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Oscillatory integrals arise in models of a wide range of physical applications, from acoustics to quantum mechanics. `PathFinder` is a Matlab/Octave package for efficient evaluation of oscillatory integrals of the form
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\begin{equation}\label{eq:I}
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I = \int_a^b f(x) \exp(\mathrm{i}\omega g(x))~\mathrm{d}x,
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I = \int_a^b f(z) \exp(\mathrm{i}\omega g(z))~\mathrm{d}z,
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\end{equation}
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where the endpoints $a$ and $b$ can be complex-valued, even infinite; $\omega>0$ is the frequency parameter, $f$ is an entire function and $g$ is a polynomial phase function. The syntax is simple:
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where the endpoints $a$ and $b$ can be complex-valued, even infinite; $\omega>0$ determines the angular frequency, $f$ is a non-oscillatory entire function and $g$ is a polynomial phase function. The syntax is simple:
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```matlab
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I = PathFinder(a, b, f, gCoeffs, omega, N);
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```
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Here, `f` is a function handle representing $f(x)$, `gCoeffs` is a vector of coefficients of $g$, `omega` is the frequency parameter $\omega$ and `N` is a parameter that controls the degree of approximation.
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Here, `f` is a function handle representing $f(z)$, `gCoeffs` is a vector of coefficients of $g$, `omega` is the frequency parameter $\omega$ and `N` is a parameter that controls the degree of approximation.
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`PathFinder` is the first black-box method that can evaluate \eqref{eq:I} accurately, robustly and efficiently for any frequency $\omega$. It will be useful across a range of scientific disciplines, for problems that were previously too computationally expensive or too mathematically challenging to solve.
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`PathFinder` is the first black-box software that can evaluate \eqref{eq:I} accurately, robustly and efficiently for any frequency $\omega$. It will be useful across many scientific disciplines, for problems that were previously too computationally expensive or too mathematically challenging to solve.
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# Statement of need
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Based on the method of Numerical Steepest Descent [@HuVa:06], `PathFinder` is an implementation of the algorithm described in @PathFinderPaper, where an earlier version of the code was used to produce numerical experiments. Since these experiments, much of the code has been rewritten in C, interfacing with Matlab/Octave via MEX (Matlab executable) functions. These files can be easily compiled using a single script.
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Based on the method of Numerical Steepest Descent [@HuVa:06], `PathFinder` is an implementation of the algorithm described in @PathFinderPaper, where an earlier version of the code was used to produce numerical experiments. Since these experiments, much of the code has been rewritten in C, interfacing with Matlab/Octave via MEX (Matlab executable) functions. These are easily compiled using a single script.
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### Ease of use
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Standard quadrature rules (midpoint rule, Gauss quadrature, etc) are easy to use, and many open-source implementations are available. However, when applied to \eqref{eq:I}, such methods become prohibitively inefficient for large $\omega$.
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On the other hand, several methods exist for the efficient evaluation of oscillatory (large $\omega$) integrals such as \eqref{eq:I}, a thorough review is given in @DeHuIs:18. However, applying these methods often requires an expert understanding of the process and a detailed analysis of the integral, making such methods inaccessible to non-mathematicians. Even with the necessary mathematical understanding, models may require hundreds or thousands of oscillatory integrals to be evaluated, making detailed analysis of each integral highly challenging or impossible.
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On the other hand, several methods exist for the efficient evaluation of oscillatory (large $\omega$) integrals such as \eqref{eq:I}; a thorough review is given in @DeHuIs:18. However, applying these methods often requires an expert understanding of the process and a detailed analysis of the integral, making such methods inaccessible to non-mathematicians. Even with the necessary mathematical understanding, models may require hundreds or thousands of oscillatory integrals to be evaluated, making detailed analysis of each integral highly challenging or impossible.
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Despite being based on complex mathematics, `PathFinder` can be easily used by non-mathematicians. The user must simply understand the definitions of the components of \eqref{eq:I}.
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### Use in academic research
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We now describe an important problem class that can be easily modelled using `PathFinder`, and new research that has been stimulated as a result.
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In many physical models, interesting physical phenomena occur in the presence of *coalescing saddle points*, (see e.g. @PathFinderPaper for a definition). Examples include chemical reactions, rainbows, twinkling starlight, ultrasound pulses, and focusing of sunlight by rippling water [@DLMF, Section 36.14].
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In many physical models, interesting physical phenomena occur in the presence of *coalescing saddle points*, (see e.g. @PathFinderPaper for a definition). Examples include chemical reactions, rainbows, twinkling starlight, ultrasound pulses, and focusing of sunlight by rippling water [@DLMF, 36.14].
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Coalescing saddle points can cause steepest descent methods to break down, even in simple cases where $g$ is a cubic polynomial [@HuJuLe:19]. Fortunately, `PathFinder` is robust for any number of coalescing saddle points. This is demonstrated in Figures \ref{fig:pearcey} and \ref{fig:swallowtail}, where `PathFinder` has been used to model well-known optics problems with coalescing saddle points.
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Coalescing saddle points can cause steepest descent methods to break down, even in simple cases where $g$ is a cubic polynomial [@HuJuLe:19]. By design, `PathFinder` is robust for any number of coalescing saddle points. This is demonstrated in Figures \ref{fig:pearcey} and \ref{fig:swallowtail}, where `PathFinder` has been used to model well-known optics problems with coalescing saddle points. Here, each point $(x_1,x_2)$ requires a separate evaluation of \eqref{eq:I} and thus a separate call to `PathFinder`.
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![PathFinder approximation to Pearcey/Cusp Catastrophe integrals [@Pe:46], which contain coalescing saddle points.\label{fig:pearcey}](../../examples/cusp.png){width=90%}
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![PathFinder approximation to Swallowtail Catastrophe integrals [@Ar:81], which contain many coalescing saddle points.\label{fig:swallowtail}](../../examples/swallowtail.png)
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In @HeOcSm:19 a new technique was described for the construction of integral solutions to the *Parabolic Wave Equation*, typically of the form \eqref{eq:I}. Via a simple change of variables, these solutions could be transformed into meaningful solutions of the Helmholtz equation. Plots of these solutions were provided using `cuspint` ([described below](#comparison-with-other-software)) in the cases that were "not too difficult", but some were excluded, for example, $A_{32}$ of equation (32) therein. This can now be easily produced using `PathFinder`, as shown in Figure \ref{fig:pwe}.
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In @HeOcSm:19 a new technique was described for the construction of integral solutions to the *Parabolic Wave Equation*, typically with coalescing saddle points. Plots of some solutions were provided using `cuspint` ([described below](#comparison-with-other-software)) in the cases that were "not too difficult", but others were excluded, for example, $A_{32}$ of equation (32) therein. This omission can now be easily produced using `PathFinder`, as shown in Figure \ref{fig:pwe}.
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{width=80%}
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The ideas of @HeOcSm:19 were combined with `PathFinder` in @OcTeHeGi:24 and applied to the famous (unsolved) Popov inflection point problem [@Popov79]. Here `PathFinder` was used to visualise a wavefield with caustic behaviour close to a curve with an inflection point (as in Figure \ref{fig:inflection}), and provided numerical validation of the asymptotic approximations therein.
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The ideas of @HeOcSm:19 were combined with `PathFinder` in @OcTeHeGi:24 and applied to the famous (unsolved) inflection point problem of @Popov79. Via a simple change of variables, these solutions to the Parabolic Wave Equation could be transformed into meaningful solutions of the Helmholtz equation. Here `PathFinder` was used to visualise a wavefield with caustic behaviour close to a curve with an inflection point (as in Figure \ref{fig:inflection}) and provided numerical validation of the asymptotic approximations therein.
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### Comparison with other software
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As was explained in @PathFinderPaper, to the author's best knowledge, the only other software packages that can efficiently evaluate oscillatory integrals are Mathematica's `NIntegrate` function, and the Fortan `cuspint` package of @KiCoHo:00. We now briefly compare these packages with`PathFinder`.
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To the author's best knowledge, the only other software packages that can efficiently evaluate oscillatory integrals are Mathematica's `NIntegrate` function, when used with the `LevinRule` option [@NIntegrate], and the Fortan `cuspint` package [@KiCoHo:00]. We now briefly compare these packages against`PathFinder`.
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One advantage of Mathematica's `NIntegrate` is that the oscillatory component does not always need to be separated explicitly, as in \eqref{eq:I}. There are three drawbacks when compared to `PathFinder`:
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An advantage of Mathematica's `NIntegrate` is that the oscillatory component does not always need to be factored explicitly; Mathematica does this symbolically. However, there are three drawbacks when compared to `PathFinder`:
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- Based on experiments, `NIntegrate` does not appear to have a frequency-independent cost for polynomial phase functions.
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-`NIntegrate` does not work in general for an unbounded contour with complex endpoints.
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-`NIntegrate` is not open source; the code cannot be seen or modified, and one must acquire a license to use it.
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-`NIntegrate` only appears to have a frequency-independent cost for monomial phase functions, i.e., $g(x)=x^\rho$ for $\rho\in\mathbb{N}$; a much narrower class than `PathFinder`, which can evaluate \eqref{eq:I} for any polynomial phase $g$.
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- It appears that `NIntegrate` requires the integration range to be finite, whereas `PathFinder` can evaluate integrals on an unbounded contour.
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The `cuspint` package is similar to `PathFinder` in that it is also based on steepest descent contour deformation. There are two drawbacks when compared with PathFinder:
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- The problem class is restricted to \eqref{eq:I} when $[a,b]=\mathbb{R}$. Therefore, it may be used to model the catastrophe integrals of Figures \ref{fig:pearcey} and \ref{fig:swallowtail}.
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-`cuspint` can experience "violent" [@KiCoHo:00,\S2] exponential growth in certain regions of the complex plane, which can lead to inaccurate results. This is because, unlike `PathFinder`, it does not attempt a highly accurate approximation of the steepest descent contours.
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- The problem class is restricted to \eqref{eq:I} when $(a,b)=\mathbb{R}$. Therefore, it may be used to model the catastrophe integrals of Figures \ref{fig:pearcey} and \ref{fig:swallowtail}, but not those of Figure \ref{fig:pwe} and \ref{fig:inflection}.
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-`cuspint` can experience "violent" exponential growth [@KiCoHo:00, Section 2], which can lead to inaccurate results. This is because, unlike `PathFinder`, it does not attempt a highly accurate approximation of the steepest descent contours.
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In summary, `PathFinder` is the only software package that can be applied in general to \eqref{eq:I} when $a$ and/or $b$ are infinite and complex, as in the problems visualised in Figures \ref{fig:pwe}-\ref{fig:inflection}.
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In summary, `PathFinder` is the only software package that can be applied in general to the class \eqref{eq:I}, for problems such as those visualised in Figures \ref{fig:pwe}-\ref{fig:inflection}.
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# Acknowledgments
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I am very grateful for the guidance of Daan Huybrechs and David Hewett throughout the development of this software. I am also grateful for financial support from KU Leuven project C14/15/05 and EPSRC projects EP/S01375X/1, EP/V053868/1.
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Some of the code in `PathFinder` relies on other projects. I am grateful to Dimas Aryo whose code is used for the Dijkstra shortest path algorithm. I am also grateful to Dirk Laurie and Walter Gautschi for writing the Golub-Welsch algorithm used to generate Gaussian quadrature rules.
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Some of the code in `PathFinder` relies on other projects. I am grateful to @DAryo whose code is used for the Dijkstra shortest path algorithm. I am also grateful to Dirk Laurie and Walter Gautschi for writing the Golub-Welsch algorithm used to generate Gaussian quadrature rules.
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