Deterministic PML Envelope

• Deterministic PML envelope is a mathematically rigorous framework that defines explicit error bounds for perfectly matched layer absorbing boundaries in wave propagation and privacy contexts.
• It employs complex coordinate stretching and smooth polynomial or exponential damping profiles to achieve exponential decay of errors, ensuring energy stability and optimal convergence rates.
• Explicit parameter selection formulas enable deterministic calibration for physical wave equations and privacy-preserving mechanisms, reducing reflection and leakage in simulations.

A deterministic PML envelope is an explicit, mathematically rigorous bound describing the error or leakage rate resulting from the use of perfectly matched layer (PML) absorbing boundaries in computational wave propagation and related contexts. The envelope quantifies, for a user-prescribed tolerance, the required thickness and damping strength profile to guarantee the desired suppression of reflections or leakage, and it extends to discrete and continuous settings with provable energy stability and optimal convergence rates. This notion appears both in physical wave equations—such as acoustics, electromagnetics, and elastodynamics—and in information-theoretic settings such as the pointwise maximal leakage (PML) metric for privacy-preserving mechanisms.

1. Mathematical Foundations: Complex Coordinate Stretching

The deterministic PML envelope is constructed via complex coordinate stretching in the Laplace or Fourier–Laplace domain. For linear hyperbolic wave equations, the solution $U(x, t)$ obeys

$$P^{-1}U_{t} = \sum_{\xi=x,y,z}A_{\xi}\,\partial_{\xi}U,$$

with truncation by PML implemented through complex-valued metrics $$S_{\xi}(s, \xi) = 1 + \sigma_{\xi}(\xi)/(s + \alpha_{\xi}(\xi))$$. The spatial derivative is replaced by $S_{\xi}^{-1}(s, \xi) \partial/\partial\xi$. Upon inversion to the time domain, auxiliary fields are introduced to preserve causality and energy structure.

For the first-order acoustic system, the un-split modal PML is derived via Laplace stretching $S_j(x_j, s) = 1 + d_j(x_j)/s$, leading to a system with auxiliary damping variables $\sigma$ and $\psi$ (Duru et al., 2018). Similarly, in Maxwell's equations and elastodynamics, a stretched coordinate yields a modified equation where the absorption profile $\sigma(r)$ or $d_j(x_j)$ is nonzero within a deterministic layer, producing exponential decay of propagating and evanescent modes (Wei et al., 2019, Duru et al., 2022).

2. Deterministic Envelope Profiles: Polynomial and Exponential Damping

The envelope function—$\sigma_{\xi}(\xi)$ or $d_j(x_j)$—is chosen to transition smoothly from zero in the interior to a maximum at the boundary over a layer thickness $L_{\rm PML}$. A standard choice is a cubic or higher-order polynomial ramp:

$$d_j(x_j) = \begin{cases} 0, & |x_j| \leq x_j^0 \ d_0 \left[ \frac{|x_j| - x_j^0}{\delta} \right]^m, & x_j^0 \leq |x_j| \leq x_j^0 + \delta \end{cases}$$

with $m \geq 2$ (typically $m = P+1$ for spectral $P$-refinement) (Duru et al., 2018, Duru et al., 2022). The coefficient $d_0$ is tuned via

$$d_0 = \frac{(m+1)\,c}{2L_{\text{PML}} \ln(1/\mathrm{tol})}$$

to ensure a target reflection $|\mathcal{R}| \leq \mathrm{tol}$ across the layer. Exponential or mixed profiles are also feasible. For envelope wave equations in laser–plasma modeling, Smilei PIC employs $\sigma_{\max} = -\frac{(m+1)\ln R}{2d}$, with empirical choices $\sigma_{\max}\sim 20$ for reflection $R\sim 10^{-6}$ (Bouchard et al., 2024).

3. Energy Estimates and Stability Guarantees

Energy stability is established by deriving continuous Laplace-domain energy estimates and showing that discretizations—e.g., DGSEM or SBP–SAT schemes—recover analogous discrete estimates. For the PML system,

$$\tilde{E}^2 + \sum_{j} \mathrm{Re}(1/S_j)\,BT_j \leq \tilde{E}\,\tilde{F}$$

ensures asymptotic stability for all nonnegative, monotone ramp profiles (Duru et al., 2018). Discretely, penalty-based upwind fluxes and stabilizing terms are required in the auxiliary equations. For discrete elements of size $\Delta$ and degree $P$, the PML reflection error converges as

$$\mathrm{tol} \simeq C_0 \left[ \frac{\Delta}{\delta(P+1)} \right]^{P+1}$$

and energy-stable time integration is achieved with order-$(P+1)$ ADER or Taylor schemes (Duru et al., 2018, Duru et al., 2022).

4. Explicit Error Envelopes and Convergence Rates

Key deterministic error envelopes take the following explicit forms:

Problem Type	Envelope Bound	Parameters
2D acoustic wave (disk)	$C e^{-2\alpha_0 R}$	Absorption $\alpha_0$, layer thickness $R$, source harmonics (Bryan et al., 9 Mar 2025)
3D electromagnetics	$$C d^2 (1+\sigma_0 T)^9 e^{-\frac{\sigma_0 d \sqrt{\mu}}{2}}$$	Layer thickness $d$, absorbing strength $\sigma_0$, time $T$, permeability $\mu$ (Wei et al., 2019)
Polynomial PML Profile	$$d_0 = \frac{(m+1)\,c}{2L_{\text{PML}}\ln(1/\mathrm{tol})}$$	$m$: profile order, $c$: max wave speed, $L$: thickness, tol (Duru et al., 2022)
DG spectral element	$$O\left(\left[\Delta / (\delta(P+1))\right]^{P+1}\right)$$	Element size $\Delta$, PML width $\delta$, polynomial degree $P$ (Duru et al., 2018)

All deterministic envelopes imply exponential decay in error with increasing damping strength or layer thickness, with explicit formulas guiding parameter selection.

5. Deterministic PML Envelope in Information-Theoretic Contexts

An analogous notion arises in privacy analysis as the deterministic pointwise maximal leakage (PML) envelope. For the Gaussian mechanism $Y = X + N$, with $X$ Gaussian and $N \sim \mathcal{N}(0, \sigma_N^2)$, the closed form

$\varepsilon_d(\delta) = \log\frac{2}{\delta}$

quantifies the minimal information leakage over all deterministic post-processings for failure probability $\delta$ (Saeidian, 13 Jan 2026). Extensions hold for general priors with posterior-variance control or strong log-concavity. In contrast to $(\varepsilon, \delta)$-differential privacy (DP), which yields $\delta \sim e^{-\Theta(\varepsilon^2/\sigma_N^2)}$, the deterministic PML envelope gives $\delta = 2e^{-\varepsilon}$ and is closed under arbitrary post-processing.

6. Practical Guidelines and Implementation Considerations

For physical wave problems:

• Choose smooth polynomial (typically cubic or higher-order) damping profiles; layer thicknesses of $2$–$4$ wavelengths are generally sufficient for tolerances $\sim 10^{-3}$.
• Upwind penalty fluxes and auxiliary equation stabilization are essential for discrete energy stability.
• In complex envelope equations, auxiliary differential equations are required to preserve explicit FDTD solvers (Bouchard et al., 2024).
• Explicit parameter selection formulas allow for deterministic calibration, avoiding trial-and-error tuning.

For privacy mechanisms:

• The PML envelope provides a fully explicit privacy–failure probability trade-off, supporting post-processing robustness, with optimal calibration given by $\delta = 2 \exp(-\bar\varepsilon)$ for target leakage $\bar\varepsilon$.

Numerical benchmarks confirm that observed error decay closely follows the deterministic envelope predictions, with exponential reduction to round-off or discretization floors for sufficient thickness and strength (Bryan et al., 9 Mar 2025, Bouchard et al., 2024).

7. Limitations, Open Problems, and Extensions

Certain scenarios present challenges for deterministic envelope guarantees:

• Anisotropic media or slowness-surface branches violating the geometric stability condition can yield instabilities unless modified PML metrics are used (Duru et al., 2022).
• Interface/guided wave modes (e.g., Scholte, Stoneley waves) lack a comprehensive stability analysis.
• Second-order (displacement-only) PML formulations in general heterogeneous media remain analytically challenging (Duru et al., 2022).
• Implementation on unstructured or curvilinear meshes and adaptive layers requires further investigation to ensure absence of discrete reflections.

A plausible implication is that ongoing research may extend deterministic envelope results to broader classes of equations, mesh structures, and interface phenomena. Practical implementation of PML layers in high-performance codes (e.g., Smilei PIC, WaveQLab3D, ExaHyPE) demonstrates resource usage reduction as high as $96\%$ for appropriate envelope configuration in multidimensional problems (Duru et al., 2022, Bouchard et al., 2024).

(Duru et al., 2018, Saeidian, 13 Jan 2026, Wei et al., 2019, Duru et al., 2022, Bouchard et al., 2024, Bryan et al., 9 Mar 2025)