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Spectrum of a symmetrically pressure-mass-preconditioned saddle system

Statement

Assume the hypotheses and Schur bounds of Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling, with constants $α, Λ, C_{b}, β > 0$ . For $γ > 0$ , use $P_{γ} = diag (A, γ M_{Q})$ from Symmetric saddle-point system, Schur complement, and block preconditioners, and set $a_{γ} = \frac{β^{2}}{Λ γ}, b_{γ} = \frac{C_{b}^{2}}{α γ}, λ_{\pm} (s) = \frac{1 \pm \sqrt{1 + 4 s}}{2} .$ Let $μ_{1}, \dots, μ_{m}$ be the eigenvalues, with multiplicity, of $(γ M_{Q})^{- 1 / 2} S (γ M_{Q})^{- 1 / 2}$ . Then $a_{γ} \leq μ_{j} \leq b_{γ}$ , and the symmetric square-root-preconditioned saddle matrix has eigenvalues ${λ_{-} (μ_{j}), λ_{+} (μ_{j}) : 1 \leq j \leq m} and 1 with multiplicity n - m .$ Consequently its spectrum is contained in $[λ_{-} (b_{γ}), λ_{-} (a_{γ})] \cup {1} \cup [λ_{+} (a_{γ}), λ_{+} (b_{γ})],$ where the eigenvalue $1$ has multiplicity zero if $n = m$ . This is a spectral estimate for the stated scaled block preconditioner, not an iteration-count or generic solver-speed claim.

Facts & Assumptions

Given: Real finite-dimensional spaces $V, Q$ , an SPD velocity matrix $A$ , full-row-rank constraint matrix $B$ , pressure mass matrix $M_{Q}$ , and the inf-sup, continuity, and energy-equivalence hypotheses of the Schur theorem.

[F1]

The saddle matrix is $K = (\begin{array}{cc} A & B^{T} \\ B & 0 \end{array})$ , the positive Schur matrix is $S = B A^{- 1} B^{T}$ , and $P_{γ} = diag (A, γ M_{Q})$ is the block preconditioner (Symmetric saddle-point system, Schur complement, and block preconditioners).

[F2]

The Schur quadratic form satisfies $(β^{2} / Λ) M_{Q} \leq S \leq (C_{b}^{2} / α) M_{Q}$ (Pressure Schur equivalence from inf-sup, continuity, and stiffness scaling).

[F3]

Every self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F4]

Rank-nullity gives $\dim V = \dim \ker T + \dim im T$ for a linear map on a finite-dimensional space (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

Proof

1.1F1F2F3algebra

By [F3], the SPD matrices $A$ and $γ M_{Q}$ have symmetric positive square roots and inverse square roots. Put $T = (γ M_{Q})^{- 1 / 2} B A^{- 1 / 2}, X = P_{γ}^{- 1 / 2} K P_{γ}^{- 1 / 2} = (\begin{array}{cc} I & T^{T} \\ T & 0 \end{array}) .$ Then $T T^{T} = (γ M_{Q})^{- 1 / 2} S (γ M_{Q})^{- 1 / 2}$ . Congruencing [F2] by $(γ M_{Q})^{- 1 / 2}$ shows $a_{γ} I \leq T T^{T} \leq b_{γ} I$ . Hence $T T^{T}$ is positive definite and all its eigenvalues $μ_{j}$ lie in $[a_{γ}, b_{γ}]$ .

2.1F1F3step 1.1algebra

By [F3], choose an orthonormal eigenbasis $y_{1}, \dots, y_{m}$ of $T T^{T}$ , with eigenvalues $μ_{j} > 0$ . Set $x_{j} = T^{T} y_{j} / \sqrt{μ_{j}}$ . Then $T x_{j} = \sqrt{μ_{j}} y_{j}$ and $x_{i}^{T} x_{j} = \frac{y_{i}^{T} T T^{T} y_{j}}{\sqrt{μ_{i} μ_{j}}} = δ_{i j},$ so the $x_{j}$ are orthonormal. On the two-dimensional space spanned by $(x_{j}, 0)$ and $(0, y_{j})$ , $X$ has matrix $(\begin{array}{cc} 1 & \sqrt{μ_{j}} \\ \sqrt{μ_{j}} & 0 \end{array}),$ whose characteristic polynomial is $λ^{2} - λ - μ_{j}$ . Its two roots are $λ_{-} (μ_{j})$ and $λ_{+} (μ_{j})$ .

3.1F1F4step 1.1step 2.1algebra

The matrix $T T^{T}$ is positive definite by step 1.1, so for every $y \in R^{m}$ , $x = T^{T} (T T^{T})^{- 1} y$ satisfies $T x = y$ . Thus $T : R^{n} \to R^{m}$ is onto, and [F4] gives $\dim \ker T = n - m$ . For $x \in \ker T$ , $X (x, 0) = (x, 0)$ , giving eigenvalue $1$ on this kernel. The $m$ orthonormal pairs from step 2.1 and this $n - m$ dimensional kernel account for all $n + m$ dimensions and hence for the full spectrum of $X$ .

4.1F1F3step 1.1step 3.1algebra∎

If $C$ is the Cholesky factor in the symmetric-preconditioning definition, then $P_{γ} = C C^{T}$ . Let $O = P_{γ}^{- 1 / 2} C$ ; step 1.1 gives $O O^{T} = I$ , so $O$ is orthogonal and $C^{- 1} K C^{- T} = O^{T} (P_{γ}^{- 1 / 2} K P_{γ}^{- 1 / 2}) O .$ Therefore the published Cholesky-symmetric preconditioning has the same spectrum as $X$ . Since $λ_{-}$ decreases and $λ_{+}$ increases on $s > 0$ , the eigenvalue intervals follow from step 1.1.

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Sources

Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems (standard reference, not scraped)