# Basic fitting tutorial

In this section we are going to go through a complete example of fitting using BornAgain. Each step will be associated with a detailed piece of code written in Python. The script can also be found in the example section.

This example uses the same sample geometry as in Basic GISAS simulation tutorial. Cylindrical and prismatic particles are deposited on a substrate layer in equal proportion, with no interference between the particles. We consider the following parameters to be unknown:

• the radius of the cylinders,
• the height of the cylinders,
• the length of the prisms’ triangular basis,
• the height of the prisms.

Our reference data are a “noisy” two-dimensional intensity map obtained from the simulation of the same geometry with a fixed value of 5nm for all sizes of cylinders and prisms.

### Importing the python libraries

 6 7  import bornagain as ba from bornagain import deg, angstrom, nm

We start with importing the BornAgain Python API and particularly the dimension units defined in BornAgain.

### Building the sample

 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  def get_sample(params): """ Returns a sample with uncorrelated cylinders and prisms on a substrate. """ cylinder_height = params["cylinder_height"] cylinder_radius = params["cylinder_radius"] prism_height = params["prism_height"] prism_base_edge = params["prism_base_edge"] # defining materials m_air = ba.HomogeneousMaterial("Air", 0.0, 0.0) m_substrate = ba.HomogeneousMaterial("Substrate", 6e-6, 2e-8) m_particle = ba.HomogeneousMaterial("Particle", 6e-4, 2e-8) # collection of particles cylinder_ff = ba.FormFactorCylinder(cylinder_radius, cylinder_height) cylinder = ba.Particle(m_particle, cylinder_ff) prism_ff = ba.FormFactorPrism3(prism_base_edge, prism_height) prism = ba.Particle(m_particle, prism_ff) layout = ba.ParticleLayout() layout.addParticle(cylinder, 0.5) layout.addParticle(prism, 0.5) # air layer with particles and substrate form multi layer air_layer = ba.Layer(m_air) air_layer.addLayout(layout) substrate_layer = ba.Layer(m_substrate, 0) multi_layer = ba.MultiLayer() multi_layer.addLayer(air_layer) multi_layer.addLayer(substrate_layer) return multi_layer

The function starting at line 12 creates a multilayered sample with cylinders and prisms randomly distributed on a surface. The function accepts a dictionary as input parameters and use its key/value pairs to define the particle sizes.

### Creating the simulation

 45 46 47 48 49 50 51 52 53 54 55  def get_simulation(params): """ Returns a GISAXS simulation with beam and detector defined """ simulation = ba.GISASSimulation() simulation.setDetectorParameters(100, -1.0*deg, 1.0*deg, 100, 0.0*deg, 2.0*deg) simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg) simulation.setBeamIntensity(1e+08) simulation.setSample(get_sample(params)) return simulation

The function starting at line 45 creates the simulation object with beam, detector and user sample defined. This function is intended to be called on every minimization iteration to provide a new simulation object for every new set of fit parameter values. In the given example the set of parameters is used to construct a new sample by calling the get_sample method described earlier.

### Creating the “experimental” data

 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79  def create_real_data(): """ Generating "experimental" data by running simulation with certain parameters. The data is saved on disk in the form of numpy array. """ # default sample parameters params = {'cylinder_height': 5.0*nm, 'cylinder_radius': 5.0*nm, 'prism_height': 5.0*nm, 'prism_base_edge': 5.0*nm} # retrieving simulated data in the form of numpy array simulation = get_simulation(params) simulation.runSimulation() real_data = simulation.result().array() # spoiling simulated data with noise to produce "real" data np.random.seed(0) noise_factor = 0.1 noisy = np.random.normal(real_data, noise_factor*np.sqrt(real_data)) noisy[noisy < 0.1] = 0.1 np.savetxt("basic_fitting_tutorial_data.txt.gz", real_data)

The experimental data here is represented by a 2D numpy array with detector intensities obtained from the same simulation. The “experimental” data file is prepared by running once a simulation with default parameters, adding noise to it and saving it on disk.

### Setting up the fit objective

 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104  def run_fitting(): """ Setup simulation and fit """ real_data = load_real_data() fit_objective = ba.FitObjective() fit_objective.addSimulationAndData(get_simulation, real_data, 1.0) # Print fit progress on every n-th iteration. fit_objective.initPrint(10) # Plot fit progress on every n-th iteration. Will slow down fit. fit_objective.initPlot(10)

The FitObjective created here is used to put into correspondence the real data, represented by a numpy array, and the simulation, represented by the get_simulation callable. On every fit iteration FitObjective

• will generate a new simulation object for a given set of fit parameter values using the get_simulation callable,
• run it to obtain simulated detector intensities,
• calculate chi2 between simulated and real data.

### Setting the fit parameters

 105 106 107 108 109   params = ba.Parameters() params.add("cylinder_height", 4.*nm, min=0.01) params.add("cylinder_radius", 6.*nm, min=0.01) params.add("prism_height", 4.*nm, min=0.01) params.add("prism_base_edge", 12.*nm, min=0.01)

The creation of fit parameters is straightforward. We give names, starting values and lower bounds to 4 fit parameters, representing the dimensions of the cylinders and prisms. The fit parameter names must coincide with the names that the get_simulation callable expects.

### Creating the minimizer and running the fit

 111 112 113 114 115 116   minimizer = ba.Minimizer() result = minimizer.minimize(fit_objective.evaluate, params) fit_objective.finalize(result) print("Fitting completed.") print("chi2:", result.minValue())

The method fit_objective.evaluate provided by the FitObjective class interface is used here as an objective function. The method accepts fit parameters and returns the chi2 value, calculated between experimental and simulated images for the given values of the fit parameters.

The method is passed to the minimizer together with the initial fit parameter values. The minimizer.minimize starts a fit that will continue further without user intervention until the minimum is found or the minimizer failed to converge. The rest of the code demonstrates how to access the fit results.

See the full script in the Fit Cylinders and Prisms example.