Inverse functions for Monte Carlo simulations with applications to hitting time distributions

Random sampling is a ubiquitous tool in simulations and modeling in a variety of applications. There are efficient algorithms for these for several known distributions, but in general, one must resort to computing or approximating the inverse to the distribution to generate random samples, given a random number generator for a uniform distribution. In certain physical and biomedical applications with which we have been particularly concerned, it has proven to be more efficient to provide random times for a walk of a fixed length, rather than the conventional random step lengths in a given time step for the walker. For these, the hitting-time distributions which have to be sampled have been computed, and proved to be complicated expressions with no efficient method to compute the inverse. In this paper, we explore a well known probability (the F-ratio distribution) - whose inverses are efficiently computable - as an alternative to generating look-up tables and interpolations to obtain the required time samples. We find that this distribution approximates the hitting-time distribution well, and report on error measures for both the approximation to the desired, and the error in the generated time samples. Future Monte Carlo simulations in a number of fields of application may benefit from methods such as we report here.

[1]  M. Freidlin,et al.  Functional Integration and Partial Differential Equations. (AM-109), Volume 109 , 1985 .

[2]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[3]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. , 1989, Microvascular research.

[4]  Karl K. Sabelfeld,et al.  Random walk on spheres method for solving drift-diffusion problems , 2016, Monte Carlo Methods Appl..

[5]  N. Henze,et al.  Cramér–von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation , 2017 .

[6]  Raghu Raghavan,et al.  Delivering therapy to target: improving the odds for successful drug development. , 2016, Therapeutic delivery.

[7]  P. Basser.  Interstitial pressure, volume, and flow during infusion into brain tissue. , 1992, Microvascular research.

[8]  Herbert Solomon,et al.  Approximations to Density Functions Using Pearson Curves , 1978 .

[9]  R. Raghavan.  Hitting time distributions for efficient simulations of drift‐diffusion processes , 2020, Engineering Reports.

[10]  Raghu Raghavan,et al.  Predictive models for pressure-driven fluid infusions into brain parenchyma , 2011, Physics in medicine and biology.

[11]  C. E. Davidson,et al.  Probability theory for 3-layer remote sensing radiative transfer model: univariate case. , 2012, Optics express.

[12]  R K Jain,et al.  Transport of fluid and macromolecules in tumors. II. Role of heterogeneous perfusion and lymphatics. , 1990, Microvascular research.

[13]  P F Morrison,et al.  High-flow microinfusion: tissue penetration and pharmacodynamics. , 1994, The American journal of physiology.

[14]  Raghu Raghavan,et al.  Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications , 2020, Engineering Reports.