Koopman-Krylov Framework

• The Koopman-Krylov framework is a computational approach that approximates infinite-dimensional Koopman operators using Krylov subspaces and companion matrices.
• It facilitates modal reconstruction, stability analysis, and data-driven control for systems ranging from deterministic dynamics to noisy and parabolic PDE environments.
• The method overcomes numerical challenges by transforming ill-conditioned Vandermonde matrices into structured Cauchy forms for accurate eigenvalue and mode computation.

The Koopman-Krylov framework denotes a class of constructions in which a Koopman operator, or its infinitesimal generator, is approximated on a Krylov subspace generated by repeated action on a seed observable or on a sequence of snapshots. In the discrete-time setting, this yields finite-section or companion-matrix representations of the Koopman operator; in continuous-time and generator-based settings, it yields Galerkin or Lanczos reductions of the Liouville/Koopman generator. Across the cited literature, the framework appears in numerical Koopman spectral analysis, time-delay and Hankel formulations of DMD, robust estimation for stochastic systems, data-driven control of parabolic PDEs, and generator-based diagnostics of integrability breaking in semiclassical strings (Mezic, 2020, Drmač et al., 2018, Wanner et al., 2020, Deutscher, 2024, Das et al., 26 Feb 2026).

1. Operator-theoretic formulation

In its most basic discrete-time form, one considers a dynamical system $T:M\to M$ with Koopman operator $U$ acting on a function space $\mathcal F$, fixes a seed observable $f\in\mathcal F$, and forms the Krylov sequence

$f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$

The associated Krylov subspace is

$\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$

and the finite-section or Galerkin approximation is

$U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$

where $P_m$ is the projection onto $\mathcal K_m$. In the ordered basis $(f_1,\dots,f_m)$, the matrix of $U$0 has Frobenius companion form, with a shift structure in the first $U$1 columns and a last column determined by the projection of $U$2 onto $U$3 (Mezic, 2020).

An equivalent state-space formulation arises from a snapshot sequence

$U$4

Writing

$U$5

one obtains

$U$6

where $U$7 is a Frobenius companion matrix and $U$8 is orthogonal to the Krylov subspace $U$9. Algebraically,

$\mathcal F$0

so $\mathcal F$1 is the representation of $\mathcal F$2 on the Krylov subspace in the nonorthonormal basis $\mathcal F$3 (Drmač et al., 2018).

This identification of Koopman approximation with a Krylov compression is also the basis of time-delay and Hankel constructions. In the stochastic setting, a Hankel matrix built from one scalar observable produces a column space approximating

$\mathcal F$4

thereby realizing a Koopman-Krylov approximation from a single observable measured over a single trajectory (Wanner et al., 2020).

2. Spectral representation, modes, and reconstruction weights

When the companion matrix $\mathcal F$5 has simple eigenvalues $\mathcal F$6, its left eigenvectors are the rows of the Vandermonde matrix

$\mathcal F$7

and its right eigenvectors are the columns of $\mathcal F$8. Hence

$\mathcal F$9

Defining the Ritz or Koopman modes by

$f\in\mathcal F$0

the snapshots admit the modal reconstruction

$f\in\mathcal F$1

If $f\in\mathcal F$2 with $f\in\mathcal F$3 and amplitudes $f\in\mathcal F$4, then

$f\in\mathcal F$5

This is the canonical finite-dimensional Koopman modal expansion in the companion-matrix realization (Drmač et al., 2018).

A reduced reconstruction uses only $f\in\mathcal F$6 modes indexed by $f\in\mathcal F$7. One then seeks coefficients $f\in\mathcal F$8 minimizing the least-squares error across all snapshots,

$f\in\mathcal F$9

If $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$0 and $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$1, the problem reduces to a structured least-squares system. The Moore-Penrose solution is $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$2, but the introduction of a reflexive $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$3-inverse $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$4 yields an explicit formula,

$f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$5

Equivalently, these coefficients minimize a weighted norm

$f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$6

The resulting approximation is

$f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$7

with $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$8 as Koopman modes and $f_1=f,\quad f_2=Uf,\quad \dots,\quad f_m=U^{m-1}f.$9 as reconstruction weights (Drmač et al., 2018).

The same paper makes the relation to Generalized Laplace Analysis precise. In matrix form, for a diagonalizable operator $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$0, the coordinate GLA is

$\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$1

Truncating to $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$2 modes gives GLA reconstruction weights

$\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$3

If all $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$4 lie on the unit circle, the GLA weights coincide exactly with the reflexive $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$5-inverse weights. If $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$6, the reflexive $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$7-inverse weights become a nonuniformly weighted ergodic average. This shows that finite-sample optimal reconstruction and GLA are not separate constructions but two limits of the same modal-weighting problem (Drmač et al., 2018).

3. Numerical conditioning and the Vandermonde-Cauchy remedy

The central numerical obstacle in companion-form Koopman-Krylov methods is the inverse Vandermonde matrix. Direct inversion of $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$8 or direct solution of Vandermonde systems suffers $\mathcal K_m=\mathrm{span}\{f_1,\dots,f_m\},$9 relative error, and $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$0 can grow like $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$1. In floating-point arithmetic, directly computing $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$2 often destroys the accuracy of the Koopman modes even when the Ritz values themselves are meaningful (Drmač et al., 2018).

The stabilization developed in "Data driven Koopman spectral analysis in Vandermonde-Cauchy form via the DFT: numerical method and theoretical insights" (Drmač et al., 2018) is based on the observation that multiplication by the unitary discrete Fourier transform $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$3 converts the Vandermonde matrix into a generalized Cauchy matrix. With

$U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$4

one has

$U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$5

or, in matrix form,

$U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$6

where $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$7 and $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$8 are diagonal and $U_m=P_mUP_m:\mathcal K_m\to\mathcal K_m,$9 is a generalized Cauchy matrix. Equivalently,

$P_m$0

This change of basis does not remove ill-conditioning in the spectral sense, but it relocates the problem to a structured Cauchy factorization for which specially tailored numerical linear algebra is available. In particular, a pivoted LDU factorization

$P_m$1

can be computed by entry-wise forward-stable algorithms, with

$P_m$2

even when $P_m$3. Once these factors are available, $P_m$4 is recovered through unitary transforms and triangular solves in $P_m$5, and $P_m$6 can be obtained to full machine precision across all entries (Drmač et al., 2018).

The same stability analysis extends to the eigenvalue computation. Rather than using a black-box eigensolver that destroys companion structure, the paper advocates companion-QR or unitary-plus-rank-one QR methods. These produce eigenvalues that are exactly the roots of a nearby companion polynomial

$P_m$7

thereby preserving backward stability in the polynomial coefficients and maintaining accuracy even when Ritz values cluster (Drmač et al., 2018).

4. Convergence theory, DMD relations, and failure modes

Theoretical analysis of Koopman-Krylov approximations in finite-section form is developed in "On Numerical Approximations of the Koopman Operator" (Mezic, 2020). Under the assumptions that the Koopman operator $P_m$8 has pure point spectrum $P_m$9, that the corresponding normalized eigenfunctions form a Riesz basis, and that the seed observable admits an expansion

$\mathcal K_m$0

the eigenpairs of the Krylov-Galerkin truncations become pseudoeigenpairs of the full operator. Concretely, if $\mathcal K_m$1 is an eigenpair of $\mathcal K_m$2 with $\mathcal K_m$3, then for every $\mathcal K_m$4 there exists $\mathcal K_m$5 such that for all $\mathcal K_m$6,

$\mathcal K_m$7

The convergence is therefore in the pseudospectral sense rather than as a direct pointwise spectral convergence statement (Mezic, 2020).

The key residual identity is

$\mathcal K_m$8

where $\mathcal K_m$9 is the coefficient of $(f_1,\dots,f_m)$0 in $(f_1,\dots,f_m)$1. Since $(f_1,\dots,f_m)$2, the approximation error is controlled by

$(f_1,\dots,f_m)$3

Under the pure-point assumption, one obtains a rate function

$(f_1,\dots,f_m)$4

which separates decaying interior-unit-disc contributions from almost-periodic unit-circle contributions (Mezic, 2020).

The same paper places DMD-type methods in the context of finite-section theory. Both Krylov-Galerkin and DMD approximate $(f_1,\dots,f_m)$5; the difference is the basis choice. Krylov-Galerkin uses the time-delay basis

$(f_1,\dots,f_m)$6

whereas standard extended DMD uses a prescribed dictionary. The Krylov choice avoids the "curse of dimensionality" in the sense stated in the paper, because the subspace dimension grows with the number of iterates rather than with the ambient state dimension (Mezic, 2020). This also clarifies why Hankel-DMD and companion-form DMD are naturally interpreted as Koopman-Krylov constructions.

The same analysis also identifies a sharp limitation. For a mixing map such as $(f_1,\dots,f_m)$7 on the unit circle, finite-section eigenvalues do not approach the continuous Koopman spectrum, and the companion matrix collapses to a nilpotent shift. By contrast, for the rotation $(f_1,\dots,f_m)$8 with $(f_1,\dots,f_m)$9, the method recovers the exact eigenvalue $U$00. This establishes that pure point spectrum is a favorable regime for Koopman-Krylov approximation, whereas continuous-spectrum settings can defeat finite-section convergence (Mezic, 2020).

5. Stochastic Koopman-Krylov and robust estimation under noise

For random dynamical systems

$U$01

with i.i.d. $U$02, the stochastic Koopman operator acts on deterministic observables by

$U$03

Its iterates form a semigroup $U$04 because the maps are i.i.d. (Wanner et al., 2020).

In this setting, standard DMD can be biased when either the dynamics or the observables are noisy. If $U$05 is a noisy observable with mean $U$06, the empirical covariance matrices built by time averaging along one trajectory satisfy

$U$07

so the usual EDMD estimator $U$08 converges to a biased matrix rather than to the true finite-rank restriction of $U$09 (Wanner et al., 2020).

The robust remedy introduces a second noisy observable $U$10 that is uncorrelated with $U$11 and with the forward dynamics. With

$U$12

the cross-covariance matrices are

$U$13

and the robust estimator is

$U$14

Under mild ergodicity and independence conditions, this estimator converges to the true restriction of $U$15 on the chosen subspace (Wanner et al., 2020).

The Koopman-Krylov extension uses time-delayed observables. For a scalar signal $U$16, the Hankel matrix

$U$17

has columns approximating

$U$18

up to noise, so $U$19 approximates the Krylov subspace. Using the shifted Hankel matrix $U$20 and a lagged dual observable $U$21, one obtains

$U$22

which converges to the finite-dimensional companion matrix of $U$23 on the Krylov subspace. For numerical stability, the same paper also proposes an SVD reduction

$U$24

as Algorithm 4 (Wanner et al., 2020).

The worked example of a random rotation on the circle illustrates the distinction. Standard EDMD yields eigenvalues biased toward the real axis, whereas robust DMD with a lagged dual observable recovers the analytic eigenvalues within sampling error. In the Hankel test with $U$25, standard Hankel DMD is highly ill-conditioned and biased, while the robust SVD-based algorithm accurately recovers six modes when enough delays are used (Wanner et al., 2020).

6. Data-driven control of parabolic PDEs

A control-theoretic extension appears in "Data-Driven Control of Linear Parabolic Systems using Koopman Eigenstructure Assignment" (Deutscher, 2024). The plant is a boundary-controlled linear parabolic PDE on $U$26,

$U$27

with boundary conditions

$U$28

The state space is $U$29, the uncontrolled generator is a self-adjoint Sturm-Liouville operator $U$30, and the semigroup is $U$31. For bounded linear functionals $U$32, the Koopman operator is

$U$33

The open-loop Koopman eigenfunctionals are

$U$34

where $U$35 and $U$36 is an orthonormal eigenbasis (Deutscher, 2024).

The paper extends classical Krylov-DMD to this infinite-dimensional setting using only finitely many point-averaged outputs sampled at times $U$37,

$U$38

From the data matrix

$U$39

one seeks coefficients $U$40 such that

$U$41

which yields a companion-DMD matrix $U$42 with least-squares residual $U$43. Solving

$U$44

produces DMD approximations of the Koopman spectrum and modes; the errors are controlled by $U$45 (Deutscher, 2024).

These approximations are then used in a Koopman eigenstructure assignment problem. With approximate open-loop data $U$46 and $U$47, and desired closed-loop eigenvalues $U$48, the gain is parameterized through vectors $U$49 and

$U$50

If the $U$51 are linearly independent, the unique real feedback gain is

$U$52

The resulting controller requires only a finite number of open-loop Koopman eigenvalues and modes of the state, extracted from output data and input samples rather than from an identified PDE model (Deutscher, 2024).

The stability result is correspondingly finite-dimensional in construction but infinite-dimensional in consequence. In the ideal case, the assigned finite block evolves under a Hurwitz matrix while the uncontrolled tail remains exponentially decaying. In the data-driven case, Theorem 4 shows that small DMD errors in $U$53 and $U$54, together with sufficiently small mode-interpolation error, preserve exponential stability of the full PDE in the $U$55 norm (Deutscher, 2024). The numerical example uses

$U$56

with $U$57 spatial points, $U$58, $U$59 snapshots, and residual $U$60; the closed-loop design shifts the unstable and slow modes to $U$61 and yields rapid stabilization in simulation (Deutscher, 2024).

7. Generator-based Koopman-Krylov space and semiclassical strings

A generator-centric variant is developed in "Integrability breaking in semiclassical strings in Koopman-Krylov space" (Das et al., 26 Feb 2026). Here the starting point is the Koopman-von Neumann formulation of classical mechanics: observables $U$62 are elements of

$U$63

the nonlinear flow $U$64 induces the unitary Koopman operator

$U$65

and for a Hamiltonian $U$66 the generator is the Liouville operator

$U$67

Writing $U$68, the autocorrelation of a seed observable has the spectral representation

$U$69

with $U$70 the spectral measure of $U$71 (Das et al., 26 Feb 2026).

Since $U$72 acts on an infinite-dimensional space, the paper uses generator-extended DMD (gEDMD) to construct a finite-dimensional Galerkin approximation. With a dictionary

$U$73

empirical Gram and generator matrices are estimated from sampled phase-space points,

$U$74

and the projected generator is

$U$75

After spectral decomposition of $U$76, one obtains a finite-resolution spectral measure

$U$77

for the observable under study (Das et al., 26 Feb 2026).

To analyze spreading under the generator, the framework then constructs a Krylov chain

$U$78

and orthonormalizes it by Lanczos recursion to obtain basis vectors $U$79 and a tridiagonal matrix

$U$80

The time-evolved observable is expanded as

$U$81

with $U$82 in the Lanczos basis (Das et al., 26 Feb 2026).

From the Krylov probabilities $U$83, the paper defines diagnostics including Krylov complexity

$U$84

the inverse participation ratio

$U$85

the Krylov entropy

$U$86

and a finite-resolution spectral measure from the Lanczos tridiagonalization,

$U$87

To compare integrable and deformed dynamics, the framework also uses the Wasserstein-1 distance

$U$88

and sector-resolved leakage fractions $U$89 and $U$90 derived from participation ratios in protected and complementary sectors (Das et al., 26 Feb 2026).

The applications cover three classes of non-integrable semiclassical string solutions: a two-loop Landau-Lifshitz $U$91 sector, a Leigh-Strassler $U$92 Landau-Lifshitz sector, and two near-Penrose-limit ansätze of $U$93. Across these examples, the reported pattern is that integrability breaking induces redistribution of spectral weight, observable-dependent delocalization and spreading in Krylov space, and nonzero Wasserstein distance between integrable and deformed spectral measures. The paper also emphasizes that the effect is not an abrupt ballistic growth in Krylov space, but a gradual, resonance-driven redistribution of spectral weight (Das et al., 26 Feb 2026).

A plausible unifying implication of these results is that "Koopman-Krylov" is not a single algorithm but a common reduction principle: approximate the Koopman operator or generator on a subspace generated by iterates of the dynamics, then read spectral, modal, control, or transport information from the resulting finite representation. The specific realization may be a companion matrix, a Hankel/SVD reduction, a robust cross-covariance estimator, a finite-section Galerkin truncation, or a Lanczos tridiagonalization, but the shared mathematical object is the Krylov subspace generated by Koopman action (Drmač et al., 2018, Mezic, 2020, Wanner et al., 2020, Deutscher, 2024, Das et al., 26 Feb 2026).