Identification of the Isotherm Function in Chromatography Using CMA-ES

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1 Identifiation of the Isotherm Funtion in Chromatography Using CMA-ES Mohamed Jebalia, Anne Auger, Mar Shoenauer, Franois James, Marie Postel To ite this version: Mohamed Jebalia, Anne Auger, Mar Shoenauer, Franois James, Marie Postel. Identifiation of the Isotherm Funtion in Chromatography Using CMA-ES. IEEE Congress on Evolutionary Computation, Sep 27, Singapour, Singapore. IEEE, pp , 27, Evolutionary Computation, 27. CEC 27. IEEE Congress on.. HAL Id: inria Submitted on 1 Ot 27 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Identifiation of the Isotherm Funtion in Chromatography Using CMA-ES M. Jebalia 1, A. Auger 1, M. Shoenauer 1, F. James 2, M.Postel 3 inria-1782, version 1-1 Ot 27 Abstrat This paper deals with the identifiation of the flux for a system of onservation laws in the speifi example of analyti hromatography. The fundamental equations of hromatographi proess are highly non linear. The state-of-theart Evolution Strategy, CMA-ES (the Covariane Matrix Adaptation Evolution Strategy), is used to identify the parameters of the so-alled isotherm funtion. The approah was validated on different onfigurations of simulated data using either one, two or three omponents mixtures. CMA-ES is then applied to real data ases and its results are ompared to those of a gradient-based strategy. I. INTRODUCTION The hromatography proess is a powerful tool to separate or analyze mixtures [6]. It is widely used in hemial industry (pharmaeutial, perfume and oil industry, et) to produe relatively high quantities of very pure omponents. This is ahieved by taking advantage of the seletive absorption of the different omponents in a solid porous medium. The moving fluid mixture is perolated through the motionless medium in a olumn. The various omponents of the mixture propagate in the olumn at different speeds, beause of their different affinities with the solid medium. The art of hromatography separation requires prediting the different proportions of every omponent of the mixture at the end of the olumn (alled the hromatogram) during the experiment. In the ideal (linear) ase, every omponent has its own fixed propagation speed, that does not depend on the other omponents. In this ase, if the olumn is suffiiently long, pure omponents ome out at the end of the olumn at different times: they are perfetly separated. But in the real world, the speed of a omponent heavily depends on every other omponent in the mixture. Hene, the fundamental Partial Differential Equations of the hromatographi proess, derived from the mass balane, are highly non linear. The proess is governed by a nonlinear funtion of the mixture onentrations, the so-alled Isotherm Funtion. This funtion omputes the amount of absorbed quantity of eah omponent w.r.t. all other omponents. Mathematially speaking, thermodynamial properties of the isotherm ensure that the resulting system of PDEs is hyperboli, and standard numerial tools for hyperboli systems an hene be applied; if the isotherm is known: The preise knowledge of the isotherm is ruial, both from the theoretial viewpoint of physio-hemial modeling and regarding the more pratial preoupation of aurately 1. TAO Projet-Team INRIA Futurs, LRI, Orsay, 2. Mathématiques, Appliations et Physique Mathématique d Orléans, 3. Laboratoire Jaques- Louis Lions UPMC, Paris, {jebalia,auger,mar}@lri.fr, Franois.James@math.nrs.fr, postel@ann.jussieu.fr ontrolling the experiment to improve separation. Speifi hromatographi tehniques an be used to diretly identify the isotherm, but gathering a few points requires several months of areful experiments. Another possible approah to isotherm identifiation onsists in solving the inverse problem numerially: find the isotherm suh that numerial simulations result in hromatograms that are as lose as possible to the atual experimental outputs. This paper introdues an evolutionary method to takle the identifiation of the isotherm funtion from experimental hromatograms. The goal of the identifiation is to minimize the differene between the atual experimental hromatogram and the hromatogram that results from the numerial simulation of the hromatographi proess. Chemial sientists have introdued several parametri models for isotherm funtions (see [6] for all details of the most important models). The resulting optimization problem hene amounts to parametri optimization, that is addressed here using the state-ofthe-art Evolution Strategy, CMA-ES. Setion II introdues the diret problem and Setion III the optimization (or inverse) problem. Setion IV-A reviews previous approahes to the problem based on gradient optimization algorithms [13], [12]. Setion IV-B details the CMA-ES method and the implementation used here. Finally, Setion V presents experimental results: first, simulated data are used to validate the proposed approah; seond, real data are used to ompare the evolutionary approah with a gradient-based method. II. PHYSICAL PROBLEM AND MODEL Chromatography aims at separating the omponents of a mixture based on the seletive absorption of hemial speies by a solid porous medium. The fluid mixture moves down through a olumn of length L, onsidered here to be one-dimensional. The various omponents of the mixture propagate in the olumn at different speeds, beause of their different behavior when interating with the porous medium. At a given time t R +, for a given z [, L] the onentration of m speies is a real vetor of R m denoted (t, z). The evolution of is governed by the following partial differential equation: z + t F() =, (, z) = (z), (1) (t, ) = inj (t). where : R R m is the initial onentration, inj : R R m the injeted onentration at the entrane of the olumn and F : R m R m is the flux funtion that an be expressed

3 in the following way F() = 1 ( + 1 ǫ ) H() u ǫ where H : R m R m is the so-alled isotherm funtion, ǫ (, 1) and u R + [12]. The Jaobian matrix of F being diagonalizable with stritly positive eigenvalues, the system (1) is stritly hyperboli and thus admits an unique solution as soon as F is ontinuously differentiable, and the initial and injetion onditions are pieewise ontinuous. The solution of Eq. 1 an be approximated using any finite differene method that is suitable for hyperboli systems []. A uniform grid in spae and time of size (K+1) (N+1) is defined: Let z (resp. t) suh that K z = L (resp. N t = T ). Then an approximation of the solution of Eq. 1 an be omputed with the Godunov sheme: n k+1 = n k z t (F(n k ) F(n 1 k )) (2) where n k is an approximation of the mean value of the solution at point (k z, n t) 1. For a fixed value of z t, the solution of Eq. 2 onverges to the solution of Eq. 1 as t and z onverge to zero. The numerial sheme given in Eq. 2 is numerially stable under the so-alled CFL ondition stating that the largest absolute value of the eigenvalues of the Jaobian matrix of F is upper-bounded by a onstant z t max Sp( F () ) CFL < 1. (3) III. THE OPTIMIZATION PROBLEM A. Goal The goal is to identify the isotherm funtion from experimental hromatograms: given initial data, injetion data inj, and the orresponding experimental hromatogram exp (that an be either the result of a simulation using a known isotherm funtion, or the result of atual experiments by hemial sientists), find the isotherm funtion H suh that the numerial solution of Eq. 1 using the same initial and injetion onditions results in a hromatogram as lose as possible to the experimental one exp. Ideally, the goal is to find H suh that the following system of PDEs has a unique solution (t, z): z + t F() =, (, z) = (z), (4) (t, ) = inj (t), (t, L) = exp (t). However, beause in most real-world ases this system will not have an exat solution, it is turned into a minimization problem. For a given isotherm funtion H, solve system 1 and define the ost funtion J as the least square differene between the omputed hromatogram H (t, L) and the experimental one exp (t): grid. J (H) = T H (t, L) exp (t) 2 dt () 1 Mean value over the volume defined by the orresponding ell of the If many experimental hromatograms are provided, the ost funtion is the sum of suh funtions J omputed for eah experimental hromatogram. B. Searh Spae hen takling a funtion identifiation problem, the first issue to address is the parametri vs non-parametri hoie [16]: parametri models for the target funtion result in parametri optimization problems that are generally easier to takle but a bad hoie of the model an hinder the optimization. On the other hand, non-parametri models are a priori less biased, but searh algorithms are also less effiient on large unstrutured searh spae. Early trials to solve the hromatography inverse problem using a non-parametri model (reurrent neural-network) have brought a proof-of-onept to suh approah [4], but have also demonstrated its limits: only limited preision ould be reahed, and the approah poorly saled up with the number of omponents of the mixture. Fortunately, hemists provide a whole zoology of parametrized models for the isotherm funtion H, and using suh models, the identifiation problem amounts to parametri optimization. For i {1,..., m}, denote H i the omponent i of the funtion H. The main models for the isotherm funtion that will be used here are the following: The Langmuir isotherm [14] assumes that the different omponents are in ompetition to oupy eah site of the porous medium. This gives, for all i = 1,...,m H i () = N 1 + m l=1 K l l K i i. (6) There are m + 1 positive parameters: the Langmuir oeffiients (K i ) i [1,m], homogeneous to the inverse of a onentration, and the saturation oeffiient N that orresponds to some limit onentration. The Bi-Langmuir isotherm generalizes the Langmuir isotherm by assuming two different kinds of sites on the absorbing medium. The resulting equations are, for all i = 1,...,m H i () = s {1,2} N s 1 + m l=1 K l,s l K i,s i. (7) This isotherm funtion here depends on 2(m + 1) parameters: the generalized Langmuir oeffiients (K i,s ) i [1,m],s=1,2 and the generalized saturation oeffiients (N s ) s=1,2. The Lattie isotherm [17] is a generalization of Langmuir isotherm that also onsiders interations among the different sites of the porous medium. Depending on the degree d of interations (number of interating sites grouped together), this model depends, additionally to the Langmuir oeffiients (K i ) i [1,m] and the saturation oeffiient N, on interation energies (E ij ) i,j [,d],2 i+j d resulting in m i=1 d+i i parameters. For instane, for one omponent (m = 1) and

4 degree 2, this gives: H 1 () = N 2 K 1 + e E 11 RT (K 1 ) K 1 + e E 11 RT (K 1 ), (8) 2 where T is the absolute temperature and R is the universal gas onstant. Note that in all ases, a Lattie isotherm with energies simplifies to the Langmuir isotherm with the same Langmuir and saturation oeffiients up to a fator 1 2. A. Motivations IV. APPROACH DESCRIPTION Previous works on parametri optimization of the hromatography inverse problem have used gradient-based approahes [13], [12]. In [13], the gradient of J is obtained by writing and solving numerially the adjoint problem, while diret differentiation of the disretized equation have also been investigated in [12]. However the fitness funtion to optimize is not neessarily onvex and no results are provided for differentiability. Moreover, experiments performed in [12] suggest that the funtion is multimodal, sine the gradient algorithm onverges to different loal optima depending on the starting point. Evolutionary algorithms (EAs) are stohasti global optimization algorithms, less prone to get stuk in loal optima than gradient methods, and do not rely on onvexity assumptions. Thus they seem a good hoie to takle this problem. Among EAs, Evolution Strategies have been speifially designed for ontinuous optimization. The next setion introdues the state of the art EA for ontinuous optimization, the ovariane matrix adaptation ES (CMA- ES). B. The CMA Evolution Strategy CMA-ES is a stohasti optimization algorithm speifially designed for ontinuous optimization [9], [8], [7], [3]. At eah iteration g, a population of points of an n- dimensional ontinuous searh spae (subset of R n ), is sampled aording to a multi-variate normal distribution. Evaluation of the fitness of the different points is then performed, and parameters of the multi-variate normal distribution are updated. More preisely, let x (g) denotes the mean value of the (normally) sampling distribution at iteration g. Its ovariane matrix is usually fatorized in two terms: σ (g) R +, also alled the step-size, and C (g), a definite positive n n matrix, that is abusively alled the ovariane matrix. The independent sampling of the λ offspring an then be written: ( = x (g) + N k, (σ (g) ) 2 C (g)) for k = 1,...,λ x (g+1) k where N k (, M) denote independent realizations of the multi-variate normal distribution of ovariane matrix M. The µ best offspring are reombined into x (g+1) = µ i=1 w i x (g+1) i:λ, (9) where the positive weights w i R are set aording to individual ranks and sum to one. The index i : λ denotes the i-th best offspring. Eq. 9 an be rewritten as x (g+1) = x (g) µ + w i N i:λ (, (σ (g) ) 2 C (g)), (1) i=1 The ovariane matrix C (g) is a positive definite symmetri matrix. Therefore it an be deomposed in C (g) = B (g) D (g) D (g) ( B (g)) T, where B (g) is an orthogonal matrix, i.e. B ( (g) B (g)) T = I d and D (g) a diagonal matrix whose diagonal ontains the square root of the eigenvalues of C (g). The so-alled strategy parameters of the algorithm, the ovariane matrix C (g) and the step-size σ (g), are updated so as to inrease the probability to reprodue good steps. The so-alled rank-one update for C (g) [9] takes plae as follows. First, an evolution path is omputed: p (g+1) = (1 ) p (g) + (2 )µ ( eff x (g+1) σ (g) ) x (g) where ], 1] is the umulation oeffiient and µ eff is a stritly positive oeffiient. This evolution path an be seen as the desent diretion for the algorithm. Seond the ovariane matrix C (g) is elongated in the diretion of the evolution path, i.e. the rank-one matrix p (g+1) ( p (g+1) ) T is added to C (g) : ( C (g+1) = (1 ov )C (g) + ov p (g+1) p (g+1) where ov ], 1[. The omplete update rule for the ovariane matrix is a ombination of the rank-one update previously desribed and the rank-mu update presented in [8]. The update rule for the step-size σ (g) is alled the path length ontrol. First, another evolution path is omputed: p (g+1) σ = (1 σ ) p (g) σ + σ (2 σ )µ eff σ (g) B (g) D (g) 1 B (g)t ( x (g+1) x (g) ) T ) (11) where σ ], 1]. The length of this vetor is ompared to the length that this vetor would have had under random seletion, i.e. in a senario where no information is gained from the fitness funtion and one is willing to keep the step-size onstant. Under random seletion the vetor p σ (g) is distributed as N(, I d ). Therefore, the step-size is inreased if the length of p (g) σ is larger than E( N(, I d ) ) and dereased if it is shorter. Formally, the update rule reads: ( ( )) σ (g+1) = σ (g) σ p (g+1) σ exp d σ E( N(, I d ) ) 1 (12) where d σ > is a damping fator.

5 The default parameters for CMA-ES were arefully derived in [7], Eqs The only problem-dependent parameters are x () and σ(), and, to some extend, the offspring size λ: its default value is 4+3 log(n) (the µ default value is λ 2 ), but inreasing λ inreases the probability to onverge towards the global optimum when minimizing multimodal fitness funtions [7]. This fat was systematially exploited in [3], where a CMA-ES restart algorithm is proposed, in whih the population size is inreased after eah restart. Different restart riteria are used: 1) RestartTolFun: Stop if the range of the best objetive funtion values of the reent generation is below than a TolFun value. 2) RestartTolX: Stop if the standard deviation of the normal distribution is smaller than a TolX value and σ p is smaller than TolX in all omponents. 3) RestartOnNoEffetAxis: Stop if adding a.1 standard deviation vetor in a prinipal axis diretion of C (g) does not hange x (g). 4) RestartCondCov: Stop if the ondition number of the ovariane matrix exeeds a fixed value. The resulting algorithm (the CMA-ES restart, simply denoted CMA-ES in the remainder of this paper) is a quasi parameter free algorithm that performed best for the CEC 2 speial session on parametri optimization [1]. An important property of CMA-ES is its invariane to linear transformations of the searh spae. Moreover, beause of the rank-based seletion, CMA-ES is invariant to any monotonous transformation of the fitness funtion: optimizing f or h f is equivalent, for any rank-preserving funtion h : R R. In partiular, onvexity has no impat on the atual behavior of CMA-ES. C. CMA-ES Implementation This setion desribes the speifi implementation of CMA-ES to identify n isotherm oeffiients. For the sake of larity we will use a single index in the definition of the oeffiients of the isotherm, i.e we will identify K a, N b and E for a [1, A], b [1, B] and [1, C] where A, B and C are integers summing up to n. Fitness funtion and CFL ondition: The goal is to minimize the fitness funtion defined in Setion III-A. In the ase where identifiation is done using only one experimental hromatogram, the fitness funtion is the funtion J defined in Eq. as the least squared differene between an experimental hromatogram exp (t) obtained using experimental onditions, inj and a numerial approximation of the solution of system (1) for a andidate isotherm funtion H using the same experimental onditions. The numerial simulation of a solution of Eq. 1 is omputed with a Godunov sheme written in C++ (see [1] for the details of the implementation). In order to validate the CMA-ES approah, first experimental hromatograms were in fat omputed using numerial simulations of Eq. 1 with different experimental onditions. Let F sim denotes the flux funtion used to simulate the experimental hromatogram. For the simulation of an approximated solution of Eq. 1, a time step t and a CFL oeffiient stritly smaller than one (typially.8) are fixed beforehand. The quantity max Sp( F sim () ) is then estimated using a power method, and the spae step z an then be set suh that Eq. 3 is satisfied for F sim. The same t and z are then used during the optimization of J. hen exp omes from real data, an initial value for the parameters to estimate, i.e. an initial guess given by the expert is used to set the CFL ondition (3). Using expert knowledge: The hoie of the type of isotherm funtion to be identified will be, in most ases, given by the hemists. Fig 1 illustrates the importane of this hoie. In Fig 1-(a), the target hromatogram exp is omputed using a Langmuir isotherm with one omponent (m = 1 and thus n = 2). In Fig 1-(b), the target hromatogram exp is omputed using a Lattie of degree 3 with one omponent (m = 1 and thus n = 4). In both ases, the identifiation is done using a Langmuir model, with n = 2. It is lear from the figure that one is able to orretly identify the isotherm, and hene fit the experimental hromatogram when hoosing the orret model (Fig 1 (a)) whereas the fit of the hromatogram is very poor when the model is not orret (Fig 1 (b)). Another important issue when using CMA-ES is the initial hoie for the ovariane matrix: without any information, the algorithm starts with the identity matrix. However, this is a poor hoie in ase the different variables have very different possible order of magnitude, and the algorithm will spend some time adjusting its prinipal diretions to those ranges. In most ases of hromatographi identifiation, however, hemists provide orders of magnitudes, bounds and initial guesses for the different values of the unknown parameters. Let [(K a ) min, (K a ) max ], [(N b ) min, (N b ) max] and [(E ) min, (E ) max ] the ranges guessed by the hemists for respetively eah K a, N b and E. All parameters are linearly saled into those intervals from [ 1, 1], removing the need to modify the initial ovariane matrix of CMA-ES. Unfeasible solutions: Two different situations an lead to unfeasible solutions: First when one parameter at least, among parameters whih have to be positive, beomes negative (remember that CMA-ES generates offspring using an unbounded normal distribution), the fitness funtion is arbitrarily set to 1 2. Seond when the CFL ondition is violated, the simulation is numerially unstable, and generates absurd values. In this ase, the simulation is stopped, and the fitness funtion is arbitrarily set to a value larger than 1 6. Note that a better solution would be to detet suh violation before running the simulation, and to penalize the fitness by some amount that would be proportional to the atual violation. But it is numerially intratable to predit in advane if the CFL is

6 Funtion Value (fval, fval minus f_min), Sigma 4 3 Chromatograms Sim Chrom Ident Chrom 1 Chromatograms Sim Chrom Ident Chrom f_min=2.449e log1(abs(value)) (a) Simulation using a Langmuir isotherm, identifiation using a Langmuir model: the hromatogram is perfetly fit. Fig. 1 (b) Simulation using a Lattie isotherm, identifiation using a Langmuir model: poor fit of the hromatogram. IMPORTANCE OF THE CHOICE OF MODEL (ONE COMPONENT MIXTURE) f_reent=2.449e 1 Fig. 2 SINGLE COMPONENT MIXTURE, 1 TIME STEPS. SIMULATE A LATTICE ( PARAMETERS) AND IDENTIFY A LATTICE OF DEGREE 4 ( PARAMETERS): BEST FITNESS VERSUS NUMBER OF EVALUATIONS. THE FIRST RUN GAVE A SATISFACTORY SOLUTION BUT TO RESTARTS HAVE BEEN PERFORMED TO REACH A FITNESS VALUE ( ) LOER THAN going to be violated (see Eq. 3), and the numerial absurd values returned in ase of numerial instability are not learly orrelated with the amount of violation either. Funtion Value (fval, fval minus f_min), Sigma f_min=1.477e 14 Initialization: The initial mean x () for CMA-ES is uniformly drawn in [ 1, 1] n, i.e., the parameters K a, N b and E are uniformly drawn in the ranges given by the expert. The initial step-size σ is set to.3. Besides we rejet individuals of the population sampled outside the initial ranges. Unfeasible individuals are also rejeted at initialization: at least one individual should be feasible to avoid random behavior of the algorithm. In both ases, rejetion is done by resampling until a good individual is got or a maximal number of sampling individuals is reahed. Initial numbers of offspring λ and parents µ are set to the default values (λ = 4+3 log(n) and µ = λ/2 ). Restarting and stopping riteria: The algorithm stops if it reahes restarts, or a given fitness value (typially a value between 1 9 and 1 1 for artifiial problems, and adjusted for real data). Restart riteria (see Setion IV-B) are RestartTolFun with TolFun= 1 12 σ (), RestartTolX with TolX= 1 12 σ (), RestartOnNoEffetAxis and Restart- CondCov with a limit upper bound of 1 14 for the ondition number. The offspring size λ is doubled after eah restart and µ is set equal to λ/2. V. RESULTS A. Validation using artifiial data A first series of validation runs was arried out using simulated hromatograms. Eah identifiation uses one or many experimental hromatograms. Beause the same disretization is used for both the identifiation and the generation of the experimental data, one solution is known (the same isotherm that was used to generate the data), and the best possible fitness is thus zero. Several tests were run using different models for the isotherm, different number of omponents, and different log1(abs(value)) f_reent=1.921e 14 Fig. 3 BINARY COMPONENT MIXTURE, TIME STEPS. SIMULATE A LANGMUIR (3 PARAMETERS) AND IDENTIFY A LATTICE OF DEGREE 3 (1 PARAMETERS): BEST FITNESS VERSUS NUMBER OF EVALUATIONS. THE FIRST RUN GAVE A SATISFACTORY SOLUTION BUT THE MAXIMAL NUMBER (HERE FIVE) OF RESTARTS HAVE BEEN PERFORMED ATTEMPTING TO REACH A FITNESS VALUE OF 1 14, THE BEST FITNESS VALUE ( ) AS REACHED IN THE FOURTH RESTART. numbers of time steps. In all ases, CMA-ES identified the orret parameters, i.e. the fitness funtion reahes values very lose to zero. In most ases, CMA-ES did not need any restart to reah a preision of (1 14 ), though this was neessary in a few ases. This happened when the whole population remained unfeasible during several generations, or when the algorithm was stuk in a loal optimum. Figures 2, 3, 4 show typial evolutions during one run of the best fitness value with respet to the number of evaluations, for problems involving respetively 1, 2 or 3 omponents. Figure 4 is a ase where restarting allowed the algorithm to esape a loal optimum. Speifi tests were then run in order to study the influene of the expert guesses about both the ranges of the variables and the starting point of the algorithm possibly given by

7 log1(abs(value)) Funtion Value (fval, fval minus f_min), Sigma f_min=9.94e f_reent=9.94e 1 Fig. 4 TERNARY COMPONENT MIXTURE, 2 TIME STEPS. SIMULATE A LANGMUIR (4 PARAMETERS) AND IDENTIFY A LANGMUIR (4 PARAMETERS): BEST FITNESS VERSUS NUMBER OF EVALUATIONS. TO RESTARTS ERE NECESSARY: BEFORE THE SECOND RESTART, CMA-ES IS STUCK IN SOME LOCAL OPTIMA (FITNESS OF ORDER OF 1 1 ), IN THE SECOND RESTART, THE ALGORITHM REACHES A FITNESS VALUE OF TABLE I ON THE USEFULNESS OF EXPERT KNOLEDGE: TARGET VALUES FOR LANGMUIR ISOTHERM ARE HERE (K 1,N ) = (.388, 17). EXPERT RANGE IS [.1,.] [, 1], IDE RANGE IS [.1, 1] [, 1]. THE EXPERT GUESS FOR THE STARTING POINT IS A BETTER INITIAL MEAN (ACCORDING TO FITNESS VALUE) THAN RANDOM. THE FIRST 3 LINES GIVE THE PROBABILITIES (COMPUTED OVER 12 RUNS) TO REACH A 1 12 FITNESS VALUE ITHIN THE GIVEN NUMBER OF RESTARTS. THE LAST LINE IS THE RATIO OF THE NUMBER OF EVALUATIONS NEEDED FOR CONVERGENCE (AVERAGED OVER THE RUNS THAT DID CONVERGE) BY THE PROBABILITY OF CONVERGENCE AFTER TO RESTARTS (LINE 3). Range Expert range ide range ide range Starting point No guess No guess Expert guess no restart restart restarts Perf the hemial engineers: In CMA-ES, in a generation g, offspring are drawn from a Gaussian distribution entered on the mean x (g). An expert guess for a good solution an hene be input as the mean of the first distribution x () that will be used to generate the offspring of the first generation. The results are presented in Table I. First 3 lines give the probabilities that a given run onverges (i.e., reahes a fitness value of 1 12 ), omputed on 12 runs, and depending on the number of restarts (this probability of ourse inreases with the number of restarts). The last line is the ratio between the average number of evaluations that were needed before onvergene (averaged over the runs that did onverge), and the probability of onvergene: this ratio measures the performane of the different experimental settings, as disussed in details in [2]. The results displayed in Table I learly demonstrate that a good guess of the range of the variables is the most prominent fator of suess: even without any hint about the starting point, all runs did reah the required preision without any restart. However, when no indiation about the range is available, a good initial guess signifiantly improves the results, without reahing the perfet quality brought by tight bounds on the ranges: saling is more important than rejeting unfeasible individuals at the beginning. Computational ost: The duration of an evaluation depends on the disretization of the numerial sheme (number of spae- and time-steps), and on the number n of unknown parameters to identify. Several runs were preisely timed to assess the dependeny of the omputational ost on both fators. The simple Langmuir isotherm was used to both generate the data and identify the isotherm. Only omputational osts of single evaluations are reported, as the number of evaluations per identifiation heavily depends on many parameters, inluding the possible expert guesses, and in any ase is a random variable of unknown distribution. All runs in this paper were performed on a 1.8GHz Pentium omputer running with a reent Linux system. For one omponent (m = 1, n = 2), and 1, and 1 time steps, the averages of the durations of a single evaluation are respetively.97,.22, and.9 seonds, fitting the theoretial quadrati inrease with the number of time steps (though 3 sample points are too few to demonstrate anything!). This also holds for the number of spae steps as the number of spae steps is proportional to the number of time steps due to the CFL ondition. For an identifiation with a 1-omponent Langmuir isotherm, the total ost of the identifiation is on average 4 seonds for a 1 time steps disretization. hen looking at the dependeny of the omputational ost on the number of unknown parameters, things are not that lear from a theoretial point of view, beause the ost of eah omputation of the isotherm funtion also depends on the number of omponents and on the number of experimental hromatograms to ompare with. Experimentally, for, 2, 3 and 4 variables, the osts of a single evaluation are respetively.9, 1.4, and 2.2 seonds (for a 1 time steps disretization). For an identifiation, the total time is roughly 1 to 2 minutes for 2 variables, 4 to 6 minutes for 3 variables, and 1 to 2 hours for 4 variables. B. Experiments on real data The CMA-ES based approah has also been tested on a set of data taken from [1]. The mixture was omposed of 3 hemial speies: the benzylalohol (BA), the 2- phenylethanol (PE) and the 2-methylbenzylalohol (MBA). Two real experiments have been performed with different proportions of injeted mixtures, with respetive proportions (1,3,1) and (3,1,). Consequently, two real hromatograms have been provided. For this identifiation, Quiñones et a.l. [1] have used a modified Langmuir isotherm model in whih

8 TABLE II COMPARING CMA-ES AND GRADIENT: THE 3-PARAMETERS CASE. SOLUTION ( LINE 1) AND ASSOCIATED FITNESS VALUES ( LINE 2) FOR THE MODIFIED LANGMUIR MODEL (EQ. 13). LINE 3: FOR CMA-ES, MEDIAN (MINIMAL) NUMBER OF FITNESS EVALUATIONS (OUT OF 12 RUNS) NEEDED TO REACH THE CORRESONDING FITNESS VALUE ON LINE 2. FOR GRADIENT, NUMBER OF FITNESS EVALUATIONS NUMBER OF GRADIENT EVALUATIONS FOR THE BEST OF THE 1 RUNS DESCRIBED CMA-ES IN [12]. Gradient N i (12.91,13.319,16.93) ( ,13.74,19.637) Fitness # Fit evals. 17 (7) 28 (23) BA:PE=1:3:1, VL=1.ml BA exp BA ident PE exp PE ident (a) 1:3:1, BA, PE MBA=1:3:1, VL=1.ml MBA exp MBA ident BA:PE=3:1, VL=.ml BA exp BA ident PE exp PE ident (b) 3:1, BA, PE MBA=3:1, VL=.ml MBA exp MBA ident eah speies has a different saturation oeffiient N i : H i () = N i l=1 K l l K i i, i = 1,..., 3. (13) Six parameters are to be identified: N i and K i, for i = 1,...,3. A hange of variable has been made for those tests so that the unknown parameters are in fat N i and K i, where K i = K i N i : those are the values that hemial engineers are able to experimentally measure. Two series of numerial tests have been performed using a gradient-based method [12]: identifiation of the whole set of 6 parameters, and identifiation of the 3 saturation oeffiients N i only, after setting the Langmuir oeffiients to the experimentally measured values (K 1,K 2,K 3) = (1.833, 3.18, 3.11). The initial ranges used for CMA- ES are [6, 2] [6, 2] [6, 2] (resp. [1., 2.] [2.7, 3.7] [3, 4] [9, 2] [1, 2] [1, 21]) when optimizing 3 parameters (resp. 6 parameters). Comparisons between the two experimental hromatograms and those resulting from CMA-ES identifiation for the two experiments are shown in Figure, for the 6-parameters ase. The orresponding plots in the 3-parameters ase are visually idential though the fitness value is slightly lower than in the 6-parameters ase (see Tables II and III). But another point of view on the results is given by the omparison between the identified isotherms and the (few) experimental values gathered by the hemial engineers. The usual way to present those isotherms in hemial publiations is that of Figure 6: the absorbed quantity H() i of eah omponent i = 1, 2, 3 is displayed as a funtion of the total amount of mixture ( ), for five different ompositions of the mixture [12]. Identified (resp. experimental) isotherms are plotted in Figure 6 using ontinuous lines (resp. disrete markers), for the 6-parameters ase. Here again the orresponding plots for the 3-parameters ase are visually idential. C. Comparison with a Gradient Method CMA-ES results have then been ompared with those of the gradient method from [12], using the same data ase of ternary mixture taken from [1] and desribed in previous Setion. Chromatograms found by CMA-ES are, aording () 1:3:1, MBA Fig (d) 3:1, MBA EXPERIMENTAL CHROMATOGRAMS (MARKERS) AND IDENTIFIED CHROMATOGRAMS (CONTINUOUS LINE) FOR THE BA, BE AND MBA SPECIES. PLOTS ON THE LEFT/RIGHT CORRESPOND TO AN INJECTION ITH PROPORTIONS (1,3,1)/(3,1,). TABLE III COMPARING CMA-ES AND GRADIENT: THE 6-PARAMETERS CASE. SOLUTIONS ( LINES 1 AND 2) AND ASSOCIATED FITNESS VALUES ( LINE 3) FOR THE MODIFIED LANGMUIR MODEL (EQ. 13). CMA-ES Gradient K i (1.861,3.12,3.63) (1.78,3.9,3.47) Ni ( ,134.86, ) ( ,141.7,168.49) Fitness to the fitness (see Tables II and III), loser to the experimental ones than those obtained with the gradient method. Moreover, ontrary to the gradient algorithm, all 12 independent runs of CMA-ES onverged to the same point. Thus, no variane is to be reported on Tables II and III. Furthermore, there seems to be no need, when using CMA-ES, to fix the 3 Langmuir oeffiients in order to find good results: when optimizing all 6 parameters, the gradient approah ould not reah a value smaller than.1, whereas the best fitness found by CMA-ES in the same ontext is (Table III). Finally, when omparing the identified isotherms to the experimental ones (figure 6), the fit is learly not very satisfying (similar deeptive results were obtained with the gradient method in [12]): Fitting both the isotherms and the hromatograms seem to be ontraditory objetives. Two diretions an lead to some improvements in this respet:

9 H() BA (g/l) (a) BA H() PE (g/l) H() MBA (g/l) () PE Fig (b) MBA ISOTHERMS ASSOCIATED TO PARAMETERS VALUES OF TABLE III (CONTINUOUS LINE) AND EXPERIMENTAL ONES (MARKERS) VERSUS TOTAL AMOUNT OF THE MIXTURE FOR DIFFERENT PROPORTIONS OF THE COMPONENT IN THE INJECTED CONCENTRATION [12]. modify the ost funtion J in order to take into aount some least-square error on the isotherm as well as on the hromatograms; or use a multi-objetive approah. Both modifiations are easy to implement using Evolutionary Algorithms (a multi-objetive version of CMA-ES was reently proposed [11]), while there are beyond what gradient-based methods an takle. However, it might also be a sign that the modified Langmuir model that has been suggested for the isotherm funtion is not the orret one. Comparison of onvergene speeds: Tables II and III also give an idea of the respetive omputational osts of both methods on the same real data. For the best run out of 1, the gradient algorithm reahed its best fitness value after 21 iterations, requiring on average 7 evaluations per iteration for the embedded line searh. Moreover, the omputation of the gradient itself is ostly roughly estimated to 4 times that of the fitness funtion. Hene, the total ost of the gradient algorithm an be onsidered to be larger than 22 fitness evaluations. To reah the same fitness value ( ), CMA-ES only needed 17 fitness evaluations (median value out of 12 runs). To onverge to its best value ( , best run out of 12) CMA-ES needed 28 fitness evaluations. Those results show that the best run of the gradient algorithms needs roughly the same amount of funtions evaluations than CMA-ES to onverge. Regarding the robustness issue, note that CMA-ES always reahed the same fitness value, while the 1 different runs of the gradient algorithm from 1 different starting points gave 1 different solutions: in order to assess the quality of the solution, more runs are needed for the gradient method than for CMA-ES! VI. CONCLUSIONS This paper has introdued the use of CMA-ES for the parametri identifiation of isotherm funtions in hromatography. Validation tests on simulated data were useful to adjust the (few) CMA-ES parameters, but also demonstrated the importane of expert knowledge: hoie of the type of isotherm, ranges for the different parameters, and possibly some initial guess of a not-so-bad solution. The proposed approah was also applied on real data and ompared to previous work using gradient methods. On this data set, the best fitness found by CMA-ES is better than that found by the gradient approah. Moreover, the results obtained with CMA-ES are far more robust: (1) CMA- ES always onverges to the same values of the isotherm parameters, independently of its starting point; (2) CMA-ES an handle the full problem that the gradient method failed to effiiently solve: there is no need when using CMA-ES to use experimental values of the Langmuir parameters in order to obtain a satisfatory fitness value. Note that the fitness funtion only takes into aount the fit of the hromatograms, resulting in a poor fit on the isotherms. The results onfirm the ones obtained with a gradient approah, and suggest to either inorporate some measure of isotherm fit in the fitness, or to try some multi-objetive method probably the best way to go, as both objetives (hromatogram and isotherm fits) seem somehow ontraditory. ACKNOLEDGMENTS This work was supported in part by MESR-CNRS ACI NIM Chromalgema. The authors would like to thank Nikolaus Hansen for the Silab version of CMA-ES, and for his numerous useful omments. REFERENCES [1] Comparison of evolutionary algorithms on a benhmark funtion set. [2] A. Auger and N. Hansen. Performane evaluation of an advaned loal searh evolutionary algorithm. In Pro. IEEE Congress On Evolutionary Computation, pages , 2. [3] A. Auger and N. Hansen. A restart CMA evolution strategy with inreasing population size. In Pro. IEEE Congress On Evolutionary Computation, pages , 2. [4] A. Fadda and M. Shoenauer. Evolutionary hromatographi law identifiation by reurrent neural nets. In D.B. Fogel and. Atmar, editors, Pro. 3 rd Annual Conferene on Evolutionary Programming, pages MIT PRESS, [] E. Godlewski and P.-A. Raviart. Hyperboli systems of onservation laws, volume 3/4 of Mathematiques et appliations. Ed. Ellipses, SMAI, [6] G. Guiohon, A. Feilinger, S. Golshan Shirazi, and A. Katti. Fundamentals of preparative and nonlinear hromatography. Aademi Press, Boston, seond edition, 26. [7] N. Hansen and S. Kern. Evaluating the CMA evolution strategy on multimodal test funtions. In Xin Yao et al., editors, Parallel Problem Solving from Nature - PPSN VIII, LNCS 3242, pages Springer, 24.

10 [8] N. Hansen, S. D. Müller, and P. Koumoutsakos. Reduing the time omplexity of the derandomized evolution strategy with ovariane matrix adaptation. Evolutionary Computation, 11(1):1 18, 23. [9] N. Hansen and A. Ostermeier. Completely derandomized selfadaptation in evolution strategies. Evolutionary omputation, 9(2):19 19, 21. [1] J.C. Ford I. Quiñones and G. Guiohon. High onentration band profiles and system peaks for a ternary solute system. Anal. Chem, pages , 2. [11] Ch. Igel, N. Hansen, and S. Roth. Covariane matrix adaptation for multi-objetive optimization. Evolutionary Computation, 1(1):1 28, 27. [12] F. James and M. Postel. Numerial gradient methods for flux identifiation in a system of onservation laws. Journal of Engineering Mathematis, 27. to appear. [13] F. James, M. Sepúlveda, I. Qui nones, F. Charton, and G. Guiohon. Determination of binary ompetitive equilibrium isotherms from the individual hromatographi band profiles. Chem. Eng. Si., 4(11): , [14] I. Langmuir. The adsorption of gases on plane surfaes of glass, mia and platinum. Jour. Am. Chem. So., 4(9): , [1] F. James (PI). CHROMALGEMA, Numerial Resolution of the Inverse Problem for Chromatography using Evolutionary Algorithms and Adaptive Multiresolution. [16] M. Shoenauer and M. Sebag. Using Domain Knowledge in Evolutionary System Identifiation. In K. Giannakoglou et al., editor, Evolutionary Algorithms in Engineering and Computer Siene. John iley, 22. [17] P. Valentin, F. James, and M. Seplveda. Statistial thermodynamis models for a multiomponent two-phases equilibrium isotherm. Math. Models and Methods in Applied Siene, 1:1 29, 1997.

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