Low-Mode Deflation Overview

• Low-mode deflation is a spectral projection technique that isolates low eigenmodes or near-null-space components to improve convergence in iterative solvers.
• It uses exact or approximate eigenvectors along with domain decomposition and oblique projectors to decouple troublesome spectral components from the main computation.
• Applications span Krylov solvers, lattice QCD propagators, non-Hermitian GMRES, and variational quantum deflation, yielding significant reductions in iteration counts and variance.

Searching arXiv for recent and foundational papers on low-mode deflation across Krylov solvers, lattice QCD, and related spectral-deflation settings. Low-mode deflation denotes a class of spectral projection techniques in which a small subspace associated with troublesome or otherwise privileged modes is isolated and treated separately from the complementary space. In its classical Krylov-solver form, the deflated subspace contains eigenvectors associated with small or isolated eigenvalues, near-null-space components, or algebraically smooth error modes, and the iteration is then performed on a projected operator with those directions removed or neutralized (Kopaničáková et al., 31 Jul 2025). In lattice QCD, the same principle appears as exact treatment of low quark modes in propagator solves and determinant reweighting (0706.2298, Kuberski, 2023, Li et al., 2010). More recent work broadens the term beyond the literal low end of the spectrum: in non-Hermitian GMRES the relevant deflation space can be defined by generalized eigenmodes of the skew-Hermitian part relative to the Hermitian part (Spillane et al., 2023), in WaveHoltz by modes nearest the driving frequency rather than smallest eigenvalues (Appelo et al., 30 Jun 2026), and in variational quantum deflation by previously computed low-energy Hamiltonian states that are penalized so that the next state can be resolved (Schaich et al., 2024).

1. Spectral principle and scope

The common algebraic idea is that a small-dimensional spectral subspace governs either convergence degradation or the physical quantity of interest. In SPD linear systems this usually means the low end of the spectrum: if the deflation basis spans exact eigenvectors corresponding to the first $k$ eigenvalues of the preconditioned operator $\Mm\Am$, then the deflated operator satisfies

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$

so the unwanted modes are annihilated in the projected problem (Kopaničáková et al., 31 Jul 2025). In lattice QCD, low modes matter because the density of eigenvalues near zero is tied to chiral symmetry breaking; the Banks–Casher relation implies that the number of low modes below a fixed physical threshold grows proportionally to the lattice volume $V$, which makes explicit global low-eigenmode deflation increasingly expensive if implemented naively (0706.2298).

The literature also shows that “low mode” is not a uniform notion across applications. In determinant reweighting for Wilson fermions, the critical objects are the lowest eigenmodes of a Hermitian even-odd preconditioned Wilson-Dirac operator because those modes dominate fluctuations of determinant ratios and their stochastic estimators (Kuberski, 2023). In overlap-fermion inversions, low modes of the massless overlap operator $D_{ov}$ and of the Hermitian Wilson kernel $H_W$ are both deflated, because the former slow the outer solve and the latter make sign-function evaluation expensive (Li et al., 2010). By contrast, in WaveHoltz the slow modes are the discrete eigenmodes satisfying $|\lambda_j-\omega^2|\ll 1$, so deflation targets a spectral packet near a shift rather than the smallest eigenvalues (Appelo et al., 30 Jun 2026). In non-Hermitian GMRES with $A=H+S$, the recommended deflation space is

$$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$

so the modes removed are those with large skew-to-Hermitian ratio, not small eigenvalues of $A$ (Spillane et al., 2023).

Setting	Modes treated	Immediate purpose
SPD DPCG	Small or isolated eigenmodes / near-null-space directions	Reduce $\Mm\Am$0 (Kopaničáková et al., 31 Jul 2025)
Lattice QCD propagators and reweighting	Lowest modes of Hermitian Wilson-Dirac or overlap-related operators	Stabilize and denoise infrared-dominated computations (Kuberski, 2023, Li et al., 2010)
Non-Hermitian GMRES	Generalized $\Mm\Am$1-eigenmodes with large $\Mm\Am$2	Improve one-step contraction bound (Spillane et al., 2023)
WaveHoltz	Near-resonant modes with $\Mm\Am$3	Remove components with iteration eigenvalues near $\Mm\Am$4 (Appelo et al., 30 Jun 2026)
VQD	Previously found low-energy eigenstates	Force the optimizer into the orthogonal complement (Schaich et al., 2024)

This diversity suggests a precise operational definition: deflation removes the subspace that most obstructs the target computation, but the spectral criterion defining that subspace depends on the operator class, objective, and solver.

2. Projectors, coarse operators, and deflated solves

The basic finite-dimensional construction is a projection-splitting of the solution into coarse and complementary parts. In the classical SPD formulation used for deflated PCG, one introduces restriction and prolongation operators

$\Mm\Am$5

with coarse operator

$\Mm\Am$6

and projector

$\Mm\Am$7

Equivalently, with $\Mm\Am$8 and $\Mm\Am$9,

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$0

The solution is decomposed as a coarse correction plus a projected solve: $\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$1 and the complementary component is obtained from

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$2

For DPCG, the search direction is explicitly deflated at each step by subtracting its coarse component: $\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$3 Because the learned subspace in that work only changes $\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$4, the theoretical guarantees are inherited from classical DPCG, with convergence estimate

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$5

(Kopaničáková et al., 31 Jul 2025).

For non-Hermitian weighted, preconditioned, deflated GMRES, the projector structure is oblique. With full-rank $\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$6, the deflation operators are

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$7

and satisfy

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$8

The exact solution is split as

$\sigma(\boldsymbol{\Pi}^T \Mm \Am) = \{0, \ldots, 0, \lambda_{k+1}, \ldots, \lambda_n \},$9

with coarse part

$V$0

and complementary singular deflated system

$V$1

In the right-preconditioned form,

$V$2

and the residual is

$V$3

Weighted GMRES does not break down provided

$V$4

(Spillane et al., 2023).

Lüscher’s domain-decomposed lattice-QCD construction uses related oblique projectors adapted to the Wilson-Dirac operator. If $V$5 spans a deflation subspace $V$6 with little Dirac operator

$V$7

then the left and right projectors are

$V$8

$V$9

with

$D_{ov}$0

The full solution of

$D_{ov}$1

splits as

$D_{ov}$2

where $D_{ov}$3 solves the deflated system

$D_{ov}$4

(0706.2298).

3. Classical lattice-QCD realizations

In lattice QCD, low-mode deflation is both an accelerator and a variance-reduction device. The infrared modes are physically distinguished by chiral symmetry breaking and numerically distinguished by their disproportionate impact on propagator inversion and determinant estimators.

A central problem is the large-volume scaling of explicit eigenmode deflation. Because the number of low quark modes below a fixed physical threshold grows like $D_{ov}$5, deflating exact global eigenmodes directly leads to the $D_{ov}$6-problem: setup and projector application become at least quadratic in the lattice volume. Lüscher’s resolution is local coherence. The claim is that low quark modes, when restricted to small blocks, can be well approximated by a low-dimensional local subspace. The lattice is partitioned into non-overlapping blocks $D_{ov}$7, one chooses $D_{ov}$8 local orthonormal fields $D_{ov}$9 per block, and the projector becomes block-local: $H_W$0 If the lattice is split into $H_W$1 blocks and the subspace is built from $H_W$2 relaxed random fields with 11 inverse-iteration steps, the resulting DFL+SAP+GCR solver exhibits flat scaling with quark mass and nearly volume-independent iteration counts on $H_W$3 and $H_W$4 lattices, while projector application accounts for about $H_W$5 to $H_W$6 of total time (0706.2298).

A second QCD realization appears in determinant reweighting for $H_W$7-improved Wilson fermions. For a general reweighting factor

$H_W$8

if $H_W$9 is the orthogonal projector onto exact low eigenmodes of $|\lambda_j-\omega^2|\ll 1$0,

$|\lambda_j-\omega^2|\ll 1$1

then

$|\lambda_j-\omega^2|\ll 1$2

In twisted-mass reweighting, the factor of interest is

$|\lambda_j-\omega^2|\ll 1$3

and the standard stochastic estimator has variance

$|\lambda_j-\omega^2|\ll 1$4

Deflating the lowest eigenmodes of the Hermitian operator $|\lambda_j-\omega^2|\ll 1$5 removes the configurations on which tiny eigenvalues drive the estimator far from unity. On CLS $|\lambda_j-\omega^2|\ll 1$6-flavor ensembles, using 4 low modes of $|\lambda_j-\omega^2|\ll 1$7 and half as many random sources as the conventional stochastic setup makes the relative standard deviation essentially independent of the value of the reweighting factor, and on the D150 ensemble changes

$|\lambda_j-\omega^2|\ll 1$8

to

$|\lambda_j-\omega^2|\ll 1$9

(Kuberski, 2023).

A third realization is overlap-fermion inversion on $A=H+S$0-flavor DWF ensembles. The massless overlap operator is

$A=H+S$1

with

$A=H+S$2

and the low-mode projector is

$A=H+S$3

The high-frequency part of the solve satisfies

$A=H+S$4

while the low part is reconstructed spectrally. Combined outer deflation and HYP smearing reduce inner/outer iteration counts from $A=H+S$5 to $A=H+S$6 on $A=H+S$7, from $A=H+S$8 to $A=H+S$9 on $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$0, and from $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$1 to $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$2 on $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$3, with total speedups of about $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$4, $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$5, and $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$6, respectively (Li et al., 2010).

These examples show three distinct but compatible QCD uses of low-mode deflation: removal of infrared error components in propagator solves, exact treatment of low determinant modes in reweighting, and reuse of the same eigensystem for downstream variance reduction such as low-mode substitution.

4. Nonstandard spectral targets and generalized meanings

A common misconception is that low-mode deflation always means “deflate the smallest eigenvalues of the operator being solved.” Several recent formulations explicitly contradict that reading.

In non-Hermitian GMRES with $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$7, $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$8 hpd, $$\mathcal Z = \operatorname{span}\{z^{(k)}:\ |\lambda_k|>\tau\}, \qquad S z^{(k)} = \lambda_k H z^{(k)},$$9 skew-Hermitian, $A$0 hpd, and $A$1, the key one-step contraction factor is

$A$2

with

$A$3

The deflation-space-dependent factor is bounded through the generalized eigenproblem

$A$4

whose eigenvalues are $A$5 or purely imaginary and whose eigenvectors may be chosen $A$6-orthonormal. Deflating

$A$7

yields, for $A$8,

$A$9

and consequently

$\Mm\Am$00

Numerically, on a $\Mm\Am$01 scaled Jordan block with $\Mm\Am$02, iteration counts decrease from 1000 at $\Mm\Am$03 to 652 at $\Mm\Am$04, 400 at $\Mm\Am$05, and 51 at $\Mm\Am$06 when the deflation space is built from the largest $\Mm\Am$07 generalized eigenvectors (Spillane et al., 2023).

WaveHoltz provides a second nonstandard target. The fixed-point or linear-system formulation involves a positive-definite WaveHoltz matrix $\Mm\Am$08, but slow convergence comes from spatial eigenmodes $\Mm\Am$09 whose eigenvalues satisfy

$\Mm\Am$10

The deflation basis

$\Mm\Am$11

contains eigenvectors of the discrete Laplacian with eigenvalues nearest $\Mm\Am$12, with coarse matrix

$\Mm\Am$13

and projectors

$\Mm\Am$14

The paper studies both direct eigenvector deflation (DEVD) and augmented-Krylov eigenvector deflation (AUKED) via DCG, AGMRES, and ABICGSTAB. For energy-conserving boundary conditions, the undeflated method requires approximately $\Mm\Am$15 iterations, while in two dimensions taking the number of deflation vectors to scale like

$\Mm\Am$16

keeps the asymptotic convergence rate essentially constant; accounting for eigenvector precomputation, the deflated solver breaks even after as few as two right-hand sides (Appelo et al., 30 Jun 2026).

Variational quantum deflation is a third extension. Here deflation is not used to accelerate a linear solve but to successively resolve the low-energy spectrum of a Hamiltonian. In the $\Mm\Am$17-th VQD stage, the operator is modified to

$\Mm\Am$18

with the practical choice $\Mm\Am$19 in the reported simulations. The variational cost is therefore the Hamiltonian expectation plus overlap penalties with previously found states. On the lattice $\Mm\Am$20 Wess–Zumino model, this allows one to inspect the pairing structure of the first few levels, using isolated lowest states plus paired excitations as a diagnostic of unbroken SUSY and paired lowest states as a diagnostic of broken SUSY (Schaich et al., 2024).

These formulations preserve the essential logic of deflation—remove a spectrally defined obstruction from the active iteration or optimization—but decouple it from the narrow notion of smallest eigenvalues of a single SPD operator.

5. Construction of the deflation space

The quality of low-mode deflation depends far more on subspace construction than on the projector formulas themselves. The recent literature spans exact eigenspaces, approximate local spaces, restricted reference modes, and learned surrogates.

One classical route is exact low-eigenmode construction. In Wilson-fermion determinant reweighting, exact low modes of $\Mm\Am$21 or $\Mm\Am$22 are computed to high precision with PRIMME, and the mixed terms in the block determinant decomposition vanish exactly because the projector is built from eigenvectors of a Hermitian operator (Kuberski, 2023). In overlap-fermion calculations, low modes of $\Mm\Am$23 are computed with Arnoldi, exploiting the normality of $\Mm\Am$24 and the pairing structure induced by $\Mm\Am$25-Hermiticity (Li et al., 2010). In WaveHoltz, the required eigenpairs are computed by EigenWave, which targets eigenvalues near a chosen frequency and, with implicit time stepping and multigrid, has cost scaling like $\Mm\Am$26 as the mesh is refined (Appelo et al., 30 Jun 2026).

A second route is structured approximation rather than exact eigenvectors. Lüscher’s domain-decomposed deflation begins from a small number of global fields $\Mm\Am$27 that satisfy

$\Mm\Am$28

with $\Mm\Am$29 in the range of low singular values. After restriction to blocks,

$\Mm\Am$30

and local Gram–Schmidt orthonormalization, one obtains a large deflation space of dimension $\Mm\Am$31 without computing $\Mm\Am$32 exact global modes (0706.2298).

A third route is reusable reference deflation under changing restricted domains. In state-constrained optimal control, the full Schur complement

$\Mm\Am$33

is restricted to inactive sets, producing SPD principal submatrices

$\Mm\Am$34

The method computes low eigenpairs

$\Mm\Am$35

once on a full-domain reference operator and then restricts them online: $\Mm\Am$36 The A-DEF2 coarse matrix and projector are

$\Mm\Am$37

The empirical justification is a spectral-coherence regime in which the leading low-mode subspace remains aligned with the low eigenspace of the current restricted system: at diagnostic rank $\Mm\Am$38, the principal angle at the cutoff stays below $\Mm\Am$39, $\Mm\Am$40, and $\Mm\Am$41 on three representative $\Mm\Am$42 problems (Kadeethum et al., 16 Jun 2026).

A fourth route is operator learning. DeepONet-based DPCG replaces eigenvector approximations or recycled Krylov spaces by a learned coarse basis. Two constructions are proposed. The trunk-basis (TB) approach forms tentative vectors by sampling DeepONet trunk outputs,

$\Mm\Am$43

motivated by the empirical claim that trunk basis functions associated with very low singular values approximate near-null-space directions. The recycling-solutions (RS) approach uses full predicted solutions as basis vectors. After forming

$\Mm\Am$44

the method imposes locality by partitioning degrees of freedom into groups $\Mm\Am$45, computing block QR factorizations

$\Mm\Am$46

and assembling a block-diagonal global prolongation $\Mm\Am$47. The paper does not prove that the learned basis converges to low eigenmodes, but it does state that both TB and RS retain the classical DPCG convergence guarantees because learning is used only to construct the deflation operator (Kopaničáková et al., 31 Jul 2025).

A final adjacent construction is the spectral low-mode reduced solver for elliptic problems. Here the coarse space is global and analytic: $\Mm\Am$48 with reduced operator

$\Mm\Am$49

This is not a deflated Krylov method, since it returns the coarse Ritz approximation

$\Mm\Am$50

without a fine-space correction, but it is directly relevant to coarse-space design because the reduced solution is energy-optimal in the selected subspace and the truncation error satisfies

$\Mm\Am$51

for $\Mm\Am$52-regular solutions (Torsu, 22 Dec 2025).

6. Numerical behavior, limits, and recurrent misconceptions

The numerical literature is consistent on one point: when the deflation space captures the genuinely slow spectral components, the effect on iteration counts or stochastic variance can be dramatic. In learned DPCG, for JumpDarcy with ASM and $\Mm\Am$53, the no-deflation method requires 2920 iterations, NICO 1824.4, TB 1066.0, and RS 1146.4; for Darcy with ASM and $\Mm\Am$54, the corresponding counts are 1830.4, 1561.2, 932.6, and about 899.3 (Kopaničáková et al., 31 Jul 2025). In inactive-set optimal control, deflation reduces CG iterations by about 55 to 98 percent across diffusion, convection-diffusion, nonlinear thermal, and conjugate-heat-transfer benchmarks (Kadeethum et al., 16 Jun 2026). In WaveHoltz, the precomputed basis can be amortized after as few as two right-hand sides (Appelo et al., 30 Jun 2026).

At the same time, the same sources delimit the scope of these gains. The explicit theory in the non-Hermitian GMRES analysis requires $\Mm\Am$55 to have hpd Hermitian part, $\Mm\Am$56 to be hpd, and $\Mm\Am$57; the bounds are one-step and pessimistic, and the prescribed deflation space should not be read as a general replacement for small-eigenvalue deflation outside that regime (Spillane et al., 2023). DeepONet-based coarse spaces require offline training and do not come with a theorem that low-singular-value trunk functions approximate small eigenvectors of $\Mm\Am$58; their justification is empirical and problem-dependent (Kopaničáková et al., 31 Jul 2025). In state-constrained optimal control, high-rank raw eigenmode restriction can become ill-conditioned in fragile 2D regimes, which motivates Rayleigh–Ritz reselection and conditioning safeguards such as

$\Mm\Am$59

and solve-time fallback when

$\Mm\Am$60

(Kadeethum et al., 16 Jun 2026).

Another misconception is that deflation is synonymous with acceleration. In VQD, the low-energy states are themselves the primary observable, and deflation is a sequential spectral-extraction mechanism rather than a preconditioner (Schaich et al., 2024). In randomized Krylov–Schur, deflation appears as Schur-vector locking: once a converged vector or block $\Mm\Am$61 is identified, subsequent expansion applies

$\Mm\Am$62

or, operationally,

$\Mm\Am$63

This is low-mode deflation in the sense of invariant-subspace extraction and restart stabilization, but not yet a full downstream deflated solver (Damas et al., 7 Aug 2025).

A final misconception is that exact eigenvectors are mandatory. The record is more mixed. Exact low modes are central in determinant factorization and overlap propagators (Kuberski, 2023, Li et al., 2010), but local coherence in lattice QCD, learned trunk bases in DPCG, restricted reference modes in optimal control, and SVD-compressed eigenvectors in WaveHoltz all indicate that approximate invariant subspaces can be sufficient when they preserve the geometry of the troublesome spectral packet (0706.2298, Kopaničáková et al., 31 Jul 2025, Kadeethum et al., 16 Jun 2026, Appelo et al., 30 Jun 2026).

In that sense, low-mode deflation is less a single algorithm than a recurring spectral design pattern. Its stable core is the separation of a small coarse space, exact or approximate treatment on that space, and projected iteration or estimation on the complement. What changes across applications is the definition of the “bad” modes: small eigenvalues for SPD Krylov solvers, low quark modes for chiral lattice observables, large skew-to-Hermitian-ratio modes for certain non-Hermitian GMRES analyses, near-resonant modes for Helmholtz-type fixed points, and already-found low-energy states for variational spectroscopy.