This project is an implementation of the Efficient Jet Marching method for solving for the quasi-potential of 2 dimensional stochastic differential equations,
dX(t)= b(X(t))dt + ε dW(t),
where W is a 2-dimensional standard Brownian motion ε is a small parameter, and b is a 2D vector field. The method is described in the paper "An efficient jet marcher for computing the quasipotential for 2D SDEs" - by Nicholas Paskal and Maria Cameron -- https://arxiv.org/abs/2109.03424.
This project requires the eigen header-only library for linear algebra. Download eigen and set the user environment variable EIGEN_PATH to the eigen-3.x.x folder. Also add this to the Path environment variable if using gcc compiler.
The visual studio solution is located in the mvs folder. Make sure EIGEN_PATH is added under EJM "Properties"->"Configuration Properties"->"VC++ Directories"->"Include Directories".
There is a gcc makefile for BASH included in the src directory. If you have the make utility for bash, run the command "make" from within the src directory. The executable main.exe will be created in the parent directory.
The project startup file is main.cpp. Instructions on how to run EJM to solve for the quasi-potential on a rectangular mesh, as well as a list of the pre-implemented drift functions b(X), are provided at the top of main.cpp.
Currently only rectangular meshes are supported. The rectangular mesh is constructed in line 78 of main.cpp on the rectangle [xleft, xright] x [yleft, yright] with nx and ny points along each axis, respectively. The implementation assumes that the point attractor lies at the origin, and hence one should have xleft < 0 < xright and yleft < 0 < yright.
Drifts should be set in the function driftFunctions.cpp. All drift functions are written as cases for a single master function. To add a new drift, one should pick an unused integer to use as a case and implement the formulas for that case for the following functions in driftFunctions.cpp.
- masterDrift -- set drift field
- (Optional) masterGradDrift -- set 1st derivative of drift field
- (Optional) masterHessDrift -- set 2nd derivative of drift field
If the first and second derivatives are not set, they will default to finite difference calculations. Currently, the coordinate system requires the point attractor to occur at the origin (0,0), so the drifts may need to be adjusted to new coordinates.
The drift functions take a pointer to void argument to contain any additional function parameters. This is cast as a pointer to a particular struct, containing the key for the above function switch, as well as 2 double parameter values (params->a and params->b). If one requires more parameters to specify the drift functions, one should construct a new struct to cast the pointer to, that contains the switch key and any other desired parameters, and adjust all the functions in driftFunctions.cpp accordingly.
The struct containing the switch key and the drift parameters is instantiated in main.cpp in lines 87-91, and particular values of drift parameters should be here.
If in addition one has an a-priori solution for the quasi-potential U and its gradient DU -- these can be set in the same manner in the functions masterSol and masterGradSol in driftFunctions.cpp.
To run a single run of the EJM for a given drift field, set the run type in line 66 of main.cpp to 's', and set debugMode to true in line 98 of main.cpp. The following txt files are created.
- solution.txt -- file containing U and DU values at termination for all mesh points (both Considered and Accepted)
- solutionAccepted.txt -- file containing U and DU values at termination for Accepted mesh point.
Outputs are written to the outputs folder (or the mvs/EJM/outputs folder if one uses Microsoft Visual Studio).
Remaining settings are described in detail in file main.cpp.
The folder prefactor contains all of the Matlab codes and files used to generate exponential prefactor and transition rate estimates. Instructions on how to use these scripts are included in the ReadMe in the prefactor folder.