An adjoint method in inverse problems of chromatography

Figures & data

Figure 1. Estimated adsorption isotherms by synthetic data with different error levels $\mathit{\delta }$ of and their corresponding relative errors $\mathrm{\Delta }$. (a) $\mathit{\delta }=0$ and $\mathrm{\Delta }=0.0154$. (b) $\mathit{\delta }=1\%$ and $\mathrm{\Delta }=0.0404$. (c) $\mathit{\delta }=5\%$ and $\mathrm{\Delta }=0.0520$. (d) $\mathit{\delta }=10\%$ and $\mathrm{\Delta }=0.0790$.

Table 1. Estimated adsorption isotherms and computational costs (the number objective function evaluation (NOFE)) of the boundary fitting method (BF) and the Kohn–Vogelius method (KV) with different error levels for both the Equilibrium-Dispersive (ED) model and the Transport-Dispersive (TD) model with the given mass transfer resistance $k_f={[95,135]}^T$.

		Exact parameters	$a_1=2$	$a_2=4$	$b_1=0.1$	$b_2=0.2$
		Initial guesses	$a_1=1$	$a_2=1$	$b_1=1$	$b_2=1$
Method	Model	Error level $\mathit{\delta }$	$a_1$	$a_2$	$b_1$	$b_2$	NOFE
BF	ED	Exact data	2.7620	4.9963	0.0896	0.3472	75
KV	ED	Exact data	2.0082	3.9898	0.0995	0.1988	76
BF	ED	$\mathit{\delta }=1\%$	3.0839	4.9961	0.0704	0.3978	75
KV	ED	$\mathit{\delta }=1\%$	2.0328	4.0250	0.1008	0.1999	105
BF	ED	$\mathit{\delta }=5\%$	3.3173	4.9911	0.0213	0.4662	66
KV	ED	$\mathit{\delta }=5\%$	1.9354	3.9432	0.0951	0.1959	114
BF	ED	$\mathit{\delta }=10\%$	3.4073	4.9936	0.0272	0.4704	85
KV	ED	$\mathit{\delta }=10\%$	1.8629	3.7265	0.0942	0.1765	104
BF	TD	Exact data	1.8381	3.6223	0.1144	0.2065	79
KV	TD	Exact data	1.9745	3.9047	0.1011	0.2017	80
BF	TD	$\mathit{\delta }=1\%$	1.6980	4.5962	0.0658	0.1959	76
KV	TD	$\mathit{\delta }=1\%$	2.1897	4.0484	0.1145	0.1606	132
BF	TD	$\mathit{\delta }=5\%$	1.3459	3.4064	0.1650	0.3968	88
KV	TD	$\mathit{\delta }=5\%$	2.3259	5.0769	0.0786	0.1811	168
BF	TD	$\mathit{\delta }=10\%$	2.9460	6.7001	0.1957	0.4217	121
KV	TD	$\mathit{\delta }=10\%$	0.9633	3.6654	0.0748	0.1407	138

Table 2. Estimated adsorption isotherm parameters and mass transfer resistance by the boundary fitting method (BF) and the Kohn–Vogelius method (KV) with different error levels in the Transport-Dispersive (TD) model.

			$a_1$	$a_2$	$b_1$	$b_2$	$k_f^{(1)}$	$k_f^{(2)}$
		Exact	2	4	0.1	0.2	95	195
		Initial	1	1	1	1	100	100
Method	Model	Error level	$a_1$	$a_2$	$b_1$	$b_2$	$k_f^{(1)}$	$k_f^{(2)}$	NOFE
BF	TD	Exact	2.3897	4.2487	0.0760	0.1801	89.5	198.7	108
KV	TD	Exact	2.0298	4.0564	0.1014	0.2013	94.8	195.4	131
BF	TD	$\mathit{\delta }=1\%$	2.2764	3.5694	0.1123	0.2361	82.7	181.3	137
KV	TD	$\mathit{\delta }=1\%$	1.9517	4.0574	0.1033	0.2020	89.8	200.0	178
BF	TD	$\mathit{\delta }=5\%$	1.6137	3.3100	0.0958	0.1576	93.1	171.4	204
KV	TD	$\mathit{\delta }=5\%$	2.2069	4.2908	0.0970	0.2059	90.7	180.7	219
BF	TD	$\mathit{\delta }=10\%$	2.9420	5.8474	0.1345	0.3480	68.3	163.9	190
KV	TD	$\mathit{\delta }=10\%$	1.6851	4.7107	0.0771	0.1572	71.8	169.1	232

Figure 2. Numerical results of the boundary fitting method for the Equilibrium-Dispersive model with different error levels $\mathit{\delta }$ at every iteration step l. (a) The value of objective function ${\mathcal{J}}_{BF,\mathit{\alpha }}$ and (b) the relative error $\Vert {\mathit{\xi }}^l{-\overline{\mathit{\xi }}\Vert }_2/{\Vert {\mathit{\xi }}^l\Vert }_2$ of recovered parameters.

Figure 3. Numerical results of KV method for the Equilibrium-Dispersive model with different error levels $\mathit{\delta }$ at every iteration step l. (a) The value of objective function ${\mathcal{J}}_{KV,\mathit{\alpha }}$ and (b) the relative error $\Vert {\mathit{\xi }}^l{-\overline{\mathit{\xi }}\Vert }_2/{\Vert {\mathit{\xi }}^l\Vert }_2$ of recovered parameters.

Figure 4. The relative error of estimated adsorption isotherm parameters for the Transport-Dispersive model with different error levels $\mathit{\delta }$ at every iteration step l. (a) The boundary fitting method and (b) the Kohn–Vogelius method.

Figure 5. The relative errors of estimated adsorption isotherms and mass transfer resistance for the Transport-Dispersive model with different error levels $\mathit{\delta }$ at every iteration step l. (a) The relative errors of estimated adsorption isotherms $\Vert {\mathit{\xi }}^l{-\overline{\mathit{\xi }}\Vert }_2/{\Vert {\mathit{\xi }}^l\Vert }_2$ and (b) The relative errors of estimated mass transfer $\Vert k_f^l-k_f{\Vert }_2/{\Vert k_f^l\Vert }_2$.

Figure 6. The reconstructed adsorption isotherms by KV method for the Equilibrium-Dispersive model.

Figure 7. The reconstructed adsorption isotherms by KV method for the Transport-Dispersive model. (a) Fixed mass transfer resistance $k_f^{(1)}=95$ and $k_f^{(2)}=195$. (b) Estimating mass transfer resistance by inverse algorithm with the constraints $20\le k_f^{(i)}\le 400$, $i=1,2$.

Figure 8. A comparison between the experimental elution profile (solid line) and the simulated total elution profile (dashed line) corresponds to the estimated parameter $\widehat{\mathit{\xi }}$ in the Equilibrium-Dispersive model. (a) Injection profile $h(t)=[0.75,0]\mathrm{mM}$. (b) Injection profile $h(t)=[0,15]\mathrm{mM}$.

Figure 9. (a) A comparison between the experimental elution profile (solid line) with injection $h(t)=[0.75,0.75]mM$ and the simulated total elution profile (dashed line) corresponds to the estimated parameter $\widehat{\mathit{\xi }}$. (b) The value of objective function ${\mathcal{J}}_{BV,\mathit{\alpha }}$ in the logarithmic scale at every iteration step l.