All-at-Once Modeling Framework

• All-at-once modeling is defined by simultaneously solving coupled state, parameter, and data equations as one global system, enhancing robustness to ill-posedness.
• The framework employs block-coupled nonlinear formulations and sophisticated regularization techniques, enabling efficient parallel computations and convergence.
• Applications span inverse problems, tensor factorization, neural PINNs, and model discovery, offering advantages in handling noise, nonlinearity, and uncertainty.

The all-at-once modeling framework refers to a broad class of methods in computational mathematics, inverse problems, tensor factorization, machine learning, and scientific computing that cast the solution of a complex, typically PDE- or data-driven system, as a single, global optimization or algebraic system involving all state variables, parameters, and occasionally data-fit or regularization terms. In contrast to classical reduced or sequential schemes, which separate state solution and parameter identification, all-at-once formulations treat all variables as unknowns, yielding a tightly coupled system that can more flexibly handle ill-posedness, parallelism, regularization, and constraints. This approach has seen rapid development in inverse problems, high-dimensional learning, model discovery, parametric surrogates, and computational mechanics (Liu et al., 2020, Cao et al., 2 Jan 2025, Acar et al., 2011, Römer et al., 25 Apr 2024, Kaltenbacher, 2016, Schlintl et al., 2021, Tabeart et al., 4 Jun 2025, Stone et al., 2023, Nguyen, 2019, Kaltenbacher, 2019).

1. Formal Definition and Core Principles

The all-at-once framework is characterized by simultaneously treating all model equations, unknown variables (including state and parameters), and observation/data equations as one large coupled system—either as an algebraic system, saddle-point problem, or unified objective in variational or Bayesian inference. The canonical abstract form is: $F(u, p) = (A(u,p),\; C(u)) = (0, y)$ where $u$ represents state variables, $p$ parameters, $A$ the model (often a discretized PDE or differential system), and $C$ the observation or data operator (Kaltenbacher, 2016). The aim is to solve for $(u^*,p^*)$ that jointly satisfy the physical model and best fit the observations, possibly under regularization.

The all-at-once setting is distinguished from the reduced (sequential) setting, where one would first eliminate the state by the parameter-to-state map $u=U(p)$, then solve for $p^*$ to minimize the data misfit. In all-at-once, no such elimination is performed: both unknowns are optimized or solved together.

2. Mathematical Structures and Algorithmic Realizations

All-at-once systems naturally lead to block-coupled nonlinear equations, large-scale linear or nonlinear saddle-point systems, or monolithic global objectives. Key structures include:

• First-order optimality/KKT systems: For constrained inverse or parameter estimation, the Lagrangian is

$$\mathcal{L}(u,p,\lambda) = J(u,p) + \langle \lambda, R(u,p) \rangle$$

yielding conditions:

$$\nabla_u \mathcal{L} = 0,\quad \nabla_p \mathcal{L} = 0,\quad \nabla_\lambda \mathcal{L} = 0$$

which must be solved as a coupled system (Römer et al., 25 Apr 2024).

• Regularization and Bayesian approaches: Regularization may be applied to both state and parameter variables. In the Bayesian setting, the all-at-once posterior is

$$\pi(u, p \mid \text{data}) \propto \exp\Big(-S(F(u,p),(0,y^)) - \alpha R(u,p)\Big)$$

allowing general joint priors and noise models (Schlintl et al., 2021, Kaltenbacher, 2016).

• Iterative solution strategies: Algorithms include block (or global) Newton-type solvers, all-at-once Landweber iteration, IRGNM, and Landweber-Kaczmarz. Each step involves linearized updates on the coupled space, often requiring only linearized PDE and adjoint solves—no repeated nonlinear forward solves (Kaltenbacher, 2019, Nguyen, 2019).
• Objective examples:
  • Tikhonov: $$\min_{(u,p)} S(F(u,p),(0,y^{\delta})) + \alpha R(u,p)$$ (Kaltenbacher, 2016)
  • Joint factorization: $$\min_{\Theta} \|X - [A^{(1)},...,A^{(N)}]\|^2_F + \sum_d \|Y^{(d)} - A^{(n(d))}V^{(d)T}\|^2_F$$ (Acar et al., 2011)
  • Neural PINNs: $$\min_\theta\, \omega_{PDE}L_{PDE}(\theta) + \omega_{BC}L_{BC}(\theta)$$ over a global architecture (Cao et al., 2 Jan 2025).

3. Computational and Numerical Aspects

All-at-once methods are well suited to high-performance and parallel architectures, as their monolithic structure allows for:

• Block structure and preconditioning: Systems involving time/parameter/space discretizations can exploit block-circulant or block-Toeplitz preconditioners; for instance, block $\alpha$-circulant preconditioners for all-at-once diffusion or evolutionary PDEs enable parallelization across temporal and spatial blocks (Tabeart et al., 4 Jun 2025, Liu et al., 2020).
• Model reduction: Online reduced-basis or ROM methods can be integrated in all-at-once solvers to accelerate the solution of large parametric blocks, as in the ROM-accelerated ParaDIAG preconditioner (Liu et al., 2020).
• Iterative methods in the presence of ill-posedness: All-at-once Landweber, Landweber-Kaczmarz, and IRGNM can offer favorable convergence and avoid the need for repeated expensive nonlinear state solves, especially when parameter-to-state maps are ill-defined or inapplicable (Kaltenbacher, 2016, Kaltenbacher, 2019, Nguyen, 2019).

Parallelism can be exploited both across sub-blocks (e.g., time steps, spatial domains) and within matrix-vector operations, essential for high-dimensional or large-scale applications.

4. Applications in Inverse Problems, Model Discovery, and Data Science

All-at-once frameworks are widely employed in:

• PDE-constrained parameter estimation: Simultaneous reconstruction of state and parameters in solid mechanics, computational mechanics, and geophysics. All-at-once approaches are especially advantageous when the parameter-to-state map is unavailable or expensive, or when noisy and partial data or strong nonlinearities are present (Römer et al., 25 Apr 2024, Kaltenbacher, 2016).
• Dynamic inverse problems: Nonlinear and time-dependent systems (e.g., parabolic or hyperbolic PDEs) benefit from all-at-once iterative regularization, especially when split over time-subdomains as in Landweber-Kaczmarz (Kaltenbacher, 2019, Nguyen, 2019).
• Tensor and coupled-matrix factorization: All-at-once optimization of coupled tensor-matrix models (CMTF-OPT) solves for all factors globally, yielding robustness to missing data and better recovery under overfactoring compared to alternating schemes (Acar et al., 2011).
• High-dimensional surrogate modeling and PINNs: Neural surrogates for parametric PDEs, such as physics-informed neural networks (PINNs), can be trained all-at-once over a continuum of shapes and parameters, enabling inference for entire families of problems in milliseconds after training (Cao et al., 2 Jan 2025).
• Astronomical imaging: AstroPhot applies all-at-once fitting of all object, instrumental, and background parameters, handling deblending and covariances directly via automatic differentiation on GPU/CPU backends (Stone et al., 2023).
• Group activity recognition: In vision, all-at-once transformer architectures fuse spatial, temporal, and textual modalities for holistic action localization and recognition (Chappa et al., 2023).

5. Advantages, Limitations, and Theory

Advantages:

• Elimination of repeated nonlinear state solves: By jointly solving for all variables, only linearized or adjoint solves (rather than full nonlinear state solutions) are needed per iteration—minimizing expensive computations for nonlinear, high-dimensional, or ill-posed inverse problems (Kaltenbacher, 2019, Kaltenbacher, 2016).
• Robustness to ill-posedness and nonlinearity: All-at-once methods often remain applicable when the parameter-to-state map fails or is not differentiable, as in strongly nonlinear or degenerate systems (Kaltenbacher, 2016).
• Unified handling of constraints and uncertainties: Regularization, Bayesian priors, and constraints can be imposed on the full space, allowing flexible encoding of prior knowledge and uncertainty quantification for both state and parameters (Schlintl et al., 2021, Römer et al., 25 Apr 2024).
• Parallelism and scalability: The global system structure invites domain and data parallelism, leveraging modern hardware (Tabeart et al., 4 Jun 2025, Liu et al., 2020, Stone et al., 2023).

Limitations:

• Larger algebraic systems: The optimization or solve dimension is typically much larger—sum of state and parameter dimensions—leading to higher memory costs and possible ill-conditioning, especially in saddle-point systems (Kaltenbacher, 2016, Römer et al., 25 Apr 2024).
• Complexity in preconditioning and solver design: Block-coupled systems require sophisticated preconditioners (block $\alpha$-circulant, ROM) and tailored iterative solvers to mitigate convergence slowdowns (Liu et al., 2020, Tabeart et al., 4 Jun 2025).
• Tuning of scaling/weighting: Appropriate scaling between data-fit, physics, and regularization terms is critical for well-conditioning; poor weighting can degrade convergence (Römer et al., 25 Apr 2024).

Theory:

• Rigorous convergence theory exists for all-at-once Tikhonov, IRGNM, Landweber, and Bayes under standard assumptions (tangential cone, source conditions, differentiability). In Hilbert/Banach settings, Bregman distance and weak convergence results guarantee stable reconstructions with regularization parameter choice (Kaltenbacher, 2016, Kaltenbacher, 2019, Schlintl et al., 2021).

6. Comparative Performance and Practical Guidelines

Empirical studies demonstrate the trade-offs of all-at-once versus reduced approaches:

• Iteration cost and count: All-at-once per-iteration cost is lower (no nonlinear solves), but may need more iterations; overall, when forward solves are expensive or S(p) is ill-conditioned, all-at-once methods are often faster (Kaltenbacher, 2019, Nguyen, 2019).
• Noise and model error handling: All-at-once systems accommodate noise and model error in both model and data equations, facilitating robust estimation in practice (Schlintl et al., 2021, Römer et al., 25 Apr 2024).
• Problem choice:
  • Use all-at-once if the problem is highly nonlinear, ill-posed, or lacks a convenient parameter-to-state map; or if the main concern is parallel scalability and full coupling of variables.
  • Prefer reduced forms if the parameter-to-state mapping is cheap, well-posed, and robust to regularization, and memory is at a premium (Kaltenbacher, 2016, Römer et al., 25 Apr 2024).

The table summarizes key application domains and computational implications:

Domain	All-at-Once Benefit	Key Challenge
PDE-constrained inverse	No repeated forward solves	Large coupled system
Tensor factorizations	Robustness, missing data	Increased memory footprint
Neural PDE surrogates	High-D parametric fitting	Training time/cost
Regularization/Bayes	Joint priors, uncertainties	Prior design, conditioning

7. Representative Algorithms and Recent Developments

Recent work introduced several notable all-at-once methodologies:

• ROM-accelerated ParaDIAG preconditioners for evolutionary PDEs: exploit online reduced basis for parametric elliptic solves, drastically reducing CPU time by 9–12× compared to multigrid (Liu et al., 2020).
• Block $\alpha$-circulant preconditioning for diffusion-based covariance operators; coupled with Chebyshev or saddle-point inner solvers for mesh-agnostic, parallel-in-time strategies (Tabeart et al., 4 Jun 2025).
• All-at-once parametric PINNs: end-to-end surrogates mapping flow parameters to steady solutions over entire shape and flow manifolds, trained jointly with transformations and TSONN-style decompositions for improved conditioning (Cao et al., 2 Jan 2025).
• CMTF-OPT: simultaneous gradient-based optimization for coupled matrix-tensor models, empirically outperforming ALS when overfactoring or with missing data (Acar et al., 2011).
• AstroPhot all-at-once fitting: global fitting of sky, object, and PSF parameters leveraging autodiff and GPU acceleration; enabling large-scale, joint uncertainty estimation in astronomical image analysis (Stone et al., 2023).
• REACT all-at-once transformer: joint spatiotemporal and multimodal action recognition in vision, leveraging simultaneous attention blocks for superior performance (Chappa et al., 2023).

These advances demonstrate the versatility and computational power of the all-at-once paradigm across domains involving coupled, high-dimensional, ill-posed, or tightly integrated model–data–parameter systems.