Numerical inverse Laplace transform

One-step algorithm (invertlaplace)

mpmath.invertlaplace(f, t, **kwargs)

Computes the numerical inverse Laplace transform for a Laplace-space function at a given time. The function being evaluated is assumed to be a real-valued function of time.

The user must supply a Laplace-space function \(\bar{f}(p)\), and a desired time at which to estimate the time-domain solution \(f(t)\).

A few basic examples of Laplace-space functions with known inverses (see references [1,2]) :

\[\mathcal{L}\left\lbrace f(t) \right\rbrace=\bar{f}(p)\]

\[\mathcal{L}^{-1}\left\lbrace \bar{f}(p) \right\rbrace = f(t)\]

\[\bar{f}(p) = \frac{1}{(p+1)^2}\]

\[f(t) = t e^{-t}\]

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> tt = [0.001, 0.01, 0.1, 1, 10]
>>> fp = lambda p: 1/(p+1)**2
>>> ft = lambda t: t*exp(-t)
>>> ft(tt[0]),ft(tt[0])-invertlaplace(fp,tt[0],method='talbot')
(0.000999000499833375, 8.57923043561212e-20)
>>> ft(tt[1]),ft(tt[1])-invertlaplace(fp,tt[1],method='talbot')
(0.00990049833749168, 3.27007646698047e-19)
>>> ft(tt[2]),ft(tt[2])-invertlaplace(fp,tt[2],method='talbot')
(0.090483741803596, -1.75215800052168e-18)
>>> ft(tt[3]),ft(tt[3])-invertlaplace(fp,tt[3],method='talbot')
(0.367879441171442, 1.2428864009344e-17)
>>> ft(tt[4]),ft(tt[4])-invertlaplace(fp,tt[4],method='talbot')
(0.000453999297624849, 4.04513489306658e-20)

The methods also work for higher precision:

>>> mp.dps = 100; mp.pretty = True
>>> nstr(ft(tt[0]),15),nstr(ft(tt[0])-invertlaplace(fp,tt[0],method='talbot'),15)
('0.000999000499833375', '-4.96868310693356e-105')
>>> nstr(ft(tt[1]),15),nstr(ft(tt[1])-invertlaplace(fp,tt[1],method='talbot'),15)
('0.00990049833749168', '1.23032291513122e-104')

\[\bar{f}(p) = \frac{1}{p^2+1}\]

\[f(t) = \mathrm{J}_0(t)\]

>>> mp.dps = 15; mp.pretty = True
>>> fp = lambda p: 1/sqrt(p*p + 1)
>>> ft = lambda t: besselj(0,t)
>>> ft(tt[0]),ft(tt[0])-invertlaplace(fp,tt[0],method='dehoog')
(0.999999750000016, -6.09717765032273e-18)
>>> ft(tt[1]),ft(tt[1])-invertlaplace(fp,tt[1],method='dehoog')
(0.99997500015625, -5.61756281076169e-17)

\[\bar{f}(p) = \frac{\log p}{p}\]

\[f(t) = -\gamma -\log t\]

>>> mp.dps = 15; mp.pretty = True
>>> fp = lambda p: log(p)/p
>>> ft = lambda t: -euler-log(t)
>>> ft(tt[0]),ft(tt[0])-invertlaplace(fp,tt[0],method='stehfest')
(6.3305396140806, -1.92126634837863e-16)
>>> ft(tt[1]),ft(tt[1])-invertlaplace(fp,tt[1],method='stehfest')
(4.02795452108656, -4.81486093200704e-16)

Options

invertlaplace() recognizes the following optional keywords valid for all methods:

method

Chooses numerical inverse Laplace transform algorithm (described below).

degree

Number of terms used in the approximation

Algorithms

Mpmath implements four numerical inverse Laplace transform algorithms, attributed to: Talbot, Stehfest, and de Hoog, Knight and Stokes. These can be selected by using method=’talbot’, method=’stehfest’, method=’dehoog’ or method=’cohen’ or by passing the classes method=FixedTalbot, method=Stehfest, method=deHoog, or method=Cohen. The functions invlaptalbot(), invlapstehfest(), invlapdehoog(), and invlapcohen() are also available as shortcuts.

All four algorithms implement a heuristic balance between the requested precision and the precision used internally for the calculations. This has been tuned for a typical exponentially decaying function and precision up to few hundred decimal digits.

The Laplace transform converts the variable time (i.e., along a line) into a parameter given by the right half of the complex \(p\)-plane. Singularities, poles, and branch cuts in the complex \(p\)-plane contain all the information regarding the time behavior of the corresponding function. Any numerical method must therefore sample \(p\)-plane “close enough” to the singularities to accurately characterize them, while not getting too close to have catastrophic cancellation, overflow, or underflow issues. Most significantly, if one or more of the singularities in the \(p\)-plane is not on the left side of the Bromwich contour, its effects will be left out of the computed solution, and the answer will be completely wrong.

Talbot

The fixed Talbot method is high accuracy and fast, but the method can catastrophically fail for certain classes of time-domain behavior, including a Heaviside step function for positive time (e.g., \(H(t-2)\)), or some oscillatory behaviors. The Talbot method usually has adjustable parameters, but the “fixed” variety implemented here does not. This method deforms the Bromwich integral contour in the shape of a parabola towards \(-\infty\), which leads to problems when the solution has a decaying exponential in it (e.g., a Heaviside step function is equivalent to multiplying by a decaying exponential in Laplace space).

Stehfest

The Stehfest algorithm only uses abscissa along the real axis of the complex \(p\)-plane to estimate the time-domain function. Oscillatory time-domain functions have poles away from the real axis, so this method does not work well with oscillatory functions, especially high-frequency ones. This method also depends on summation of terms in a series that grows very large, and will have catastrophic cancellation during summation if the working precision is too low.

de Hoog et al.

The de Hoog, Knight, and Stokes method is essentially a Fourier-series quadrature-type approximation to the Bromwich contour integral, with non-linear series acceleration and an analytical expression for the remainder term. This method is typically one of the most robust. This method also involves the greatest amount of overhead, so it is typically the slowest of the four methods at high precision.

Cohen

The Cohen method is a trapezoidal rule approximation to the Bromwich contour integral, with linear acceleration for alternating series. This method is as robust as the de Hoog et al method and the fastest of the four methods at high precision, and is therefore the default method.

Singularities

All numerical inverse Laplace transform methods have problems at large time when the Laplace-space function has poles, singularities, or branch cuts to the right of the origin in the complex plane. For simple poles in \(\bar{f}(p)\) at the \(p\)-plane origin, the time function is constant in time (e.g., \(\mathcal{L}\left\lbrace 1 \right\rbrace=1/p\) has a pole at \(p=0\)). A pole in \(\bar{f}(p)\) to the left of the origin is a decreasing function of time (e.g., \(\mathcal{L}\left\lbrace e^{-t/2} \right\rbrace=1/(p+1/2)\) has a pole at \(p=-1/2\)), and a pole to the right of the origin leads to an increasing function in time (e.g., \(\mathcal{L}\left\lbrace t e^{t/4} \right\rbrace = 1/(p-1/4)^2\) has a pole at \(p=1/4\)). When singularities occur off the real \(p\) axis, the time-domain function is oscillatory. For example \(\mathcal{L}\left\lbrace \mathrm{J}_0(t) \right\rbrace=1/\sqrt{p^2+1}\) has a branch cut starting at \(p=j=\sqrt{-1}\) and is a decaying oscillatory function, This range of behaviors is illustrated in Duffy [3] Figure 4.10.4, p. 228.

In general as \(p \rightarrow \infty\) \(t \rightarrow 0\) and vice-versa. All numerical inverse Laplace transform methods require their abscissa to shift closer to the origin for larger times. If the abscissa shift left of the rightmost singularity in the Laplace domain, the answer will be completely wrong (the effect of singularities to the right of the Bromwich contour are not included in the results).

For example, the following exponentially growing function has a pole at \(p=3\):

\[\bar{f}(p)=\frac{1}{p^2-9}\]

\[f(t)=\frac{1}{3}\sinh 3t\]

>>> mp.dps = 15; mp.pretty = True
>>> fp = lambda p: 1/(p*p-9)
>>> ft = lambda t: sinh(3*t)/3
>>> tt = [0.01,0.1,1.0,10.0]
>>> ft(tt[0]),invertlaplace(fp,tt[0],method='talbot')
(0.0100015000675014, 0.0100015000675014)
>>> ft(tt[1]),invertlaplace(fp,tt[1],method='talbot')
(0.101506764482381, 0.101506764482381)
>>> ft(tt[2]),invertlaplace(fp,tt[2],method='talbot')
(3.33929164246997, 3.33929164246997)
>>> ft(tt[3]),invertlaplace(fp,tt[3],method='talbot')
(1781079096920.74, -1.61331069624091e-14)

References

1. [DLMF] section 1.14 (http://dlmf.nist.gov/1.14T4)

2. Cohen, A.M. (2007). Numerical Methods for Laplace Transform Inversion, Springer.

3. Duffy, D.G. (1998). Advanced Engineering Mathematics, CRC Press.

Numerical Inverse Laplace Transform Reviews

1. Bellman, R., R.E. Kalaba, J.A. Lockett (1966). Numerical inversion of the Laplace transform: Applications to Biology, Economics, Engineering, and Physics. Elsevier.

2. Davies, B., B. Martin (1979). Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of Computational Physics 33:1-32, http://dx.doi.org/10.1016/0021-9991(79)90025-1

3. Duffy, D.G. (1993). On the numerical inversion of Laplace transforms: Comparison of three new methods on characteristic problems from applications. ACM Transactions on Mathematical Software 19(3):333-359, http://dx.doi.org/10.1145/155743.155788

4. Kuhlman, K.L., (2013). Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches, Numerical Algorithms, 63(2):339-355. http://dx.doi.org/10.1007/s11075-012-9625-3

Specific algorithms

Fixed Talbot algorithm

class mpmath.calculus.inverselaplace.FixedTalbot(ctx)

calc_laplace_parameter(t, **kwargs)

The “fixed” Talbot method deforms the Bromwich contour towards \(-\infty\) in the shape of a parabola. Traditionally the Talbot algorithm has adjustable parameters, but the “fixed” version does not. The \(r\) parameter could be passed in as a parameter, if you want to override the default given by (Abate & Valko, 2004).

The Laplace parameter is sampled along a parabola opening along the negative imaginary axis, with the base of the parabola along the real axis at \(p=\frac{r}{t_\mathrm{max}}\). As the number of terms used in the approximation (degree) grows, the abscissa required for function evaluation tend towards \(-\infty\), requiring high precision to prevent overflow. If any poles, branch cuts or other singularities exist such that the deformed Bromwich contour lies to the left of the singularity, the method will fail.

Optional arguments

calc_laplace_parameter recognizes the following keywords

tmax

maximum time associated with vector of times (typically just the time requested)

degree

integer order of approximation (M = number of terms)

r

abscissa for \(p_0\) (otherwise computed using rule of thumb \(2M/5\))

The working precision will be increased according to a rule of thumb. If ‘degree’ is not specified, the working precision and degree are chosen to hopefully achieve the dps of the calling context. If ‘degree’ is specified, the working precision is chosen to achieve maximum resulting precision for the specified degree.

\[p_0=\frac{r}{t}\]

\[p_i=\frac{i r \pi}{Mt_\mathrm{max}}\left[\cot\left( \frac{i\pi}{M}\right) + j \right] \qquad 1\le i <M\]

where \(j=\sqrt{-1}\), \(r=2M/5\), and \(t_\mathrm{max}\) is the maximum specified time.

calc_time_domain_solution(fp, t, manual_prec=False)

The fixed Talbot time-domain solution is computed from the Laplace-space function evaluations using

\[f(t,M)=\frac{2}{5t}\sum_{k=0}^{M-1}\Re \left[ \gamma_k \bar{f}(p_k)\right]\]

where

\[\gamma_0 = \frac{1}{2}e^{r}\bar{f}(p_0)\]

\[\gamma_k = e^{tp_k}\left\lbrace 1 + \frac{jk\pi}{M}\left[1 + \cot \left( \frac{k \pi}{M} \right)^2 \right] - j\cot\left( \frac{k \pi}{M}\right)\right \rbrace \qquad 1\le k<M.\]

Again, \(j=\sqrt{-1}\).

Before calling this function, call calc_laplace_parameter to set the parameters and compute the required coefficients.

References

1. Abate, J., P. Valko (2004). Multi-precision Laplace transform inversion. International Journal for Numerical Methods in Engineering 60:979-993, http://dx.doi.org/10.1002/nme.995

2. Talbot, A. (1979). The accurate numerical inversion of Laplace transforms. IMA Journal of Applied Mathematics 23(1):97, http://dx.doi.org/10.1093/imamat/23.1.97

Gaver-Stehfest algorithm

class mpmath.calculus.inverselaplace.Stehfest(ctx)

calc_laplace_parameter(t, **kwargs)

The Gaver-Stehfest method is a discrete approximation of the Widder-Post inversion algorithm, rather than a direct approximation of the Bromwich contour integral.

The method abscissa along the real axis, and therefore has issues inverting oscillatory functions (which have poles in pairs away from the real axis).

The working precision will be increased according to a rule of thumb. If ‘degree’ is not specified, the working precision and degree are chosen to hopefully achieve the dps of the calling context. If ‘degree’ is specified, the working precision is chosen to achieve maximum resulting precision for the specified degree.

\[p_k = \frac{k \log 2}{t} \qquad 1 \le k \le M\]

calc_time_domain_solution(fp, t, manual_prec=False)

Compute time-domain Stehfest algorithm solution.

\[f(t,M) = \frac{\log 2}{t} \sum_{k=1}^{M} V_k \bar{f}\left( p_k \right)\]

where

\[V_k = (-1)^{k + N/2} \sum^{\min(k,N/2)}_{i=\lfloor(k+1)/2 \rfloor} \frac{i^{\frac{N}{2}}(2i)!}{\left(\frac{N}{2}-i \right)! \, i! \, \left(i-1 \right)! \, \left(k-i\right)! \, \left(2i-k \right)!}\]

As the degree increases, the abscissa (\(p_k\)) only increase linearly towards \(\infty\), but the Stehfest coefficients (\(V_k\)) alternate in sign and increase rapidly in sign, requiring high precision to prevent overflow or loss of significance when evaluating the sum.

References

1. Widder, D. (1941). The Laplace Transform. Princeton.

2. Stehfest, H. (1970). Algorithm 368: numerical inversion of Laplace transforms. Communications of the ACM 13(1):47-49, http://dx.doi.org/10.1145/361953.361969

de Hoog, Knight & Stokes algorithm

class mpmath.calculus.inverselaplace.deHoog(ctx)

calc_laplace_parameter(t, **kwargs)

the de Hoog, Knight & Stokes algorithm is an accelerated form of the Fourier series numerical inverse Laplace transform algorithms.

\[p_k = \gamma + \frac{jk}{T} \qquad 0 \le k < 2M+1\]

where

\[\gamma = \alpha - \frac{\log \mathrm{tol}}{2T},\]

\(j=\sqrt{-1}\), \(T = 2t_\mathrm{max}\) is a scaled time, \(\alpha=10^{-\mathrm{dps\_goal}}\) is the real part of the rightmost pole or singularity, which is chosen based on the desired accuracy (assuming the rightmost singularity is 0), and \(\mathrm{tol}=10\alpha\) is the desired tolerance, which is chosen in relation to \(\alpha\).`

When increasing the degree, the abscissa increase towards \(j\infty\), but more slowly than the fixed Talbot algorithm. The de Hoog et al. algorithm typically does better with oscillatory functions of time, and less well-behaved functions. The method tends to be slower than the Talbot and Stehfest algorithsm, especially so at very high precision (e.g., \(>500\) digits precision).

calc_time_domain_solution(fp, t, manual_prec=False)

Calculate time-domain solution for de Hoog, Knight & Stokes algorithm.

The un-accelerated Fourier series approach is:

\[f(t,2M+1) = \frac{e^{\gamma t}}{T} \sum_{k=0}^{2M}{}^{'} \Re\left[\bar{f}\left( p_k \right) e^{i\pi t/T} \right],\]

where the prime on the summation indicates the first term is halved.

This simplistic approach requires so many function evaluations that it is not practical. Non-linear acceleration is accomplished via Pade-approximation and an analytic expression for the remainder of the continued fraction. See the original paper (reference 2 below) a detailed description of the numerical approach.

References

1. Davies, B. (2005). Integral Transforms and their Applications, Third Edition. Springer.

2. de Hoog, F., J. Knight, A. Stokes (1982). An improved method for numerical inversion of Laplace transforms. SIAM Journal of Scientific and Statistical Computing 3:357-366, http://dx.doi.org/10.1137/0903022

Cohen acceleration algorithm

class mpmath.calculus.inverselaplace.Cohen(ctx)

calc_laplace_parameter(t, **kwargs)

The Cohen algorithm accelerates the convergence of the nearly alternating series resulting from the application of the trapezoidal rule to the Bromwich contour inversion integral.

\[p_k = \frac{\gamma}{2 t} + \frac{\pi i k}{t} \qquad 0 \le k < M\]

where

\[\gamma = \frac{2}{3} (d + \log(10) + \log(2 t)),\]

\(d = \mathrm{dps\_goal}\), which is chosen based on the desired accuracy using the method developed in [1] to improve numerical stability. The Cohen algorithm shows robustness similar to the de Hoog et al. algorithm, but it is faster than the fixed Talbot algorithm.

Optional arguments

degree

integer order of the approximation (M = number of terms)

alpha

abscissa for \(p_0\) (controls the discretization error)

The working precision will be increased according to a rule of thumb. If ‘degree’ is not specified, the working precision and degree are chosen to hopefully achieve the dps of the calling context. If ‘degree’ is specified, the working precision is chosen to achieve maximum resulting precision for the specified degree.

References

1. P. Glasserman, J. Ruiz-Mata (2006). Computing the credit loss distribution in the Gaussian copula model: a comparison of methods. Journal of Credit Risk 2(4):33-66, 10.21314/JCR.2006.057

calc_time_domain_solution(fp, t, manual_prec=False)

Calculate time-domain solution for Cohen algorithm.

The accelerated nearly alternating series is:

\[f(t, M) = \frac{e^{\gamma / 2}}{t} \left[\frac{1}{2} \Re\left(\bar{f}\left(\frac{\gamma}{2t}\right) \right) - \sum_{k=0}^{M-1}\frac{c_{M,k}}{d_M}\Re\left(\bar{f} \left(\frac{\gamma + 2(k+1) \pi i}{2t}\right)\right)\right],\]

where coefficients \(\frac{c_{M, k}}{d_M}\) are described in [1].

1. H. Cohen, F. Rodriguez Villegas, D. Zagier (2000). Convergence acceleration of alternating series. Experiment. Math 9(1):3-12

Manual approach

It is possible and sometimes beneficial to re-create some of the functionality in invertlaplace. This could be used to compute the Laplace-space function evaluations in a different way. For example, the Laplace-space function evaluations could be the result of a quadrature or sum, solution to a system of ordinary differential equations, or possibly computed in parallel from some external library or function call.

A trivial example showing the process (which could be implemented using the existing interface):

>>> from mpmath import *
>>> myTalbot = calculus.inverselaplace.FixedTalbot(mp)
>>> t = convert(0.25)
>>> myTalbot.calc_laplace_parameter(t)
>>> fp = lambda p: 1/(p + 1) - 1/(p + 1000)
>>> ft = lambda t: exp(-t) - exp(-1000*t)
>>> fpvec = [fp(p) for p in myTalbot.p]
>>> ft(t)-myTalbot.calc_time_domain_solution(fpvec,t,manual_prec=True)
mpf('1.928300179528890061756872185e-21')

This manual approach is also useful to look at the Laplace parameter, order, or working precision which were computed.

>>> myTalbot.degree
34

Credit

The numerical inverse Laplace transform functionality was contributed to mpmath by Kristopher L. Kuhlman in 2017. The Cohen method was contributed to mpmath by Guillermo Navas-Palencia in 2022.