codes

Codes for all examples in the book "A Toolbox for Digital Twins". When multi-language versions are available, they are indicated by a * after the example description.

Code examples for Chapter 1.

Example 1.3 underfitting and overfitting

Code examples for Chapter 3.

Availability of multi-language versions is indicated by a *.

nonlinear, linear and difference equations

Example 3.4 solution of a nonlinear equation by various methods.*
Example 3.11 solution of an ill-conditioned linear system.*
Example 3.15 cobweb plot of a difference equation.*

ode

Example 3.25 solve an unstable ordinary differential equation by different methods.*
Example 3.28 – see Example 3.25.

finite difference methods

Example 3.29 solution of a transport equation by different methods.*
Example 3.30 Laplace equation with Neumann boundary condition.*
Example 3.31 2D Poisson equation in L-shaped geometry.
Example 3.32 2D Poisson equation for electrostatics with different solvers.
Example 3.33 1D heat equation with Crank-Nicolson scheme.
Example 3.34 1D heat equation with explicit and implicit schemes.
Example 3.35 2D wave equation with absorbing boundary conditions.
Example 3.36 2D wave equation with very efficient implementation.

finite element methods

Example 3.39 2D elctrostatics with Dirichlet conditions.
Example 3.40 2D Poisson equation convergence analysis.
Example 3.41 2D Poisson equation with Dirichlet and Neumann conditions.
Example 3.43 nonlinear elliptic equation with Picard iteration.
Example 3.44 nonlinear elliptic equation with Newton iteration.
Example 3.45 nonlinear elliptic equation with additive term.

stochastic simulation (monte-carlo methods)

Example 3.46 Monte-Carlo integration.*
Example 3.47 Monte-Carlo integration–convergence study.
Example 3.48 importance sampling for variance reduction.
Example 3.49 rejection sampling for variance reduction.
Example 3.56 Metropolis-Hasting MCMC for computing a posterior.
Example 3.57 – see Example 3.56.
Example 3.58 Simple MCMC for a Gaussian posterior.

stochastic differential equations

Example 3.60 white noise simulation.*
Example 3.62 Brownian motion simulation.*
Example 3.63 Ito and Stratonovich stochastic integral evaluation.*
Example 3.82 Euler-Maruyama method for a linear SDE.*
Example 3.85 Ornstein-Uhlenbeck equation.*
Example 3.87 Milstein method for a nonlinear SDE.*
Example 3.88 Nagumo’s SPDE.

Code examples for Chapter 4.

Example 4.10 stochastic gradient descent

Code examples for Chapter 5.

supervised learning

Example 5.5 simple linear regression.*
Example 5.16 k-nn classifier for iris data.
Example 5.19 SVM classifier for iris data.
Example 5.20 SVM regression for lidar data.
Example 5.21 neural network for squares data.
Example 5.22 saving and reloading a previously learned model.
Example 5.23 neural network for fraud detection.
Example 5.25 naive Bayes classifier.

unsupervised learning

Example 5.26 simple k-means clustering.
Example 5.27 simple k-means clustering with tuning.

cross-validation and tuning

Example 5.30 cross-validation with caret.
Example 5.31 cross-validation for tuning of k-nn model.
Example 5.32 tuning and cross-validation of SVM model.
Example 5.33 further tuning of SVM model.
Example 5.34 tuning neural networks for regression.
Example 5.35 tuning neural networks for classification.

Code examples for Chapter 6.

Example 6.2 underitting and overfitting
Example 6.3 - see Example 6.2

Code examples for Chapter 8.

Example 8.5 Duffing’s equation
Example 8.17 Gauss-Newton method for nonlinear inversion
Example 8.18 Levenberg-Marquardt method for nonlinear inversion

Code examples for Chapter 9.

Example 9.8 Gaussian state space model.
Example 9.12 Kalman filter for Gaussian state space model.
Example 9.14 extended Kalman filter for noisy pendulum.
Example 9.18 unscented Kalman filter for 2D navigation.
Example 9.21 ensemble Kalman filter for Lorenz-63 chaotic system.
Example 9.22 ensemble Kalman filter for Kuramoto-Sivashinsky turbulent system.

Code examples for Chapter 10.

Example 10.2 SVD for optimal thresholding
Example 10.8 optimal matched pursuit for binary signal

Code examples for Chapter 11.

Bayesian estimation

Example 11.5 Bayesian inference for binomial random variable (influence of priors).
Example 11.7 Bayesian inference for binomial random variable with beta conjugate priors.
Example 11.8 Bayesian inference for epidemics.

Bayesian regression

Example 11.10 Bayesian inference with Gaussian products.
Example 11.12 Bayesian inference of a mean for air quality data.
Example 11.13 Bayesian inference for a multivariate Gaussian (linear regression).
Example 11.14 Bayesian inference for parameters of a noisy pendulum.
Example 11.15 Bayesian regression and model reduction for diabetes data.
Example 11.17 Gaussian process regression for a sine function.

Bayesian filters

Example 11.18 Bayesian sequential regression for a multivariate Gaussian.
Example 11.19 Bayesian sequential estimation of 2d coordinates.

Bayesian inverse problems

Example 11.21 Bayesian inversion for 1d heat equation.

Bayesian optimization

Example 11.22 Bayesian optimization of a 1d function.

probabilistic programming

Example 11.23 Bayesian inference of a proportion.
Example 11.24 Bayesian linear regression.
Example 11.25 Bayesian inverse problems for ordinary differential equations.

Code examples for Chapter 12.

Example 12.9 basic example from pcalg package