Controlled Accuracy Relaxation

• Controlled Accuracy Relaxation is a framework of algorithmic strategies that dynamically adjust numerical parameters to guarantee uniform error bounds, independent of stiffness or discretization schemes.
• It is implemented in IMEX time-stepping, auxiliary variable methods, and iterative solvers to enforce conservation, energy stability, and high-order accuracy.
• The paradigm employs explicit relaxation or correction steps that couple provisional updates with control mechanisms, enabling cost-efficient and robust simulations across multiscale regimes.

Controlled Accuracy Relaxation is a unifying methodological paradigm for enforcing a target level of accuracy in numerical discretizations, control, and optimization of systems with relaxation, auxiliary variables, or hierarchical subproblem structure. Its core objective is to maintain or enforce prescribed accuracy bounds—often independently of problem stiffness, discretization regime, or numerical auxiliary variables—by coupling the primary algorithm with an explicit control or correction (relaxation) stage, theoretically justified by uniform a priori error estimates and/or discrete functional constraints.

1. Fundamental Principles and Definitions

Controlled Accuracy Relaxation refers to algorithmic strategies that dynamically adjust numerical, algebraic, or control step parameters to guarantee error bounds or discrete-invariant enforcement irrespective of stiff or multiscale features in the underlying model. Examples include:

• Implicit-explicit (IMEX) time-stepping for hyperbolic equations with stiff relaxation terms, designed and analyzed to achieve uniform order-of-accuracy independent of the relaxation parameter $\varepsilon$ (Hu et al., 2019, Hu et al., 2023).
• Relaxation (correction) steps in energy quadratization (EQ) and scalar auxiliary variable (SAV) schemes, which enforce the correct decay of the original (not just modified) energy functional in the presence of discretization errors (Zhao, 2021, Jiang et al., 2021).
• Relaxed control or rounding in mixed-integer optimal control problems (MIOCP), so that solutions to convexified approximations can be rounded/controlled to integer-feasible solutions with explicitly controlled error (Hante et al., 2012).
• Relaxed tolerances for inexact Krylov and inner–outer iterative solvers, coupling inner solve accuracy to the outer-convergence state to minimize overall cost while enforcing outer solution fidelity (Wang et al., 2015, Darrigrand et al., 2022).
• General relaxation-based multistep or Runge–Kutta schemes that adjust step size or a relaxation parameter to ensure discrete conservation or dissipation of chosen functionals at any order of accuracy (Ranocha et al., 2020).

The common feature is an explicit algorithmic stage—relaxation, projection, convex combination, or error control—whose parameters are determined by a rigorous error analysis and are adapted (locally, per step) to preserve accuracy or invariants.

2. Algorithmic Realizations in Time-Stepping and Multiscale PDEs

In time integration of stiff PDEs and ODEs, Controlled Accuracy Relaxation is central to the construction of uniformly accurate schemes:

• IMEX-BDF and IMEX-RK schemes for stiff hyperbolic relaxation systems are shown to provide a priori error bounds of the form $\|u(t_n)-U^n\|+\|v(t_n)-V^n\| \leq C\Delta t^q$, where $C$ is independent of the relaxation parameter $\varepsilon$ (Hu et al., 2019, Hu et al., 2023).
• Such schemes typically rely on multiplier-energy arguments or specific coupling of implicit-explicit discretization coefficients to ensure stability and accuracy across all stiffness regimes, including the intermediate regime $\varepsilon = O(\Delta t)$ where classical analyses may fail and order reduction is a risk (Hu et al., 2019).

The design principles involve selecting time-stepping coefficients and formulating auxiliary equations—sometimes with correctors for energy or invariant functionals—such that uniform error bounds are achieved independently of stiff terms.

3. Auxiliary Variable Methods and Correction by Relaxation

Controlled-accuracy relaxation is essential in auxiliary variable approaches (EQ, SAV, and their relatives) for dissipative and conservative PDEs:

• In baseline EQ and SAV schemes, energy-stability is often guaranteed for a modified energy in auxiliary variables, but the original energy law is corrupted by O$(\Delta t)$ errors due to mismatch between discrete auxiliary variables and their theoretical definitions (Zhao, 2021, Jiang et al., 2021).
• By appending a local relaxation step—e.g., resetting the auxiliary variable as a convex combination of its computed value and the analytic value as a function of the current primary variable, with the weights determined to minimally perturb monotonicity—the original energy dissipation or conservation law is restored step-by-step, up to discretization accuracy (Zhao, 2021, Jiang et al., 2021).
• The stability analysis typically leads to a scalar quadratic inequality in the relaxation parameter, ensuring that the corrected scheme preserves unconditional energy stability, original energy dissipation laws, and formal temporal order, with negligible additional computational cost.

4. Adaptive Relaxation in Iterative Linear Algebra Solvers

In Krylov subspace and inner–outer solvers for linear systems, Controlled Accuracy Relaxation targets the total cost by dynamically relaxing inner or mat–vec accuracy based on outer iteration progress:

• When solving linear systems via inexact GMRES or inner-outer Golub–Kahan bidiagonalization, the error in the inner solve or mat–vec is relaxed in proportion to the norm of the outer residual or the current outer error estimate (Wang et al., 2015, Darrigrand et al., 2022).
• Classical results show convergence is maintained if the inner error at iteration $k$ is bounded by the outer target accuracy scaled by the residual, e.g., $\varepsilon_k\le \eta/\min(\|r_{k-1}\|, 1)$ for a target tolerance $\eta$ (GMRES/FMM context) (Wang et al., 2015).
• For inner–outer GKB, the allowed inner tolerance increases as the dominant singular components are captured, with parameters extrapolated from the decay of the bidiagonal coefficients (Darrigrand et al., 2022).
• These strategies typically yield 25–60% reduction in inner iterations or computational time while provably guaranteeing controlled accuracy of the final solution.

5. Controlled Rounding and Relaxation in Mixed-Integer and Optimal Control Problems

Controlled-accuracy relaxation appears in mixed-integer optimal control for the approximation of nonconvex problems by convex relaxations:

• For semilinear evolutions with integer-restricted controls, a convexified ("relaxed") control problem is solved first; then, rounding and controlled projection algorithms produce integer-feasible solutions whose cost differs from the relaxed optimum by an explicit bound in terms of time-mesh size and solver accuracy (Hante et al., 2012).
• Under suitable regularity, Lipschitz, and boundedness conditions for data and operators, one obtains explicit a priori estimates:

$$\|y^*(t)-z^k(t)\|\leq C_1\varepsilon^k+C_2(N-1)\Delta t^k, \quad |J(\omega^*,\alpha^*,y^*)-J(u^k,v^k,z^k)| \leq C_3\varepsilon^k+C_4(N-1)\Delta t^k,$$

confirming the controlled-accuracy property (Hante et al., 2012).

6. General Multistep and Runge–Kutta Relaxation Techniques

A general unifying framework treats relaxation as a scalar parameter adjustment in numerical integrators to enforce discrete invariants:

• Given a provisional update $\tilde u^n$ and an estimate $\tilde \eta$ of a smooth functional $\eta(u)$, the next step is set as a convex combination, $u^n = \bar u^n + \gamma(\tilde u^n - \bar u^n)$, where the relaxation factor $\gamma$ is chosen to enforce conservation or monotonic decay of $\eta$ (Ranocha et al., 2020).
• Analytical results demonstrate that, under mild nondegeneracy or convexity, this relaxation step preserves the baseline order of accuracy and produces unique $\gamma=1+O(h^{p-1})$.
• This structure is flexible, encompassing Runge–Kutta, multistep, general linear methods, and variable step sizes, and supports both conservation and dissipation constraints without extra order conditions (Ranocha et al., 2020).

7. Uniformity and Multiscale Robustness

A defining property of Controlled Accuracy Relaxation is uniformity with respect to multiscale or stiff parameters:

• For IMEX schemes and auxiliary variable methods, error estimates, stability, and conservation/dissipation are maintained uniformly in the stiffness parameter (e.g., $\varepsilon$ in relaxation problems), enabling use of large time steps set by accuracy alone rather than stiffness-induced restrictions (Hu et al., 2019, Hu et al., 2023, Zhao, 2021, Jiang et al., 2021).
• This uniform behavior bridges disparate regimes—hyperbolic ($\varepsilon \gg \Delta t$), parabolic ($\varepsilon \sim \Delta t$), and asymptotic-preserving ($\varepsilon \ll \Delta t$)—eliminating order reduction scenarios and simplifying simulation design (Hu et al., 2019, Hu et al., 2023).

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The Controlled Accuracy Relaxation paradigm is broadly applicable across time integration for stiff systems, energy-stable auxiliary variable schemes, iterative linear algebra, and mixed-integer optimal control, providing a foundation for stable, high-order, and cost-efficient numerical simulation with precision guarantees independent of multiscale stiffness or auxiliary variable drift (Hu et al., 2019, Hu et al., 2023, Zhao, 2021, Jiang et al., 2021, Wang et al., 2015, Darrigrand et al., 2022, Hante et al., 2012, Ranocha et al., 2020).