Documentation

The FUNCLE module

Functional UNcertainty Constrained by Law and Experiment

User’s Guide

Definitions

Physics Process

“A physics process is a relation between two or more physical quantities, e.g. the rate at which plasma ion and electron temperatures equilibrate as a function of the individual temperatures, particle masses, and number densities. Note that a physics process need not be time dependent, e.g. the shear modulus as a function of temperature and density.”:cite:`VaughanPUBS2014`

Physics Model

“A physics model is an approximate mathematical representation of a physics process. As an example of a physics model, consider the plastic constitutive relation of a metal. This relates the basic quantities governing plastic deformation, e.g. the flow stress, plastic strain, strain rate, temperature, and density. It is an approximate representation of the very complicated mesoscale physics responsible for the flow of a solid, and is therefore a physics model.”:cite:`VaughanPUBS2014`

Degree of Freedom

The mathematical representation of a physics model is dependant on a set of variables whose values can alter the behaviour of the model. These variables are refered to as the degrees of freedom or DOF

Experiment

An experiment represents data measured from a real physical experiments. Experiments used in F_UNCLE are chosen so that thier results are dominated by a single physics process which can be represented by a physics model Simulation

Simulation

A simulation which represents the physical system from which the experimental data were obtained. The simulation is dependent on one or more physics models and acts as a map between the model degree of freedom space and the experimental result space

F_UNCLE Parent Classes

These concepts are implemented in the following three fundemental F_UNCLE classes

Physics Model

F_UNCLE.Utils.PhysicsModel.PhysicsModel. This class represents a physics model. The degrees of freedom of a PhysicsModel are immutable. Setting the degrees of freedom of a PhysicsModel will return a independent copy of itself with updated DOF.

  • F_UNCLE.Utils.PhysicsModel.PhysicsModel.get_dof(): Returns a list of the model degrees of freedom
  • F_UNCLE.Utils.PhysicsModel.PhysicsModel.set_dof(): Generates a new Physics Model with the specified degrees of freedom
  • F_UNCLE.Utils.PhyscisModel.PhysicsModel.get_sigma(): Returns a square matrix representingthe covariance of the model degrees of freedom
  • F_UNCLE.Utils.PhyscisModel.PhysicsModel.shape(): Returns an integer representing the number of model degrees of freedom

Experiment

F_UNCLE.Utils.Experiment.Experiment. This class represents the results for a single experiment performed under a single set of conditions.

  • F_UNCLE.Utils.Experiment.Experiment.__call__(): Returns a tuple with the folliwing elements:

    1. The independent varialbe of the simulation (i.e. time)
    2. List of arrays of dependant variables. Element zero is the element for comparisson, the last element is the labels for the previous elements
    3. Dictionary of additonal simulation result attribures. Contains at least the key mean_fn which is a functional representation of the relationship between the independant variable and the dependent variable for comparisson

The Experiment object provides some internal routines to ease comparisson of simulations and experiments

  • F_UNCLE.Utils.Experiment.Experiment.align(): This takes a set of simulation data and returns a copy of it with the simulation data aligned so it is evaluated at each independent data value of the experiment. This ‘aligned’ tuple of simulation data has a new key tau which is the shift in independant variable (likely time) required to align the simulation to the experiment. In adition, the mean_fn attribute has been modified so it returns values aligned to the experimental data.
  • F_UNCLE.Utils.Experiment.Experiment.compare(): This takes a set of simulation data and returns the error between the experiment and the simulation. The simulation can either be aligned or not. The return value is the experimental value less the aligned simulation value

Sumulation

F_UNCLE.Utils.Simulation.Simulation. This class wraps some computer simulatin which is dependent on a PhysicsModel

  • F_UNCLE.Utils.Simulation.Simulation.compare(): Compares two simulations results and returns the difference

  • F_UNCLE.Utils.Simulations.Simulations.__call__(): Returns the results of the simulation as a tuple

    1. The independent varialbe of the simulation (i.e. time)
    2. List of arrays of dependant variables. Element zero is the element for comparisson, the last element is the labels for the previous elements
    3. Dictionary of additonal simulation result attributes. Contains at least the key mean_fn which is a functional representation of the relationship between the independant variable and the dependent variable for comparisson

  • F_UNCLE.Utils.Simulations.Simulations.get_sens(): This returns a sensitvity matrix of the simulation response to changes in the required model degrees of freedom. The response is evaluated at the independent data points of the provided initial_data and the deltas for each finite difference step are evaluated using the compare method.

Optimizaiton Methods

The goal of F_UNCLE is to determine the set of model degrees of freedom which minimizes the error between a set of Experiments and associated Simulations.

F_UNCLE.Opt.Bayesian.Bayesian: