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Merge pull request #116 from ComputationalScienceLaboratory/lorenz-96-documentation
Lorenz '96 documentation
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‎docs/references.bib‎

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@@ -105,5 +105,18 @@ @inproceedings{Lor96
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volume={1},
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number={1},
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year={1996},
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organization={Reading}
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organization={Reading},
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doi={10.1017/CBO9780511617652.004}
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}
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@article{PS19,
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title={A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions},
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author={Popov, Andrey A and Sandu, Adrian},
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journal={Nonlinear Processes in Geophysics},
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volume={26},
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number={2},
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pages={109--122},
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year={2019},
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publisher={Copernicus Publications G{\"o}ttingen, Germany},
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doi={10.5194/npg-26-109-2019}
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}

‎toolbox/+otp/+lorenz63/+presets/Canonical.m‎

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classdef Canonical < otp.lorenz63.Lorenz63Problem
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% Original Lorenz '63 preset presented in :cite:p:`Lor63`
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% which uses time span $t \in [0, 60]$, $\sigma = 10$, $\rho = 28$,
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% $\beta = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.
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% which uses time span $t \in [0, 60]$, $σ = 10$, $ρ = 28$,
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% $β = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.
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methods
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function obj = Canonical(varargin)
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% varargin
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% A variable number of name-value pairs. The accepted names are
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%
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% - ``sigma`` – Value of $\sigma$.
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% - ``rho`` – Value of $\rho$.
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% - ``beta`` – Value of $\beta$.
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% - ``sigma`` – Value of $σ$.
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% - ``rho`` – Value of $ρ$.
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% - ``beta`` – Value of $β$.
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%
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% Returns
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% -------
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% obj : Lorenz63Problem
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% The constructed problem.
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p = inputParser;
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p.addParameter('sigma', 10);

‎toolbox/+otp/+lorenz63/+presets/LimitCycle.m‎

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classdef LimitCycle < otp.lorenz63.Lorenz63Problem
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% Lorenz '63 preset limit cycle, a non-chaotic preset, from :cite:p:`Str18`
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% which uses time span $t \in [0, 60]$, $\sigma = 10$, $\rho = 350$,
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% $\beta = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.
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% which uses time span $t \in [0, 60]$, $σ = 10$, $ρ = 350$,
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% $β = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.
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methods
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function obj = LimitCycle
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% Create the LimitCycle Lorenz '63 problem object.
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%
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sigma = 10;
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rho = 350;
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beta = 8/3;

‎toolbox/+otp/+lorenz63/+presets/Surprise.m‎

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classdef Surprise < otp.lorenz63.Lorenz63Problem
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% Lorenz '63 preset 'surprise' from :cite:p:`Str18` which uses time span $t \in [0, 60]$,
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% $\sigma = 10$, $\rho = 100$, $\beta = 8/3$, and intial conditions $y_0 = [2, 1, 1]^T$.
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% $σ = 10$, $ρ = 100$, $β = 8/3$, and intial conditions $y_0 = [2, 1, 1]^T$.
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function obj = Surprise
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% Create the Surprise Lorenz '63 problem object.
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%
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sigma = 10;
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rho = 100;

‎toolbox/+otp/+lorenz63/Lorenz63Problem.m‎

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% The three variable Lorenz '63 problem :cite:p:`Lor63` of the form,
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%
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% $$
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% x' &= \sigma(y - x),\\
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% y' &= \rho x - y - xz,\\
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% z' &= xy - \beta z,
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% x' &= σ(y - x),\\
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% y' &= ρx - y - xz,\\
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% z' &= xy - βz,
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% $$
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%
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% exhibits chaotic behavior for certain values of the parameters.
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% +---------------------+-----------------------------------------------------------+
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% | Number of Variables | 3 |
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% +---------------------+-----------------------------------------------------------+
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% | Stiff | not typically, depending on $\sigma$, $\rho$, and $\beta$ |
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% | Stiff | not typically, depending on $σ$, $ρ$, and $β$ |
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% +---------------------+-----------------------------------------------------------+
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%
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% Example

‎toolbox/+otp/+lorenz96/+presets/Canonical.m‎

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classdef Canonical < otp.lorenz96.Lorenz96Problem
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%CANONICAL This is the original problem presented in the literature.
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% The initial condition is a slight perturbation of a critical point.
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%
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% See:
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% Lorenz, E. N. (1996, September). Predictability: A problem partly solved.
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% In Proc. Seminar on predictability (Vol. 1, No. 1).
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% Original Lorenz '96 preset presented in :cite:p:`Lor96`
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% which uses time span $t \in [0, 720]$, $N = 40$, $F=8$, and initial
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% conditions of $y_i = 8$ for all $i$ except for $y_{\lfloor N/2 \rfloor}=8.008$.
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function obj = Canonical(varargin)
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% Create a Canonial problem object.
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%
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% Parameters
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% ----------
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% varargin
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% A variable number of name-value pairs. The accepted names are
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%
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% - ``Size`` – The size of the problem as a positive integer.
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% - ``Forcing`` – The forcing as a scalar, vector of N constants, or as a
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% function.
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%
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p = inputParser;
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p.addParameter('Size', 40, @isscalar);

‎toolbox/+otp/+lorenz96/+presets/PopovSandu.m‎

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classdef PopovSandu < otp.lorenz96.Lorenz96Problem
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%POPOVSANDU
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% Used in https://doi.org/10.5194/npg-26-109-2019
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% A preset that has a cyclic forcing function that is different for
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% every variable. This preset was created for :cite:p:`PS19`.
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% This preset uses time span $t \in [0, 720]$, $N = 40$, and initial
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% conditions of $y_i = 8$ for all $i$ except for $y_{\lfloor N/2 \rfloor}=8.008$.
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% The forcing, as a function of time is given by
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%
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% $$
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% f(t) = 8 + 4\cos(2 \pi \omega (t+\text{mod}(i - 1, q)/q))
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% $$
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%
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% where by default $q=4$ is the number of partitions, and $\omega = 1$
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% is a forcing period of one time unit.
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function obj = PopovSandu(varargin)
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% Create a PopovSandu problem object.
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%
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% Parameters
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% ----------
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% varargin
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% A variable number of name-value pairs. The accepted names are
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%
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% - ``Size`` – The size of the problem as a positive integer.
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% - ``Partitions`` – The number of partitions into which to divide the variables.
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% - ``ForcingPeriod`` – The period of the forcing function in radians per unit time.
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%
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p = inputParser;
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‎toolbox/+otp/+lorenz96/Lorenz96Parameters.m‎

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classdef Lorenz96Parameters
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%LORENZ96PARAMETERS User-configurable parameters for the Lorenz 96 problem
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%
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% See also otp.lorenz96.Lorenz96Problem
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% Parameters for the Lorenz '96 problem.
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properties
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%F is a forcing function, scalar, or vector
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% A forcing function, scalar, or vector.
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F %MATLAB ONLY: {mustBeA(F, {'numeric','function_handle'})}
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end
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end

‎toolbox/+otp/+lorenz96/Lorenz96Problem.m‎

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classdef Lorenz96Problem < otp.Problem
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%LORENZ96PROBLEM The Lorenz 96 model is a classic model for testing data assimilation techniques.
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% A chaotic system modeling nonlinear transfer of a dimensionless
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% quantity along a cyclic one dimensional domain.
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%
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% See also otp.loren96.Lorenz96Parameters
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% The $N$ variable dynamics :cite:p:`Lor96` are represented by the equation,
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%
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% $$
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% y_i' = -y_{i-1}\left(y_{i-2} - y_{i+1}\right) - y_i + f(t), \quad i = 1, \dots, N,
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% $$
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%
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% where $y_0 = y_N$, $y_{-1} = y_{N - 1}$, and $y_{N + 1} = y_2$, exhibits
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% chaotic behavior for certain pairs of values of the dimension $N$ and
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% forcing function $f$.
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%
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% Notes
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% -----
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% +---------------------+-----------------------------------------------------------+
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% | Type | ODE |
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% +---------------------+-----------------------------------------------------------+
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% | Number of Variables | $N$ for any postive integer four or greater |
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% +---------------------+-----------------------------------------------------------+
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% | Stiff | no |
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% +---------------------+-----------------------------------------------------------+
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%
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% Example
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% -------
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% >>> problem = otp.lorenz96.presets.Canonical('Forcing', @(t) 8 + 4*sin(t));
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% >>> sol = problem.solve();
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% >>> problem.movie(sol);
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%
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% See also
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% --------
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% :doc:`Lorenz '63 Problem </problems/Lorenz63Problem>`
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properties (SetAccess = private)
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DistanceFunction
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end
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function obj = Lorenz96Problem(timeSpan, y0, parameters)
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% Create a Lorenz '96 problem object.
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%
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% Parameters
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% ----------
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% timeSpan : numeric(1, 2)
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% The start and final time.
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% y0 : numeric(:, 1)
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% The initial conditions.
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% parameters : Lorenz96Parameters
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% The parameters.
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%
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obj@otp.Problem('Lorenz 96', [], timeSpan, y0, parameters);
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end
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end
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'Vectorized', 'on');
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% We also provide a canonical distance function as is standard for
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% localisation in Data Assimilation. This is heavily tied to this
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% localization in Data Assimilation. This is heavily tied to this
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% problem.
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obj.DistanceFunction = @(t, y, i, j) otp.lorenz96.distanceFunction(t, y, i, j);

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