DOW Experimental Design - Revision history

RobertLampel: /* Parameters */

2025-08-26T06:19:50Z

Parameters

← Older revision		Revision as of 06:19, 26 August 2025
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	</tr>		</tr>
	</table>		</table>
	Note that for the calculations all temperatures given in <math>^{\circ}C</math> have to be rescaled to <math>K</math> by adding <math>273.15</math>.		Note that for the calculations all temperatures given in<math>{\ }^{\circ}C</math> have to be rescaled to <math>K</math> by adding <math>273.15</math>.

	There are three datasets for different temperatures <math>T</math>, with corresponding starting values		There are three datasets for different temperatures <math>T</math>, with corresponding starting values

2025-04-04T12:56:48Z

Mathematical formulation

← Older revision		Revision as of 12:56, 4 April 2025
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	Let <math>f_k(\cdot)</math> denote the right hand side of equation <math>(k)</math> for <math>k=1,\ldots,6</math>.		Let <math>f_k(\cdot)</math> denote the right hand side of equation <math>(k)</math> for <math>k=1,\ldots,6</math>.
	~~We reformulate the last four algebraic equations as differential ones:~~
	~~<p>~~		The right hand sides of <math>(1)-(10)</math> are summarized as the vector-valued function <math>f</math>.
	~~<math>~~
	~~\begin{align}~~
	~~\dot{y}_7 &= f_7 = f_6 + f_8 + f_9 + f_{10} && \quad (7') \\~~
	~~\dot{y}_8 &= f_8 = \frac{\theta_8 f_1 \cdot (\theta_8 + y_7) - \theta_8 y_1 f_7}{(\theta_8 + y_7)^2} && \quad (8') \\~~
	~~\dot{y}_9 &= f_9 = \frac{\theta_9 f_3 \cdot (\theta_9 + y_7) - \theta_9 y_3 f_7}{(\theta_9 + y_7)^2} && \quad (9')\\~~
	~~\dot{y}_{10} &= f_{10} = \frac{\theta_7 f_5 \cdot (\theta_7 + y_7) - \theta_7 y_5 f_7}{(\theta_7 + y_7)^2} && \quad (10')~~
	~~\end{align}~~
	~~</math>~~
	~~</p>~~
	The right hand sides of <math>(1)-(~~6)</math> and <math>(7')-(9'~~)</math> are summarized as the vector-valued function <math>f</math>.
	Moreover, let		Moreover, let
	<p>		<p>

2025-02-18T10:31:29Z

Parameters

← Older revision		Revision as of 10:31, 18 February 2025
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	\begin{array}{l}		\begin{array}{l}
	y_5 = 0 \\		y_5 = 0 \\
	y_6 = ~~0.00131~~ \\		y_6 = [Q^+] \\
	y_7 = \frac{1}{2} \cdot \left( -K_2 + \sqrt{K_2^2 + 4K_2 y_1(0)} \right) \\		y_7 = \frac{1}{2} \cdot \left( -K_2 + \sqrt{K_2^2 + 4K_2 y_1(0)} \right) \\
	y_8 = y_7 \\		y_8 = y_7 \\

2024-12-18T07:34:23Z

Chemical background

← Older revision		Revision as of 07:34, 18 December 2024
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	<math>		<math>
	\begin{align}		\begin{align}
	\frac{d[MBMH]}{dt} &= k_1 \left[ M^- \right] \left[ BM \right] + k_{-1} \left[ MBM^- \right] && \quad (g)\\		\frac{d[MBMH]}{dt} &= k_1 \left[ M^- \right] \left[ BM \right] - k_{-1} \left[ MBM^- \right] && \quad (g)\\
	\frac{d[HA]}{dt} &= k_2 \left[ A^- \right] \left[ BM \right] && \quad (h)\\		\frac{d[HA]}{dt} &= k_2 \left[ A^- \right] \left[ BM \right] && \quad (h)\\
	\frac{d[HABM]}{dt} &= k_2 \left[ A^- \right] \left[ BM \right] + k_3 \left[ M^- \right] \left[ AB \right] - k_{-3} \left[ ABM^- \right] && \quad (i)		\frac{d[HABM]}{dt} &= k_2 \left[ A^- \right] \left[ BM \right] + k_3 \left[ M^- \right] \left[ AB \right] - k_{-3} \left[ ABM^- \right] && \quad (i)

2024-12-18T07:29:03Z

Mathematical formulation

← Older revision		Revision as of 07:29, 18 December 2024
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	</math>		</math>
	</p>		</p>
	Here the letters in parentheses stand for the corresponding chemical process and the quantity <math>\left[Q^+\right]=0.~~00131~~</math> is a constant during the reaction.		Here the letters in parentheses stand for the corresponding chemical process and the quantity <math>\left[Q^+\right]=0.0131</math> is a constant during the reaction.
	The nine parameters form the vector		The nine parameters form the vector
	<p>		<p>

2024-12-16T14:57:57Z

Parameters

← Older revision		Revision as of 14:57, 16 December 2024
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	</p>		</p>

	To reduce the intercorrelation between the parameters in the rate constants, we apply the following reparametrization (cf. [[#Kristensen2004\|[1]]].):		To reduce the intercorrelation between the parameters in the rate constants, we apply the following reparametrization (cf. [[#Kristensen2004\|[4]]].):
	<p>		<p>
	<math>		<math>

2024-12-16T14:57:44Z

Parameters

← Older revision		Revision as of 14:57, 16 December 2024
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	</p>		</p>

	To reduce the intercorrelation between the parameters in the rate constants, we apply the following reparametrization:		To reduce the intercorrelation between the parameters in the rate constants, we apply the following reparametrization (cf. [[#Kristensen2004\|[1]]].):
	<p>		<p>
	<math>		<math>

2024-12-16T14:56:44Z

References

← Older revision		Revision as of 14:56, 16 December 2024
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	<span id="OEDUDE">[2]</span> "Optimal Experimental Design for Universal Differential Equations" by C. Plate, C.J. Martensen and S. Sager <br>		<span id="OEDUDE">[2]</span> "Optimal Experimental Design for Universal Differential Equations" by C. Plate, C.J. Martensen and S. Sager <br>
	<span id="Stortelder1998">[3]</span> "Parameter estimation in nonlinear systems" by W.J.H. Stortelder <br>		<span id="Stortelder1998">[3]</span> "Parameter estimation in nonlinear systems" by W.J.H. Stortelder <br>
			<span id="Kristensen2004">[4]</span> "Parameter Estimation in Nonlinear Dynamical Systems" by Morten Rode Kristensen <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 -->

2024-12-16T10:14:03Z

Parameters

← Older revision		Revision as of 10:14, 16 December 2024
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	</tr>		</tr>
	</table>		</table>
			Note that for the calculations all temperatures given in <math>^{\circ}C</math> have to be rescaled to <math>K</math> by adding <math>273.15</math>.

	There are three datasets for different temperatures <math>T</math>, with corresponding starting values		There are three datasets for different temperatures <math>T</math>, with corresponding starting values

2024-12-16T09:42:34Z

Parameters

@@ Line 281: / Line 281: @@
 </math>
 </p>
-in which <math>k_{0,i} = \alpha_i \cdot \exp\left( - \frac{E_i}{RT_0} \right)</math>.
+in which <math>k_{0,i} = \alpha_i \cdot \exp\left( - \frac{E_i}{RT_0} \right)</math>. The reference temperature in <math>T_0</math> is chosen as the average over all performed experiments, i.e., <math>T_0 = 69^\circ C</math>. Additionally, we add a logarithmic transformation, which gives rise to the following transformed starting values:
 == Optimal Experimental Design Problem ==