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Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches
Kristopher L. Kuhlman
TL;DR
Laplace-space numerical methods need efficient inversion strategies because simulation-based function evaluations are costly and prior reviews focused mainly on single-time accuracy. The paper couples a two-dimensional BEM diffusion simulation with five inversion algorithms and finds Fourier-series approaches especially robust and economical for common time behaviors, while identifying boundaries for Weeks and fixed Talbot methods.
Problem
Prior inversion reviews focused on accurate single-time solutions rather than minimizing Laplace-space model evaluations for simulation-based approaches.
Method
The study uses a two-dimensional BEM solution of the Laplace-transformed diffusion equation to compare five numerical inverse Laplace transform algorithms and implementation approaches.
Results
Fourier-series inversion is most economical for common time behaviors, more robust to non-optimal free parameters, and supports reuse of image-function evaluations across at least a log cycle of time.
Takeaways & Limitations
Fourier-series methods are recommended when broad time behavior, parameter robustness, and efficient sequence evaluation are priorities.
Takeaways & Limitations
The fixed Talbot method may not work for non-zero step-time behavior, while the Weeks method is not robust for non-optimal free-parameter values.
Abstract
from arXiv · showhide
A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time.
11 Introduction
Laplace-space numerical approaches can avoid time marching but require effective numerical inversion, especially when many forward simulations are needed. This study couples a two-dimensional BEM model with five inversion algorithms to compare sequence evaluation efficiency and implementation choices.
- A two-dimensional BEM solves the Laplace-transformed diffusion equation, and five inversion methods are compared for the resulting solution.
- Laplace-space methods permit evaluation at arbitrary times without evolving from an initial condition, while image-function calculations are readily parallelized.
- Forward-model efficiency is critical because parameter estimation may require hundreds or thousands of simulations.
- Published inversion reviews emphasized accurate single-time solutions rather than minimizing simulation-based Laplace-space function evaluations.
- The study evaluates inversion algorithms that compute sequences of time-domain values using as few Laplace-space model evaluations as possible.
- The comparison uses a common BEM modified Helmholtz solution with four boundary conditions and leads to recommendations for Laplace-space numerical approaches.
552 Governing Equation and Laplace Transform
The paper transforms a two-dimensional diffusion problem with mixed boundary conditions into a Laplace-space modified Helmholtz problem solved using BEM. Numerical inversion is required because the numerical model supplies only samples of the transformed solution.
- The BEM formulation considers a two-dimensional diffusion equation with constant diffusivity and mixed Dirichlet and Neumann boundary conditions.
- Homogeneous initial conditions are assumed, and the potential can represent quantities such as temperature or hydraulic head.
- Laplace transformation converts the transient diffusion equation into a homogeneous Yukawa or modified Helmholtz equation with q^2 = p/α.
- Boundary time and spatial behavior are separated in Laplace space, with arbitrary time behavior represented through convolution and multiplication of image functions.
- The inverse transform is defined by the Bromwich contour integral, whose convergence abscissa lies to the right of all singularities.
- Because the numerical model provides only samples of the transformed function, analytical inversion is unavailable and a numerical algorithm is required.
823 Numerical Inverse Laplace Transform Methods
The paper reviews five numerical inversion strategies spanning quadrature and functional-expansion approaches, emphasizing their assumptions, parameter sensitivity, precision behavior, and computational cost. Reuse of image-function evaluations is important for simulation-based applications.
- The methods divide into quadrature-based Fourier-series and Talbot approaches and functional expansions using complex- or real-valued basis functions.
- Numerical inverse Laplace transformation is ill-posed, and no single method is optimal for all temporal behaviors.
- Gaver-Stehfest Method: Gaver-Stehfest discretizes the Post-Widder formula, but its alternating, increasingly large coefficients can cause cancellation and precision demands.
- Schapery’s Method: Schapery expands deviations from steady state with exponential basis functions, requiring a steady-state solution in its stated formulation and trial-and-error parameter selection.
- Weeks’ Method: Weeks maps the image function into the unit circle for a Laguerre expansion, but its parameter rules depend on the maximum transformed time.
- Fixed Talbot Method: Fixed Talbot deforms the Bromwich contour and is simple to implement, but step changes can cause precision, stability, or convergence problems.
- Algorithm Properties Summary: Complex-p methods generally converge better than real-only methods, while the computational effort of expanding the image function was insignificant relative to the BEM solve.
2054 Numerical Comparison
The BEM tests compare five inverse Laplace transform methods across steady, sinusoidal, and step-change boundary behaviors, emphasizing accuracy, parameter sensitivity, and reuse of Laplace-space evaluations. Fourier-series inversion is generally robust, while shared evaluation sets can substantially reduce computational effort over time ranges.
- Test setup: Four test problems used a two-dimensional BEM domain with specified potentials at two ends, insulated remaining boundaries, and solutions sampled near one boundary.The domain was a 3 × 2 rectangle with homogeneous initial conditions; results were evaluated at x = 1/3, y = 1.
- Evaluation strategy: Using optimal p separately at each time was accurate but inefficient, especially when many time values were required.The study therefore examined reuse of the same Laplace-space function evaluations across times and sensitivity to non-optimal free parameters.
- Steady boundary conditions: 135 total ¯f(p) evaluations were required when nine terms per time produced accurate results across the tested time range.All methods performed well in this setting, although Schapery’s method was worst and the finite-difference approach required at least an order of magnitude more computational effort.
- Steady boundary conditions: One shared vector of nine ¯f(p) values accurately covered only the last log-cycle of times, whereas 51 terms removed most convergence problems.Grouping times by log-cycles required one-third as many model evaluations as the broader shared-vector comparison while remaining nearly as accurate.
- Sinusoidal boundary conditions: For sinusoidal boundary conditions, Schapery failed because no steady-state solution existed, while the other methods performed well with 19 terms across one log-cycle and 51 terms across at least two.The sinusoidal behavior was smooth and had singularities at p = ±4i; Fourier series, Talbot, and Weeks were nearly coincident for potential.
- Step-change boundary conditions: For step-change boundaries, Talbot failed for t < τ in double precision, while Weeks and Schapery performed worse than Fourier series even with N = 51.Discontinuous time behavior was difficult to invert accurately, although further parameter optimization could improve Weeks and Schapery.
- Numerical results summary: The Fourier series method was more robust to non-optimal p values, while the p sizes required by Fourier series and Weeks grew more slowly than those for fixed Talbot.Weeks was also sensitive to parameter choices, and the reported results used its rule-of-thumb parameters rather than further optimization.
2865 Conclusions
The review finds several viable inverse Laplace transform strategies, with trade-offs governed by time behavior, parameter robustness, implementation complexity, and the cost or availability of complex evaluations. Practical recommendations therefore depend on the simulation’s evaluation budget and modeling constraints.
- Method trade-offs: Robust methods require complex arithmetic and Laplace-space function evaluations, but handle broader time behaviors and remain relatively simple to implement.The Fourier series method handles a broader class of time behaviors, while fixed Talbot remains relatively simple to implement.
- Evaluation reuse: Grouping times by log cycle allows one Laplace-space function vector to serve multiple inversions, leading to much faster simulations despite added implementation complexity.This strategy is especially useful when many observations span several time log cycles.
- Method trade-offs: When function evaluations are expensive and numerical dispersion is acceptable, the Fourier series approach is most economical and automatic regarding free-parameter selection.The recommendation excludes problems requiring sharp-front resolution.
- Parameter selection: For a single time-domain value, Weeks’ or Piessen’s methods may benefit from optimized free parameters or asymptotic information, but Weeks’ method is not robust with non-optimal values.Selecting the optimum b parameter is far from automatic.
- Implementation constraints: Fixed Talbot is simple to code and automatic, but may fail for non-zero step-time behavior without extended precision.Its implementation simplicity is therefore paired with a numerical scope boundary.
- Implementation constraints: When complex-valued evaluations are unavailable, Schapery or Piessen’s methods can reuse the same p values across different times, unlike Gaver-Stehfest.This makes real-only methods suitable for settings limited to real matrix or special-function libraries.