ROM construction for batch chromatography

Recall that the fully discrete system for batch chromatographic model is given in (2.15). That is, the FOM reads

Acn+1_z = Bcn_z + dn_z − 1 −   ∆th n z, (6.3) q_zn+1= q_zn+ ∆thn_z, z = a, b, (6.4) where cn_z := cn_z(µ), q_zn:= q_zn(µ), d_zn := dn_z(µ), hn_z := hn_z(µ) ∈ RN, A, B ∈ RN ×N. Detailed description can be found in Section 2.2.1. The parameter µ characterizes the operating conditions, i.e., µ := (Q, tin) in this work.

We now construct a RB for each field variable. Let N ∈ N be the number of the RB vectors for cz and qz, and M ∈ N be the number of the CRB vectors for the operators

ha and hb. Here for simplicity of analysis, we use the same dimension N of the RB for

ca, cb, qa and qb, but one can certainly take different dimensions for the RB. This also

applies to haand hb. Assume that Gz ∈ RN ×M is the CRB for the nonlinear operator hz,

and Vcz, Vqz ∈ R

N ×N _VT

czVcz = I, V T

qzVqz = I
are the RB for the field variables cz and qz,

respectively, i.e., hn_z ≈ Gzβzn, cnz ≈ ˆcnz := Vcza n cz, q n z ≈ ˆqzn:= Vqza n qz, n = 0, . . . , K. (6.5)

Applying Galerkin projection and empirical operator interpolation, we formulate the ROM for the FOM (6.3)–(6.4) as follows:

ˆ Acza n+1 cz = ˆBcza n cz+ d n 0dˆcz − 1 −   ∆t ˆHczβ n z, (6.6) an+1_qz = an_qz+ ∆t ˆHqzβ n z, z = a, b, (6.7)

where an_cz := an_cz(µ), an_qz := an_qz(µ) ∈ RN are the unknown vectors of the ROM, and ˆ Acz = V T czAVcz, ˆBcz = V T czBVcz, ˆdcz = V T cze1, ˆHcz = V T czGz, and ˆHqz = V T qzGz are the reduced matrices.

Note that βn_z := βn_z(µ) = (βz1n, . . . , βznM)T ∈ RM are the vectors of coefficients for the

empirical interpolation of the nonlinear operator hn_z, and they are parameter- and time- dependent. The evaluation of β_zn is essentially the same as the computation of the coef- ficients σi(µ) in (3.30) in Algorithm 3. More specifically, βnz are obtained by solving the

following system of equations:

M

X

i=1

βzniGz i(xj) = hznj, j = 1, . . . , M.

Here, the evaluation of hznj only needs the jth entries (canj, cbnj and qznj) of the solution

vectors (can, cbn and qzn), i.e., hznj = hz(canj, cbnj, qznj). For the general operator empirical

interpolation, the value of the operator at the interpolation point (e.g., xj) may depend

on more entries of the solution vectors (e.g., the jth entries and their neighbors) [49, 87]. As discussed in Section 3.5, the efficiency of the RB approximation is ensured by a strategy of suitable offline-online decomposition. During the offline stage, given the training sets Pcrb

train and Ptrain(they can be chosen differently), Algorithm 3 is implemented to generate

the CRB Gz for the nonlinear operator hz. Then Algorithm 5 is used to generate the

RB matrices Vcz and Vqz for cz and qz, respectively. As a result, all N -dependent and

µ-independent terms are precomputed and assembled to construct the reduced matrices (e.g., ˆAcz, ˆBcz, ˆdcz, ˆHcz, and ˆHqz). For a newly given parameter µ ∈ P, the small-sized

ROM (6.6)–(6.7) is rapidly assembled and solved online so that the solution to the FOM (6.3)–(6.4) can be recovered by (6.5).

Performance of ASS To investigate the performance of the technique of ASS, we compare the runtime for RB and CRB generation with different threshold values εASS. As

shown in Algorithm 5 in Chapter 5, the ASS technique is combined with the POD-Greedy algorithm and is used for RB generation. The error indicator ψN(µ?) in Algorithm 5

involves the contribution from EI. To efficiently generate a CRB for EI, the ASS technique is also employed. The training set for CRB generation is a sample set with 25 sample points of µ = (Q, tin), uniformly distributed in the parameter domain. For each sample point,

Algorithm 4 is used to adaptively choose the snapshots for the generation of the CRB. The runtime of CRB generation with different choices of εASS is shown in Table 6.3. It is seen

that the larger threshold εASS is used, the more the runtime is saved. A lot of redundant

information is discarded due to the adaptive selection process. Particularly, with the tolerance εASS= 1.0 × 10−4, the computational time is reduced by 90.3% compared with

that of the original algorithm without ASS. It should be mentioned that the tolerance εASS

cannot be taken too large; otherwise too much information from the chosen parameter will be discarded, and the parameter might be selected again afterward due to the bad approximation at this parameter. A repeated selection requires one more full simulation at

6.1 MOR for batch chromatography

this parameter, which probably takes more time. Nevertheless, how to choose an optimal threshold is empirical and problem-dependent.

Table 6.3: Comparison of runtime of the generation of CRBs (Ga, Gb) at the same error

tolerance (εCRB= 1.0 × 10−7) with different thresholds for ASS. M0 = 1 is the number of

the basis vectors for error estimation.

εASS kξM +M0_,ak kξ_{M +M}0_,bk M (G_aG_b) Runtime (h)

no ASS – 9.2 × 10−8 8.5 × 10−8 146 152 62.50 (-) ASS 1.0 × 10−4 9.6 × 10−8 8.1 × 10−8 147 152 6.05 (−90.3%) ASS 1.0 × 10−3 8.7 × 10−8 9.9 × 10−8 147 152 3.62 (−94.2%) ASS 1.0 × 10−2 9.4 × 10−8 6.2 × 10−8 144 150 2.70 (−95.7%) Table 6.4 shows the comparison of the runtime for RB generation by using the POD- Greedy algorithm with and without ASS. Note that the CRB is precomputed with εASS=

1.0 × 10−4 for the ASS, and the corresponding runtime for CRB generation is not included here. The training set is a sample set with 60 points uniformly distributed in the parameter domain. Here and in the following, the tolerances are chosen as εCRB = 1.0 × 10−7,

εROM= 1.0 × 10−4, εASS= 1.0 × 10−4. It is seen that the runtime for generating the ROM

with ASS is reduced by 54.1% compared with that without ASS at the same tolerance εRB. Moreover, the accuracy of the resulting ROM with ASS is almost the same as that

without ASS.

Table 6.4: Comparison of the runtime for RB generation using the POD-Greedy algorithm with and without ASS.

Algorithms Runtime (h)1

POD-Greedy 14.8

ASS-POD-Greedy 6.8 (−54.1%)

1_{Due to memory limitations of the PC, this computation was done on a workstation}

with 4 Intel Xeon E7-8837 CPUs (8 cores per CPU) 2.67 GHz, 1 TB RAM.

It is worth noting that the ASS technique is devised for effectively collecting snapshots, and it is independent of the error indicator (true error or error bound) employed for the basis construction. Thus, the ASS is also applicable to other snapshot-based MOR methods, e.g., the POD method and the POD-DEIM method [43].