Multiresolution Algorithm

This package is an implementation of the standard and hybrid multiresolution algorithm as proposed in Computation of the Distribution of the Absorption Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions by Turin, Magalang, Aguilar, Colombani, Sanchez-Taltavull, and Gatto¹

Installation

Install the package from PyPI:

pip install multires

Usage

To run a simulation, begin by first choosing the scheme that you want:

1. smra for the standard MRA
2. hmra for the hybrid MRA

By default, either simulation scheme will produce a first passage time, rather than a full trajectory.

SMRA

The smra function has the following syntax:

smra(v, D, r, x0, eps, kmax = 30, kmin = 25, tmax_in = 10**5, trajectory = False)

Here is a description of each argument:

• v : Float. Drift constant.
• D : Float. Diffusion constant. Must be positive.
• r : Float. Resetting constant. Must be non-negative.
• x0 : Float. Initial position. Must be from $0 \leq x_0 < 1$
• eps : Float. Error threshold. Must be positive.
• kmax : Integer, optional. Simulation parameter to determine the maximum resolution $k$ considered before stopping the simulation. The default is 30.
• kmin : Integer, optional. Simulation parameter to determine the minimum resolution $k$ considered before computing the next resetting interval given that the stopping condition has not been reached prior. The default is 25.
• tmax_in : The time at the endpoint of the Brownian trajectory when r = 0. The default is 10**5.
• trajectory : Boolean, optional. If True, function returns the full arrays of position and time of the trajectory. If False, function returns only the first passage time. The default is False.

Running the function will yield either three or four outputs, depending on the argument trajectory:

If trajectory is False (by default):

• T : Float. First passage time.
• k : Integer. Resolution at end of simulation
• has_reached_eps : Boolean. If False, simulation has failed to reach the error threshold before reaching k = kmax

If trajectory is True:

• x_arr : Array. Positions of the Brownian trajectory at the end of the simulation.
• t_arr : Array. Times of the Brownian trajectory at the end of the simulation.
• k : Integer. Resolution at end of simulation
• has_reached_eps : Boolean. If False, simulation has failed to reach the error threshold before reaching k = kmax

This is an example of the usage of the smra function:

For a single first passage time:

import multires as mra

T, k, has_reached_eps = mra.smra(-10**(-3), 10**(-4), (1/3)*(1/365), 0.8, 10**(-4))

Sample output:

>>> T
np.float64(370.1612953761)
>>> k
24
>>> has_reached_eps
True

For a trajectory:

import multires as mra

xvalues, tvalues, k, has_reached_eps = mra.smra(-10**(-3), 10**(-4), (1/3)*(1/365), 0.8, 10**(-4), trajectory = True)

Sample output:

>>> xvalues
array([ 0.8       ,  0.80002268,  0.8000829 , ..., -0.03714226, -0.03716605, -0.03718239])
>>> tvalues
 array([0.00000000e+00, 5.02717020e-05, 1.00543404e-04, ..., 8.43419103e+02, 8.43419153e+02, 8.43419204e+02]),
>>> k
24
>>> has_reached_eps
True

HMRA

The hmra function has the following syntax:

hmra(v, D, r, x0, eps, eul_dt, mul_thresh = 0.1, kmax = 30, tmax_in = 10**5, trajectory = False)

Here is a description of each argument:

• v : Float. Drift constant.
• D : Float. Diffusion constant. Must be positive.
• r : Float. Resetting constant. Must be non-negative.
• x0 : Float. Initial position. Must be from $0 \leq x_0 < 1$
• eps : Float. Error threshold. Must be positive.
• eul_dt : Float. Euler-Maruyama simulation timestep.
• mul_thresh : Float, optional. Position threshold, the multiresolution algorithm will begin to compute if the Euler trajectory is below this threshold. The default is 0.1.
• kmax : Integer, optional. Simulation parameter to determine the maximum resolution $k$ considered before stopping the simulation. The default is 30.
• tmax_in : The time at the endpoint of the Brownian trajectory when r = 0. The default is 10**5.
• trajectory : Boolean, optional. If True, function returns the full arrays of position and time of the trajectory. If False, function returns only the first passage time. The default is False.

Running the function will yield either three or four outputs, depending on the argument trajectory:

If trajectory is False (by default):

• T : Float. First passage time.
• k : Integer. Resolution at end of simulation
• has_reached_eps : Boolean. If False, simulation has failed to reach the error threshold before reaching k = kmax

If trajectory is True:

• x_arr : Array. Positions of the Brownian trajectory at the end of the simulation.
• t_arr : Array. Times of the Brownian trajectory at the end of the simulation.
• k : Integer. Resolution at end of simulation
• has_reached_eps : Boolean. If False, simulation has failed to reach the error threshold before reaching k = kmax

This is an example of the usage of the smra function:

For a single first passage time:

import multires as mra

T, k, has_reached_eps = mra.hmra(-10**(-3), 10**(-4), (1/3)*(1/365), 0.8, 10**(-4), 0.1)

Sample output:

>>> T
np.float64(638.0523437500765)
>>> k
10
>>> has_reached_eps
True

For a trajectory:

import multires as mra

xvalues, tvalues, k, has_reached_eps = mra.hmra(-10**(-3), 10**(-4), (1/3)*(1/365), 0.8, 10**(-4), 0.1, trajectory = True)

Sample output:

>>> xvalues
array([ 0.8       ,  0.79679549,  0.80113966, ..., -0.00413409, -0.0042602 , -0.00409163])
>>> tvalues
 array([0.00000000e+00, 1.00000000e-01, 2.00000000e-01, ..., 7.48199805e+02, 7.48199902e+02, 7.48200000e+02]),
>>> k
10
>>> has_reached_eps
True

Auxiliary Functions

This package also contains auxiliary functions:

1. increase_resolution increases the resolution of a trajectory from level $k-1$ to $k$ given that the input array has length $2^{k-1}+1$.
2. euler is an implementation of the Euler-Maruyama algorithm which outputs a full trajectory and is required for the HMRA.

Written with StackEdit.

1. Turin, R., Magalang, J., Aguilar, J., Colombani, L., Sanchez-Taltavull, D., & Gatto, R. (2024). Computation of the distribution of the absorption time of the drifted diffusion with stochastic resetting and mixed boundary conditions (arXiv:2311.03939). arXiv. https://doi.org/10.48550/arXiv.2311.03939 ↩