Lotka Shared OED - Revision history

RobertLampel: /* Mathematical formulation */

2026-03-26T10:16:25Z

Mathematical formulation

← Older revision		Revision as of 10:16, 26 March 2026
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	</math>		</math>
	</p>		</p>
	where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>~~w^i~~(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.		where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w_i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

	Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:		Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

RobertLampel: /* Mathematical formulation */

2026-03-26T09:58:11Z

Mathematical formulation

← Older revision		Revision as of 09:58, 26 March 2026
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	\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\		\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
	\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\		\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
	\quad \dot{z}(t) & = & w(t), \\		\quad \dot{z}(t) & = & w(t) \\
	\quad x(0) & = & (1.5, 0.5, 1)^T \\		\quad x(0) & = & (1.5, 0.5, 1)^T \\
	\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\		\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\

RobertLampel: /* Mathematical formulation */

2026-03-26T09:57:09Z

Mathematical formulation

← Older revision		Revision as of 09:57, 26 March 2026
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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), \\		\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), && t \in [0,t_f], \quad x_0(0) = 1.5, \\
	\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), \\		\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), && t \in [0,t_f], \quad x_1(0) = 0.5, \\
	\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t),		\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t), && t \in [0,t_f], \quad x_2(0) = 1,
	\end{array}		\end{array}
	</math>		</math>

RobertLampel: /* Parameters */

2026-03-26T09:56:11Z

Parameters

← Older revision		Revision as of 09:56, 26 March 2026
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	\| align=center \| <math>\varepsilon_\mathrm{reg}</math> \|\| align=right \| 0.1 \|\| Regularization of Fisher matrix		\| align=center \| <math>\varepsilon_\mathrm{reg}</math> \|\| align=right \| 0.1 \|\| Regularization of Fisher matrix
	\|-		\|-
	\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| [0,1] \|\| Bounds of control function		\| align=center \| <math>\mathcal{U}</math> \|\| align=right \| <math>[0,1]^3</math> \|\| Bounds of control function
	\|-		\|-
	\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function		\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| <math>[0,1]^3</math> \|\| Bounds of measurement function
	\|-		\|-
	\| align=center \| <math>M_0, M_1, M_2</math> \|\| align=right \| 4 \|\| Maximum measurement time		\| align=center \| <math>M_0, M_1, M_2</math> \|\| align=right \| 4 \|\| Maximum measurement time

RobertLampel: /* Parameters */

2026-03-26T09:28:52Z

Parameters

← Older revision		Revision as of 09:28, 26 March 2026
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	\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function		\| align=center \| <math>\mathcal{W}</math> \|\| align=right \| [0,1] \|\| Bounds of measurement function
	\|-		\|-
	\| align=center \| <math>M_1, M_2~~, M_3~~</math> \|\| align=right \| 4 \|\| Maximum measurement time		\| align=center \| <math>M_0, M_1, M_2</math> \|\| align=right \| 4 \|\| Maximum measurement time
	\|}		\|}

RobertLampel: /* Mathematical formulation */

2026-03-26T09:28:11Z

Mathematical formulation

← Older revision		Revision as of 09:28, 26 March 2026
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	\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\		\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
	\quad \dot{z}(t) & = & w(t), \\		\quad \dot{z}(t) & = & w(t), \\
	\quad x(0) & = & ~~x_0~~ \\		\quad x(0) & = & (1.5, 0.5, 1)^T \\
	\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\		\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
	\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\		\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\

RobertLampel: /* Mathematical formulation */

2026-03-26T09:14:48Z

Mathematical formulation

← Older revision		Revision as of 09:14, 26 March 2026
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	== Mathematical formulation ==		== Mathematical formulation ==

	We are interested in estimating the parameters <math>~~p_2~~</math> and <math>~~p_4~~</math> of the Lotka-Volterra type predator-prey ~~fish~~ initial value problem		We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem
	<p>		<p>
	<math>		<math>

RobertLampel: /* Variants */

2026-03-26T09:13:23Z

Variants

← Older revision		Revision as of 09:13, 26 March 2026
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	* [[:Category:Casadi \| CasADi]] at https://github.com/rlampel/PySHeRLOC		* [[:Category:Casadi \| CasADi]] at https://github.com/rlampel/PySHeRLOC

	~~== Variants ==~~

	~~There are several alternative formulations and variants of the above problem, in particular~~

	* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
	* no fishing, i.e., <math>u \equiv 0</math>,
	* different fishing control functions for the two species,
	* different parameters and start values.

	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==

RobertLampel: /* Source Code */

2026-03-26T09:13:13Z

Source Code

← Older revision		Revision as of 09:13, 26 March 2026
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	Model descriptions are available in		Model descriptions are available in

	* [[:Category:~~AMPL~~ \| ~~AMPL~~]] at ~~[[Lotka Experimental Design (AMPL)]]~~		* [[:Category:Casadi \| CasADi]] at https://github.com/rlampel/PySHeRLOC
	* [[:~~Category: VPLAN \| VPLAN code]] at [[Lotka Experimental Design (VPLAN)]]~~

	== Variants ==		== Variants ==

RobertLampel: /* Reference Solutions */

2026-03-26T09:08:33Z

Reference Solutions

← Older revision		Revision as of 09:08, 26 March 2026
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="2">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:~~vplotkaUStates~~.png\| States and ~~fishing control.~~		Image:Lotka_Shared_OED.png\| States, control, and sampling functions for a local optimum.
	~~Image:vplotkaUSensitivities.png\| Sensitivities G().~~
	~~Image:vplotkaUMeas1.png\| Sampling function for first state.~~
	~~Image:vplotkaUMeas2.png\| Sampling function~~ for ~~second state~~.
	</gallery>		</gallery>