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Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems

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Pith's one-line read For nonsymmetric and indefinite systems, the best possible transfer operators in a two-level method are the left and right generalized eigenvectors of the matrix pencil $(A,M)$.

desk verdict Strong complex-valued optimality theory for two-level transfer operators, but the real-valued optimality theorem is false when n_c splits a conjugate eigenvalue pair. read the letter →

arxiv 2505.05598 v2 pith:IAM2RMLS submitted 2025-05-08 math.NA cs.NA

Oliver A. Krzysik , Ben S. Southworth , Golo A. Wimmer , Ahsan Ali , James Brannick , Karsten Kahl This is my paper · ORCID
classification math.NAcs.NA MSC 65N5565F1065F1565F3565F50
keywords two-levelmethodsalgebraicmultigridoptimaltransferoperatorsnonsymmetricproblemsindefinitegeneralizedeigenvaluereal-valuedconvergencebounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

For nonsymmetric and indefinite linear systems, the paper answers a sharp design question: given a fixed fine-space preconditioner $M$ and a coarse-space dimension $n_c$, which restriction $R$ and interpolation $P$ give the best two-level method? The answer is that $P$ and $R$ should be built from the $n_c$ right and left generalized eigenvectors of the matrix pencil $(A,M)$ whose eigenvalues $\lambda$ have the largest values of $|1-\lambda|$. In a carefully chosen norm, these transfer operators minimize the error-propagation norm over all possible transfer operators, and the minimum is exactly $|1-\lambda_{n_c+1}|^{\nu_1+\nu_2}$, where $\nu_1+\nu_2$ is the number of smoothing steps. This gives a necessary and sufficient test for whether any convergent two-level method exists at a given coarsening. The paper also shows that when $A$ and $M$ are real, optimal real-valued transfer operators exist with the same convergence guarantees.

What carries the argument

The central object is the generalized eigen-decomposition of the matrix pencil $(A,M)$, with right and left eigenvectors $V_r,V_l$ satisfying $V_l^*AV_r=D_a$ and $V_l^*MV_r=D_m$. The analysis runs in the $N$-norm with $N=V_r^{-*}D^*DV_r^{-1}$, where $D$ is any diagonal, full-rank, CF-split scaling; in this norm the coarse-space projection becomes the block-diagonal operator $\mathrm{diag}(I,0)$ in the eigenvector basis, so the error propagator has an explicit eigenvalue decomposition and the diagonal $D$ commutes with the smoothing factors. The optimality proof applies the generalized Courant-Fischer-Weyl min-max principle to the diagonal pencil formed from the values $|1-\lambda_j|$, which yields the lower bound $|1-\lambda_{n_c+1}|^{\nu_1+\nu_2}$. For real $A$ and $M$, a block-diagonal similarity $T$ maps conjugate complex eigenvector pairs to real columns $W_l,W_r$ while preserving the orthogonality relations, which is what allows real-valued transfer operators to inherit the same theory.

What would settle it

Take a small diagonalizable but non-normal pencil, for instance a $3\times 3$ nonsymmetric $A$ with $M=I$, order the eigenvalues so $n_c=1$, and compute $\|E_{\mathrm{TG}}^{(1,1)}(P_\sharp,R_\sharp)\|_N$ with $D=I$. The central claim is wrong if this number differs from $|1-\lambda_2|^{2}$ or if a specific alternative pair $(P,R)$ gives a smaller $N$-norm. A boundary check: unstabilized upwind DG advection produces a non-diagonalizable $M^{-1}A$, so the assumed eigenvector bases do not exist and the theorem does not apply.

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Extended reading notes

Core claim

The paper proves that, for the norm induced by $N=V_r^{-*}D^*DV_r^{-1}$, the transfer operators $P_\sharp$ and $R_\sharp$---whose ranges are the $n_c$ right and left generalized eigenvectors of $(A,M)$ selected by the $n_c$ largest values of $|1-\lambda|$---minimize the two-level error-propagation norm over all possible transfer operators. The minimum value is $|1-\lambda_{n_c+1}|^{\nu_1+\nu_2}$ when $n_c<n$, and the spectral radius and the geometrically averaged $N$-norm coincide with this same number. When $A$ and $M$ are real, a block-diagonal similarity transform turns complex conjugate eigenvector pairs into real columns, and the resulting real-valued transfer operators $P^R_\sharp$ and $R^R_\sharp$ achieve the identical bounds and optimality. A direct corollary is that a convergent two-level method of coarse dimension $n_c$ exists if and only if $|1-\lambda_{n_c+1}|<1$.

Load-bearing premise

The load-bearing premise is that $M^{-1}A$ and $M^{-*}A^*$ are diagonalizable, so the $n$ left and right generalized eigenvectors form complete invertible bases; for a defective eigenvalue this fails, and the paper itself notes that unstabilized upwind DG advection gives such a case.

Editorial extensions

If this is right

  • For a fixed smoother $M$, the best possible two-level convergence factor in the $N$-norm is exactly $|1-\lambda_{n_c+1}|^{\nu_1+\nu_2}$, realized by the generalized-eigenvector transfer operators.
  • A convergent two-level method of coarse dimension $n_c$ exists if and only if $|1-\lambda_{n_c+1}|<1$, so no interpolation and restriction pair can beat that threshold.
  • For real nonsymmetric or indefinite $A$ and $M$, real-valued transfer operators attain the same optimal bounds as complex ones, making the result implementable in real arithmetic.
  • The theory contains the classical Hermitian positive-definite optimal interpolation result as a special case, giving $A$- and $M$-norm optimality for V($\nu_1,\nu_2$) cycles.
  • The optimal error propagator has equal spectral radius, $N$-norm, and geometric average in the $N$-norm, so the optimal iteration cannot show transient divergence in that norm.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One can pre-screen a candidate smoother $M$ by checking the generalized eigenvalue $|1-\lambda_{n_c+1}|$ before designing a coarse space; if it exceeds 1, no transfer operators can produce a convergent two-level method at that coarsening.
  • Because the optimal transfer operators are dense and require a global solve, their practical role is as a benchmark: any local or sparse approximation can be measured against the exact floor $|1-\lambda_{n_c+1}|^{\nu_1+\nu_2}$.
  • The block-diagonal similarity used to produce real transfer operators suggests that analogous structure-preserving changes of basis could yield optimal transfer operators with other desired algebraic forms, such as complex-symmetric or banded real representations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies algebraic two-level methods for linear systems Ax=b with a fine-space preconditioner M and transfer operators P,R. For a non-Hermitian pencil (A,M), it constructs complex-valued 'optimal' transfer operators P♯ and R♯ whose ranges are spanned by the first n_c right and left generalized eigenvectors, ordered by decreasing |1−λ|. It characterizes all HPD norms in which the resulting coarse-space correction Π(P♯,R♯) is orthogonal (Theorem 3.1), proves that in the N=V_r^{-*}D^*DV_r^{-1} norms the error-propagation norm, spectral radius, and geometrically averaged norm all coincide and equal |1−λ_{n_c+1}|^{ν1+ν2} (Theorem 3.3), and claims that P♯,R♯ minimize the N-norm over all possible transfer operators (Theorem 3.5). It then introduces real-valued generalized eigenvectors W_l,W_r for real A,M and claims that the corresponding real transfer operators P_R♯,R_R♯ satisfy the same convergence bounds and optimality in related real norms (Theorem 4.3). Numerical experiments for advection-reaction and mixed wave-equation discretizations compare the predicted factor |1−λ_{n_c+1}|^{ν1+ν2} with computed convergence factors.

Significance. If the complex-valued results are correct, this is a substantial theoretical contribution: it upgrades the pseudo-optimality of Ali et al. to genuine norm optimality, gives tight norm bounds for nonnormal two-level error propagators, recovers the HPD results of Brannick et al. as a special case, and yields an explicit condition for convergence in the N-norm. The derivations are largely transparent and parameter-free, and the numerical section provides direct verification of the equality in Theorem 3.3. However, the real-valued transfer-operator theorem, which is a central advertised practical contribution, is false as stated when n_c splits a conjugate eigenvalue pair, and the Courant-Fischer theorem used in the optimality proof is misstated. These issues need to be repaired before the paper's main claims can be accepted.

major comments (2)
  1. [§4.1, Eq. (4.14), Theorem 4.3(3)] The identity range(P_R♯)=range(P♯B_c)=range(W_{r,1:n_c}) in (4.14) is valid only when n_c does not split a conjugate eigenvalue pair. If n_c cuts a 2×2 block of T^*, then B_c=(T^*)_{cc} is not a complete block of the similarity transformation, so P♯B_c involves only one complex eigenvector of the pair while W_{r,1:n_c} contains a real combination of both; Lemma 4.2 cannot be invoked and Theorem 4.3(3) is not established. A concrete counterexample is M=I, A=[[1,-2],[2,1]], n_c=1, whose eigenvalues are 1±2i. With the construction in (4.3), W_{r,1}=[1,1]^T and \hat N=(1/2)I, so the \hat N-norm is the Euclidean norm. Taking P_R♯=R_R♯=[1,1]^T and ν1=ν2=1 gives E_TG=[[-6,-6],[2,2]] and ||E_TG||_{\hat N}=sqrt(80)≈8.944, while (4.18) predicts |1−λ_2|^2=4; the complex optimal operators P=R=[1,-i]^T attain 4 in the same norm. The theorem therefore needs an explicit no-split condition or a different real-valued optimality statement for coarse spaces whose dimension cuts a conjugate pair.
  2. [§3.3, Theorem 3.4 and Eq. (3.29c)] The generalized Courant-Fischer formula in Theorem 3.4 is misstated: with eigenvalues ordered α_1≤...≤α_n, the minimum over subspaces of dimension n−k+1 equals α_{n−k+1}, not α_k (for n=2 and k=1, the displayed expression is the maximum over the whole space and equals α_2). The proof of Theorem 3.5 uses the displayed formula with dim null(\hat R^*)=n_f=n−n_c and concludes k=n_c+1, which would give α_{n_c+1}=|1−λ_{n_f}|^{2(ν1+ν2)} rather than the claimed |1−λ_{n_c+1}|^{2(ν1+ν2)}. The intended lower bound can be recovered from the correct min-max statement (the minimum over an n_f-dimensional subspace gives the n_f-th smallest eigenvalue, which is μ_{n_c+1} in the descending ordering μ_i=|1−λ_i|^{2(ν1+ν2)}), but as written the proof is not valid.
minor comments (5)
  1. [Abstract] In the abstract, 'in the case of that A and M are real valued' should be 'in the case that A and M are real valued'.
  2. [Eq. (4.17)] The condition d_{i+1}=d_i if λ_{i+1}=λ_i does not enforce the needed equality for a conjugate pair, since λ_{i+1}=\overline{λ_i}\neq λ_i; it should state that d_i=d_{i+1} when λ_i and λ_{i+1} form a conjugate pair.
  3. [Proof of Lemma 3.2] There is a typo 'identifty' in the sentence introducing Eq. (3.5).
  4. [Remark 5.1] There is a typo 'eignevector matrices' in Remark 5.1.
  5. [Section 5.4 and Corollary 3.7] The sentence in Section 5.4 that |1−λ_{n_c+1}|<1 is necessary 'regardless of the interpolation and restriction used' should be qualified as necessity for contraction in the N-norm; the N-norm lower bound in Corollary 3.7 does not by itself rule out asymptotic convergence in spectral radius for some other P,R.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimality claim is proved by an independent Courant-Fischer lower bound over all transfer operators, and self-citations are to parameter-free auxiliary results.

full rationale

The paper's central claim is a genuine derivation rather than a fit or a definitional restatement. Theorem 3.3 proves the upper bound by explicitly constructing P_sharp and R_sharp from the first n_c generalized eigenvectors, and Theorem 3.5 proves the matching lower bound for arbitrary P,R via equations (3.28)-(3.31), which transform the minimization into a generalized Courant-Fischer-Weyl min-max problem whose value is |1 - lambda_{n_c+1}|^{nu_1+nu_2}. This lower bound does not use the special form of P_sharp and R_sharp, so it is independent of the construction that attains it. The N-norm is defined through V_r and a diagonal D, but N is fixed once D is chosen and does not depend on the transfer operators being optimized; choosing a norm adapted to the eigenbasis of (A,M) is a legitimate metric choice, not a definitional identification of the claim with its input. The citations to the authors' prior work [5] for Lemma 2.1 and Theorem 2.2 concern parameter-free results with stated assumptions (diagonalizability of M^{-1}A) that do not include the target optimality claim, so they are independent support rather than load-bearing self-citation. The paper also discloses the diagonalizability assumption in Section 2.1 and Remark 5.1; this is an applicability limitation, not a circular step. No fitted parameter is later renamed as a prediction, no ansatz is smuggled in via citation, and no known result is merely relabeled. The external critique of Theorem 4.3 for coarse dimensions that split a conjugate eigenpair is a correctness concern, not a circularity, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on diagonalizability, invertibility of the coarse matrix, standard min-max theory, and a prior compatibility lemma. No free parameters are fitted to data. The N-norm family is a mathematical construction, not an invented physical entity.

assumptions (5)
  • domain assumption M^{-1}A and M^{-*}A* are diagonalizable
    Stated in Section 2.1; needed for the sets of right and left generalized eigenvectors V_l and V_r to be complete and invertible. Remark 5.1 notes that upwind DG advection matrices can violate it.
  • domain assumption R*AP is invertible so the coarse-space projection is well defined
    Stated in Section 2.1; singular coarse spaces are excluded. The authors argue this does not affect optimality because a singular coarse space would lower the projection rank.
  • standard math Generalized Courant-Fischer-Weyl min-max principle for Hermitian pencils
    Theorem 3.4; used for the lower bound on the minimal N-norm in Theorem 3.5.
  • standard math Compatibility condition for N-orthogonal projections from Manteuffel and Southworth [36, Lemma 4]
    Used in Lemma 3.2 to characterize all inner products in which the coarse-space correction is orthogonal; the condition is cited from prior literature.
  • domain assumption Conjugate-pair normalization of real generalized eigenvectors
    Theorem 4.1 assumes eigenvectors for conjugate eigenvalue pairs appear as conjugate columns and that a similarity transform maps the complex diagonal Λ to the real block-diagonal Λ_R.

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Pith. "Pith review of Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems." pith.science (2026). https://pith.science/paper/IAM2RMLS

@misc{pith2026250505598,
  author       = {Pith},
  title        = {Pith review of: Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAM2RMLS}},
  note         = {Machine review of arXiv:2505.05598}
}
abstract

Consider an algebraic two-level method applied to the $n$-dimensional linear system $A \mathbf{x} = \mathbf{b}$ using fine-space preconditioner (i.e., ``relaxation'' or ``smoother'') $M$, with $M \approx A$, restriction and interpolation $R$ and $P$, and algebraic coarse-space operator ${A_c := R^*AP}$. Then, what are the the best possible transfer operators $R$ and $P$ of a given dimension $n_c < n$? Brannick et al. (2018) showed that when $A$ and $M$ are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the $n_c$ smallest generalized eigenvectors of the matrix pencil $(A, M)$. Recently, in Ali et al. (2025) we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of $(A, M)$ and defining corresponding nonsymmetric transfer operators $\{R_\#,P_\#\}$. Tight convergence bounds for $\{R_\#,P_\#\}$ are derived in spectral radius, as well as a proof of pseudo-optimality. Note, $\{R_\#,P_\#\}$ are typically complex valued, which is not practical for real-valued problems. Here we build on Ali et al. (2025), first characterizing all inner products in which the coarse-space correction defined by $\{R_\#,P_\#\}$ is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators $\{R_\#,P_\#\}$ are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from Brannick et al. (2018). Finally, we show how to construct optimal, real-valued transfer operators in the case of that $A$ and $M$ are real valued, but are not HPD. Numerical examples arising from discretized advection and wave-equation problems are used to verify and illustrate the theory.

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    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.check...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.