^{\frac{1}{2}}A^{-1}X^{\frac{1}{2}}$). Thus, the $u_{i}$, $i=1,\ldots,r$ are orthogonal to the $w_{r+1},\ldots,w_{n}$ in the Euclidean inner product and the interpolation operator $P_{\mathrm{opt}}$ given by (2.7) is the corresponding minimizer. ∎

Now, we consider $E_{TG}$ and $E_{STG}$ defined in (2.1) and (2.2). Again $E_{STG}$ and $E_{TG}$ can be written as

[TABLE]

for some matrices $B_{STG}$ and $B_{TG}$ in $\mathbb{C}^{n\times n}$. A straightforward computation shows that $B_{STG}$ is Hermitian, and by [1, Lemma 2.11] we have

[TABLE]

Moreover, the maximal eigenvalue of $B_{STG}A$ satisfies $\lambda_{\max}(B_{STG}A)\leq 1$, see e.g. [8, Theorem 3.16]. We then obtain

[TABLE]

The matrix $X$ in (1.3) is given by

[TABLE]

With (2.3) we have that $X_{STG}$ is Hermitian positive definite. We obtain the following corollary.

Corollary 4**.**

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive definite. Let $M\in\mathbb{C}^{n\times n}$ such $M+M^{H}-A$ is Hermitian positive definite, and let $X_{STG}^{-1}$ be as in (2.9), and let $\{(\lambda_{i},u_{i})\}_{i=1}^{n}$ be the eigenpairs of $X_{STG}^{-1}A$, where $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$, Then

[TABLE]

An optimal interpolation operator is given by

[TABLE]

Proof.

We have that $X_{STG}$ is positive definite and $\lambda_{\max}(B_{STG}A)\leq 1$. By Theorem 3 we obtain the desired result. ∎

Next, let us consider the non-symmetric multigrid method defined implicitly by $E_{TG}$, in (2.1). We use a Hermitian positive definite smoother $M^{-1}$. The matrix $X$ in (1.3) is given by

[TABLE]

Hence

[TABLE]

Therefore, it is not clear which of $\lambda_{\min}(B_{TG}A)$ or $\lambda_{\max}(B_{TG}A)$ equals the spectral radius. One way to overcome this problem is scaling. Note that we have for all Hermitian positive defnite matrices $X$ and $A$ and for all matrices $\tilde{U}\in\mathbb{C}^{n\times n-r}$

[TABLE]

Hence, the Hermitian smoother

[TABLE]

satisfies

[TABLE]

With Theorem 1 and $X^{-1}=\hat{M}^{-1}$ we then have

[TABLE]

thus

[TABLE]

Note that (2.12) is equivalent to $\hat{M}-A$ being positive semidefinite. This discussion leads to the following corollary.

Corollary 5**.**

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive definite. Let $M\in\mathbb{C}^{n\times n}$ such $M-A$ is Hermitian positive definite. Let $X_{TG}^{-1}=M^{-1}$. Let $\tilde{\lambda}_{1}\leq\tilde{\lambda}_{2}\leq\ldots\leq\tilde{\lambda}_{n}$ be the eigenvalues of $X_{TG}^{-1}A$ and let $x_{i}$, $i=1,\ldots,n$, be the corresponding eigenvectors. Then

[TABLE]

An optimal interpolation operator is given by

[TABLE]

Proof.

The matrix $X_{TG}^{-1}=M^{-1}$ is Hermitian positive definite. Moreover, since $M-A$ is also Hermitian positive definite the eigenvalues of $X_{TG}^{-1}A$ are less then one. Thus, with Theorem 1, $\lambda_{\max}(B_{TG}A)=1$. So, with Theorem 3 we obtain (2.13) and (2.14). ∎

Now we will compare the optimal interpolation with respect to the $A$-norm as given in Corollary 4, with the optimal interpolation with respect to the spectral radius as given in Corollary 5. Using $M=M^{H}$ and $M-A$ Hermitian positive definite, the vectors used in Corollary 4 are eigenvectors of

[TABLE]

while in Corollary 4 we use the eigenvectors of

[TABLE]

But $X^{-1}_{STG}A$ is just a polynomial in $M^{-1}A$ , where the polynomial is given by

[TABLE]

Thus, the eigenvectors of both matrices are the same. Moreover, the eigenvalues are related by the above polynomial. Hence, the eigenvectors corresponding to the smallest eigenvalues of $X^{-1}_{STG}A$ are the same eigenvectors that correspond to the smallest eigenvalues of $X^{-1}_{TG}A$. In consequence, the optimal interpolation in Corollary 4 and Corollary 5 are the same, if we assume that $M-A$ is Hermitian positive definite.

Next, let us have a closer look to the non-symmetric two-grid method and avoid scaling. We assume that the smoother $M$ is Hermitian and leads to a convergent scheme, i.e.

[TABLE]

which implies $\sigma(M^{-1}A)\subset(0,2).$ Thus, for the matrix $E_{TG}$ we have as above

[TABLE]

Let

[TABLE]

Then we have $\sigma(Z)\subset(0,2)$ and with Theorem 1

[TABLE]

But $\sigma(I-Z)\subset(-1,1)$. To get an upper bound for the minimal spectral radius of $E_{TG}$ over all interpolation we consider the matrix $(I-Z)^{2}$. Our next theorem deals with this case.

Theorem 6**.**

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive definite, and let $M\in\mathbb{C}^{n\times n}$ be Hermitian such that $\rho(I-M^{-1}A)<1$. Let $X_{TG}^{-1}=M^{-1}$, and let $\{(\lambda_{i},y_{i})\}_{i=1}^{n}$ be the eigenpairs of $(I-X_{TG}^{-1}A)^{2}$ with $\hat{\lambda}_{1}\leq\hat{\lambda}_{2}\leq\ldots\leq\hat{\lambda}_{n}$. Then

[TABLE]

The spectral radius $(\hat{\lambda}_{n-r})^{\frac{1}{2}}$ can be achieved by the interpolation operator

[TABLE]

Proof.

The proof follows immediately from the above arguments. ∎

Note that the above Theorems correspond to clear statements: the optimal interpolation operators are given by those eigenvectors of $X^{-1}A$ for which the smoothing is slowest to converge.

3 The optimal interpolation with respect to the condition number

Note that for symmetric multigrid where $M+M^{H}-A$ is Hermitian positive definite the largest eigenvalue of $B_{STG}A$ is one (see e.g. [7]). As seen in the proof of Corollary 5, the same holds for $B_{TG}A$ when we assume that $M-A$ is Hermitian positive definite. The later assumption can be obtained by scaling, however, this scaling affects the spectral radius of the error propagation matrix. But for the condition number of the multigrid preconditioned system, this scaling has no effect.

Theorem 1 characterizes the spectrum of $B_{STG}A$ and $B_{TG}A$. Following the arguments above, where we found optimal interpolation operators, such that $\lambda_{\min}(B_{STG}A)$ and $\lambda_{\min}(B_{TG}A)$ are maximal, we obtain that the same interpolation operators are optimal with respect to the condition number $\kappa$ of the preconditioned system. This leads to the next result.

Theorem 7**.**

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive definite. Let $M\in\mathbb{C}^{n\times n}$ such $M+M^{H}-A$ is Hermitian positive definite. Let $X_{STG}^{-1}$ be as in (2.9). Let $\{(\lambda_{i},v_{i})\}_{i=1}^{n}$ be the eigenpairs of $X_{STG}^{-1}A$, where $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$. Then

[TABLE]

An optimal interpolation operator is given by

[TABLE]

Our final result gives the optimal interpolation operator for the non-symmetric two-grid method with respect to the condition number $\kappa$.

Theorem 8**.**

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive definite. Let $M\in\mathbb{C}^{n\times n}$ be Hermitian positive definite such that $\rho(I-M^{-1}A)<1.$ Let $X_{TG}^{-1}=M^{-1}$, and let $\{(\tilde{\lambda}_{i},x_{i})\}_{i=1}^{n}$ be the eigenpairs of $X_{TG}^{-1}A$ where $\tilde{\lambda}_{1}\leq\tilde{\lambda}_{2}\leq\ldots\leq\tilde{\lambda}_{n}$. Then

[TABLE]

An optimal interpolation operator is given by

[TABLE]

Note, that in all cases of the previous sections any other interpolation operator $\tilde{P}$ with ${\cal R}(\tilde{P})={\cal R}(P_{\mathrm{opt}})$ is also optimal.

4 Conclusion

As mentioned in [9], the $A$ in AMG methods can also be understood as an $A$ for Abstract Multigrid Methods. Here we contributed to the theory of abstract multigrid methods by presenting alternate derivations of previously known results and by establishing new results. Building on a result from [6] which gives a characterization of the spectrum of the error propagation operator and the preconditioned system of two-grid methods, we derived optimal interpolation operators with respect to the $A$-norm and the spectral radius of the error propagation operator matrix in a short way. We also showed that these interpolation operators are optimal with respect to the condition number of the preconditioned system.