Waveform Iteration: Methods & Applications

Updated 17 November 2025

Waveform Iteration is a family of iterative methods that update entire time-continuous signal trajectories to address multiphysics and dynamic problems.
It enables high-order multiphysics coupling, domain decomposition, and advanced signal processing through operator splitting, relaxation, and interpolation techniques.
Recent implementations leverage acceleration strategies like Anderson acceleration and IQN-ILS, achieving superlinear or finite-step convergence with reduced iteration counts.

Waveform iteration comprises a family of iterative methods where entire waveforms (i.e., functions of time, often vector-valued and possibly distributed across spatial or parameter domains) are the objects of iteration. These methods generalize classical fixed-point, domain decomposition, or operator-splitting algorithms by operating on time-continuous or time-discretized signal trajectories or state histories, rather than pointwise or single-step variables. Waveform iteration is foundational in high-order multiphysics coupling, domain decomposition for evolution equations, nonlinear operator splitting, phase demodulation, signal processing, waveform design for radar/communications, fast waveform construction for gravitational-wave analysis, and neural vocoding.

1. Mathematical Foundations and Variants

Waveform iteration (WI) refers to iterative schemes where the iterates are entire functions on an interval (waveforms). The generic fixed-point form reads: $x^{k+1}(t) = F\left(x^{k}\right)(t) \,,\quad t\in [0,T]$ where $F$ acts on suitable function spaces. This framework arises in multiple domains:

Operator splitting/dynamic iteration: In ODEs/PDEs, one often writes $u'(t) = F(u(t))$ . Nonlinear WI (Botchev, 2023) splits $F$ into a linear part and a correction $F(u) = -A_k u + f_k(u) + g(t)$ ; each iteration solves a (possibly block-Krylov) linear inhomogeneous IVP over $[0,T]$ .
Multiphyics and coupled systems: Each physical subsystem "sees" a continuous (interpolated) coupling waveform from the other (Rodenberg et al., 10 Nov 2025, Rüth et al., 2020, Kotarsky et al., 5 Feb 2025).
Domain decomposition: In waveform relaxation (WR), interface values $h^{k}(x,t)$ or $w^{k}(x,t)$ over an interface $\Gamma\times[0,T]$ are iteratively updated based on boundary value solves in subdomains over the full time window (Gander et al., 2014, Gander et al., 2013).
Signal demodulation: The phase of a modulated waveform $x(t)$ is iteratively unwrapped via Hilbert-transform-based operator on the whole trajectory (Gengel et al., 2019).

Structural commonalities include:

Iteration of entire time histories.
Use of time-continuous (or fine time-discretized/interpolated) interface, coupling, or observation variables.
Frequent coupling with operator splitting, Dirichlet–Neumann/Neumann–Neumann update, or optimization subloops.
The possibility of combining with higher-order time integration, non-uniform (adaptive) grids, and nonlinear acceleration (quasi-Newton, Anderson acceleration).

2. Key Algorithms and Fixed-Point Structures

2.1. Coupled Multi-Physics and Domain Decomposition

Partitioned multiphysics WI defines functional fixed-points. Let $A$ and $B$ denote subsystems (each a black box), with $\Omega_A, \Omega_B$ spatial domains and interface $\Gamma_C = \Omega_A \cap \Omega_B$ . For each window $[t_{\text{ini}}, t_{\text{ini}}+\Delta T]$ , each solver advances its own time grid, exports time-stamped interface data vectors $c_i^j$ , and reads interpolated waveforms $C_{\text{other}}(t)$ at arbitrary evaluation times. The coupled system satisfies

$C_A = \mathcal{I}_A \circ \mathcal{A} [C_B]\,,\quad C_B = \mathcal{I}_B \circ \mathcal{B} [C_A]$

and waveform iteration updates $C_A^{k+1} = F(C_A^{k}) = \mathcal{I}_A \circ \mathcal{A} \circ \mathcal{I}_B \circ \mathcal{B}(C_A^{k})$ (Rodenberg et al., 10 Nov 2025).

2.2. Waveform Relaxation in Domain Decomposition

Dirichlet–Neumann (DNWR) and Neumann–Neumann WR (NNWR) for parabolic/wave equations:

At iteration $k$ , solve subdomain PDEs over $[0,T]$ with time-dependent interface boundary conditions derived from previous outer iterates.
Update interface values $h^{k}$ (Dirichlet data) or $w^{k}$ (Neumann data) via relaxation:

$h^k(x,t) = \theta u_2^k|_\Gamma(x,t) + (1-\theta) h^{k-1}(x,t)$

or for NNWR,

$w^k = w^{k-1} - \theta (\psi_1^k|_\Gamma + \psi_2^k|_\Gamma)$

Laplace-transform and functional contraction analyses characterize convergence, with finite-step convergence for optimal $\theta$ and window size (Gander et al., 2014, Gander et al., 2013).

2.3. Nonlinear Waveform Relaxation

Nonlinear waveform relaxation (“dynamic iteration”) solves

$u^{(k+1)}(t) = e^{-tA_k}u_0 + \int_0^t e^{-(t-s)A_k} [f_k(u^{(k)}(s)) + g(s)]\,ds$

with explicit update for the entire time interval. Inner solution may use a block-Krylov basis (EBK), with outer residual contraction (Botchev, 2023).

3. Acceleration and Convergence Analysis

WI methods display convergence ranging from linear (underrelaxation or suboptimal parameter choices) to superlinear or even finite-step under optimal settings.

Spectral analysis: For linear problems, Laplace-transform symbols (e.g., in DNWR/NNWR) give precise contraction radii, with optimal $\theta$ yielding zero spectral radius (finite-step, error support vanishing after $k$ iterations for $T$ small enough relative to subdomain width) (Gander et al., 2014).
Nonlinear contraction: For Lipschitz $f_k$ and suitable bounds on $e^{-tA}$ , error sequences $\|\epsilon_{k}\|$ decay linearly or, via a superlinear bound, as $(C L)^k t^k / k!$ (Botchev, 2023).
Quasi-Newton/Anderson acceleration: Many recent WI implementations combine classical waveform relaxation with interface quasi-Newton (IQN-ILS) or Anderson acceleration (Rüth et al., 2020, Rodenberg et al., 10 Nov 2025, Kotarsky et al., 5 Feb 2025, Aghazade et al., 2021). These methods
- Construct approximate inverse Jacobians from iteration histories.
- Achieve superlinear or finite-step convergence for linear interface problems (termination within $d+1$ steps for $d$ -dimensional interface).
- Require minimal additional storage (secant table or past $m$ iterates) and have low overhead.

Scheme	Convergence (linear)	Superlinear/finite-step	Acceleration Mechanism
WR (scalar θ)	Generic θ	Optimal θ (e.g., ½)	None or underrelaxation
QN/Anderson	Generic initializations	Yes, for regular problems	IQN-ILS, AA secant tables

4. Practical Implementations and Algorithmic Pseudocode

4.1. Implementation Architecture

Modern frameworks such as preCICE (v3) implement WI by:

Subdividing the time domain into “windows” over which sampling and iteration are performed.
Each participant collects time-stamped “stamples” and builds interpolants (e.g., B-splines) for waveform exchange.
The READ_DATA/WRITE_SAMPLE/ADVANCE interface is used to synchronize and transfer waveform data (Rodenberg et al., 10 Nov 2025).
Non-matching mesh support is handled via projection/interpolation at each stample.
Parallel, peer-to-peer communication ensures scalability.

4.2. Pseudocode Example (Serial-Implicit Multi-Rate WI)

INITIALIZE: cA⁰, cB⁰               # endpoint values from prior window
k ← 0; C_A^k(t) ≡ cA⁰              # waveform initial guess

while not converged:
    # Subsolver B
    for j = 1 to nB:
        fA ← READ_DATA("Data_A", time = t_B^{j-1} - t_ini)
        # Advance B using fA
        cB^j = B.step(fA, t_B^{j-1}, δt_B^j)
        WRITE_SAMPLE(cB^j)
    C_B^{k+1}(t) ← INTERPOLATE(samples of B)

    # Subsolver A
    for i = 1 to nA:
        fB ← READ_DATA("Data_B", time = t_A^{i-1} - t_ini)
        cA^i = A.step(fB, t_A^{i-1}, δt_A^i)
        WRITE_SAMPLE(cA^i)
    TILDE_cA ← [cA^1, ..., cA^{nA}]

    # Convergence check and Quasi-Newton acceleration
    r^{k+1} ← norm(TILDE_cA^{nA} - cA^{nA, k})
    if r^{k+1} < tol: break
    cA^⋆ ← ACCELERATE(history of cA, r)
    C_A^{k+1}(t) ← INTERPOLATE(cA^⋆)
    k ← k + 1

(Rodenberg et al., 10 Nov 2025)

4.3. Discretization and Interpolation

For high-order accuracy, polynomial B-spline waveform interpolants of degree $p$ match the per-window local order, provided solvers' algorithms are also of order $q \geq p + 1$ .
Multi-rate support is achieved by interpolating between non-uniform step grids.

5. Application Domains

5.1. Multiphysics Coupling and Partitioned PDE Solvers

Waveform iteration is now standard in partitioned multiphysics simulation, enabling:

Black-box solver coupling without requiring shared time-stepping or detailed knowledge of internal states (Rodenberg et al., 10 Nov 2025, Kotarsky et al., 5 Feb 2025).
Arbitrary time-step ratios (multi-rate) and full reuse of native integration schemes in each code.
Retention of high-order convergence of monolithic solvers provided the interpolation and partitioning match the accuracy requirements.

Empirical studies show that for high-order time-integrators and B-spline degree matching, WI can preserve expected convergence order and reduce coupling errors by orders of magnitude (Rodenberg et al., 10 Nov 2025, Rüth et al., 2020), with total synchronization counts and wall time often much lower than single-value schemes.

5.2. Domain Decomposition for Evolution Equations

In parabolic and hyperbolic PDEs, WR (including DNWR and NNWR) can achieve superlinear or finite-step convergence for optimal relaxation, independent of spatial dimension—provided the time-window is not too long relative to subdomain size (Gander et al., 2013, Gander et al., 2014).

5.3. Inverse Problems and Wave Equation Reconstruction

“Back-and-Forth” waveform-iteration schemes serve in large-scale time-reversible inverse problems for wave equations (Marinesque, 2011). Extending the single-step Kalman or nudging filters to time-interval sequences, waveforms are propagated forward and backward, with feedback or covariance updates at each trajectory, rapidly converging to minimum-variance or least-squares solutions even with incomplete or noisy data.

5.4. Signal Processing and Demodulation

Iterative Hilbert-transform embedding applies waveform iteration to phase demodulation: rather than a single analytic-signal extraction, iteratively recomposing and remapping the waveform sharply increases phase recovery precision. The normalized periodicity error decays exponentially with iteration, limited only by numerical/discretization artifacts (Gengel et al., 2019).

5.5. Waveform Synthesis and Optimization

Iterative update schemes based on alternating minimization or Lagrangian optimization, applied to waveform synthesis/design (e.g., NLFM, radar, or communications waveform shaping), are built on waveform iteration between time and frequency domains, enforcing spectral constraints and waveform modulus/correlation goals (Ghavamirad et al., 2018, Setlur et al., 2018).

5.6. Neural Vocoding and Data-Driven Generative Models

Recent neural vocoders implement waveform iteration via learned fixed-point denoising mappings. WaveFit iterates a DNN-based denoiser, trained over all iterations with adversarial losses, resulting in high-fidelity speech within a handful of steps and orders-of-magnitude inference acceleration compared to autoregressive models (Koizumi et al., 2022).

5.7. Fast Waveform Generation for Physical Models

In gravitational-wave parameter estimation, evolutionary algorithm-based waveform iteration determines optimal frequency subsampling for approximant evaluation, filling the full spectrum by interpolation for order-unity speedups at acceptably low SNR error (Meijer et al., 12 Apr 2024).

6. Extensions, Limitations, and Current Directions

Higher-order and adaptive time integration: Extensive research supports WI in both fixed and adaptively refined time grids, with time-adaptive multirate quasi-Newton algorithms available for stiff or highly nonlinear problems (Kotarsky et al., 5 Feb 2025).
Scalability and parallelism: WI is naturally parallel across both time and space domains. PARAEXP and similar frameworks leverage time-parallelization in the block-Krylov inner solves of nonlinear WI (Botchev, 2023).
Robust acceleration: Anderson acceleration (AA) and IQN-ILS-based updates reduce iteration counts by 2–60% in challenging inverse or PDE-constrained optimization problems (Aghazade et al., 2021), at minimal cost.
Error control and limitations: In all domains, discretization (temporal/spatial), interpolation errors, and step ratio choices define the ultimate achievable accuracy and efficiency. For very high-accuracy goals, low-rank approximations or ill-conditioned subproblems can trigger stagnation in nonlinear WI.
Black-box applicability: Many recent WI approaches require only "coupling" API hooks (e.g., reading/writing waveforms), with no codebase modification to the core solvers.

7. Impact and Representative Numerical Results

WI methods have demonstrated substantial error reduction and speed improvements in a broad class of benchmarks:

Parabolic/wave PDEs: Superlinear or finite-step convergence is attainable for optimal relaxation parameter and window/geometry choice—e.g., in 1D DNWR with $\theta=1/2$ , convergence in $k$ iterations when $T \leq 2k \min(a,b)/c$ (Gander et al., 2014, Gander et al., 2013).
Black-box multiphysics coupling: In partitioned fluid-structure interaction (OpenFOAM+FEniCS), cubic B-spline WI remains stable for window sizes up to $500 \times \delta t$ , and iteration counts grow sublinearly with window size (Rodenberg et al., 10 Nov 2025).
Nonlinear ODE/PDE systems: Nonlinear EBK WI achieves 20–50× speedup (CPU time) vs Rosenbrock/ode15s in 3D benchmark problems for the same accuracy (Botchev, 2023).
Neural vocoding: WaveFit achieves speech MOS no different from human ground truth in five iterations, at >240× speedup over WaveRNN (Koizumi et al., 2022).
Waveform design: Iterative NLFM method reduces autocorrelation PSL by ~5 dB on average, up to 17 dB (Poisson window) (Ghavamirad et al., 2018).
Inverse problems: Back-and-forth SEEK waveform iteration produces sub-percent RMS error in a handful of forward/backward loops, even with sparse/noisy data, outperforming time reversal or full Kalman filter methods (Marinesque, 2011).

In summary, waveform iteration provides a unifying, highly scalable, and high-accuracy framework for time-dependent problems across computational physics, signal processing, optimization, and data-driven modeling. Its flexibility underpins contemporary advances in black-box multiphysics software, high-resolution domain decomposition, fast inverse algorithms, and neural generative models.