Exponential OED - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:44:33Z

Reference Solutions

← Older revision		Revision as of 13:44, 28 November 2025
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	situation is different, due to the decoupling of the integration		situation is different, due to the decoupling of the integration

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with <math>x(t_0) = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.		Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with <math>x(t_0) = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
	</gallery>		</gallery>

RobertLampel: /* References */

2025-08-22T13:40:08Z

References

← Older revision		Revision as of 13:40, 22 August 2025
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	== References ==		== References ==
	<~~biblist~~ />		<span id="Koerkel2000">[1]</span> "A Multiple Shooting Formulation for Optimum Experimental Design" by S. Körkel, A. Potschka, H.G. Bock, and Sebastian Sager <br>

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-22T13:38:31Z

Miscellaneous and Further Reading

← Older revision		Revision as of 13:38, 22 August 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	The Toy OED problem was introduced by ~~Sebastian Sager~~ in ~~<bib id="Sager2013" />~~, which contains further details.		The Toy OED problem was introduced by Körkel et al. in [[#Koerkel \| [1]]], which contains further details.

	== References ==		== References ==

RobertLampel: /* Reference Solutions */

2025-08-22T13:36:53Z

Reference Solutions

← Older revision		Revision as of 13:36, 22 August 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with		Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with <math>x(t_0) = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
	<math>x = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2025-08-22T13:36:25Z

Reference Solutions

← Older revision		Revision as of 13:36, 22 August 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with		Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with
	<math>x = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.		<math>x = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2025-08-22T13:35:35Z

Reference Solutions

← Older revision		Revision as of 13:35, 22 August 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Exponential_OED.png\| ~~States and measurement control for different choices~~ of <math>p</math>.		Image:Exponential_OED.png\| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with
			<math>x = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2025-08-22T13:33:18Z

Reference Solutions

← Older revision		Revision as of 13:33, 22 August 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:~~Toy OED~~.png\| States and measurement control for different choices of <math>p</math>.		Image:Exponential_OED.png\| States and measurement control for different choices of <math>p</math>.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2025-08-22T13:25:33Z

Reference Solutions

← Older revision		Revision as of 13:25, 22 August 2025
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	As can be seen, the optimal solution will maximize the value of <math>G(t_f)</math> and hence also of <math>x(t_f)</math>.		As can be seen, the optimal solution will maximize the value of <math>G(t_f)</math> and hence also of <math>x(t_f)</math>.
	For <math>t_f = 0.6</math> the optimal initial value is given by <math>q^∗ = 1.203</math> leading to state		For <math>t_f = 0.6</math> the optimal initial value is given by <math>q^* = 1.203</math> leading to state
	values of <math>x(t_f) = 200</math> and <math>G(t_f) = 6088</math> and an objective value of </math>φ^∗ = 2.7 \cdot 10^{−8}</math>.		values of <math>x(t_f) = 200</math> and <math>G(t_f) = 6088</math> and an objective value of <math>\varphi^* = 2.7 \cdot 10^{-8}</math>.
	The main problem with direct single shooting is that a large part of the feasible		The main problem with direct single shooting is that a large part of the feasible
	domain <math>\mathcal{D}</math> of <math>q</math> will cause the integrator to run into a singularity before <math>t_f</math>. Hence only initial guesses for the optimization variable <math>q</math> that are below a critical value		domain <math>\mathcal{D}</math> of <math>q</math> will cause the integrator to run into a singularity before <math>t_f</math>. Hence only initial guesses for the optimization variable <math>q</math> that are below a critical value
	of <math>≈ 1.23</math> will give rise to a successful optimization. For multiple shooting the		of <math>\approx 1.23</math> will give rise to a successful optimization. For multiple shooting the
	situation is different, due to the decoupling of the integration		situation is different, due to the decoupling of the integration

RobertLampel: /* Reference Solutions */

2025-08-22T13:21:39Z

Reference Solutions

← Older revision		Revision as of 13:21, 22 August 2025
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	== Reference Solutions ==		== Reference Solutions ==

	~~Here~~ is ~~one local solution~~ to the ~~above control problem~~.		As can be seen, the optimal solution will maximize the value of <math>G(t_f)</math> and hence also of <math>x(t_f)</math>.
			For <math>t_f = 0.6</math> the optimal initial value is given by <math>q^∗ = 1.203</math> leading to state
			values of <math>x(t_f) = 200</math> and <math>G(t_f) = 6088</math> and an objective value of </math>φ^∗ = 2.7 \cdot 10^{−8}</math>.
			The main problem with direct single shooting is that a large part of the feasible
			domain <math>\mathcal{D}</math> of <math>q</math> will cause the integrator to run into a singularity before <math>t_f</math>. Hence only initial guesses for the optimization variable <math>q</math> that are below a critical value
			of <math>≈ 1.23</math> will give rise to a successful optimization. For multiple shooting the
			situation is different, due to the decoupling of the integration

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">

RobertLampel: /* Parameters */

2025-08-22T13:19:11Z

Parameters

← Older revision		Revision as of 13:19, 22 August 2025
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	<p>		<p>
	<math>		<math>
	~~x_0~~ = 1; \quad t_f = 0.2; \quad \~~mathcal{W}~~ = ~~[0,1]; \quad M = 0.2; \quad p \in \~~{-0.~~5, -2\}~~		p = 1; \quad t_f = 0.6; \quad \varepsilon = 10^{-3}.
	</math>		</math>
	</p>		</p>