Systematic Parameter Selection for FFT-based Numerical Inverse Laplace Transform: CFL-informed Tuning Rules and Quality Diagnostics

• Derives CFL-like feasibility condition for FFT-based NILT parameter selection • Proposes adaptive acceptance criteria combining ε Im and N -doubling convergence tests • Reduces parameter sensitivity from trial-and-error to deterministic selection • Improves accuracy by 2–3 orders of magnitude over naive defaults across benchmarks • For uniform-grid inversion: 3–4 × better accuracy than de Hoog, 4–5 × faster ( N = 2048 ) • Provides NumPy, JAX, and PyTorch implementations; GPU achieves 9–15 × speedup at N = 16384 The FFT-based numerical inverse Laplace transform (NILT) introduced by Hsu and Dranoff (1987) offers O ( N log N ) efficiency for recovering time-domain solutions from Laplace-domain transfer functions. However, practical application remains limited by parameter sensitivity, requiring expert trial-and-error tuning of the Bromwich shift, sampling frequency, and integration period. This work develops a systematic parameter selection framework based on three necessary conditions analogous to the Courant–Friedrichs–Lewy (CFL) stability constraint in numerical PDEs: (i) dynamic-range feasibility, (ii) spectral placement correctness, and (iii) aliasing suppression. We derive an explicit feasibility condition relating spectral abscissa, time horizon, and floating-point precision, with distinct bounds for stable ( α c ≤ 0) and unstable ( α c > 0) systems. A dimensionless stiffness ratio σ NILT quantifies proximity to the feasibility boundary. The imaginary leakage metric ε Im combined with N -doubling convergence tests enables a posteriori quality assessment without reference solutions. Computational experiments on five distributed-parameter transfer functions—including semi-infinite diffusion, packed-bed dispersion, and first-order-plus-dead-time systems—demonstrate that CFL-informed tuning automatically achieves near-optimal accuracy without user intervention, improving accuracy by 2–3 orders of magnitude over naive defaults across the tested benchmarks. The framework is validated by cross-comparison with Method of Lines time-domain solutions and the de Hoog algorithm. For uniform-grid inversion on these benchmarks, CFL-informed FFT-NILT with N = 2048 achieves 3–4 × better accuracy than de Hoog ( M = 20 ) while remaining 4–5 × faster in our Python implementations. The proposed methodology transforms NILT from a specialist technique into a routine computational tool for frequency-domain analysis of distributed-parameter systems.