**6.1 Study for a particular parameter couple** ð Þ *Ca***,** *a=***ℓ**

The objective of this subsection being to illustrate the methods on an example and show how to apply them, we only consider the snapshot FOM solutions for ð Þ¼ *Ca*, *a=*ℓ ð Þ 0*:*1, 0*:*9 for the sake of simplicity and brevity. **Figure 1** shows both membrane shapes and positions in the channel at different instants.

At each time *t <sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*Δ*t*, where the time step is equal to <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>0</sup>*:*04, a snapshot is saved and stored in the database. Note that all time quantities are nondimensionalized by the factor ℓ*=V*. The resulting generated data matrix is used as the entry matrix for the ROM learning process. Then a truncated SVD is applied to get the spatial POD modes. In **Figure 2**, the four first eigenmodes Φ*<sup>k</sup>* are plotted (more precisely this is a superposition of each mode onto the original spherical shape for a better visualization and understanding of their influence). Based on, the graphics *k*↦1 � *RIC k*ð Þ plotted in log scale in **Figure 3**, we decide to use a truncation rank *K* equal to *<sup>K</sup>* <sup>¼</sup> 10, returning a relative information content of about 1 � <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> .

Then a reduced-order dynamical system for the capsule time evolution is searched. In this example, we compare two models: the first one is the affine approximation (denoted by ROM-A)

$$\mathfrak{a}\_{\mu}^{n+1} = A\_{\mu}\mathfrak{a}\_{\mu}^{n} + \mathfrak{b}\_{\mu} \tag{39}$$

with the matrix *A* and the vector *b* to identify. It is equivalent to consider the vector of features *<sup>η</sup>*ð Þ¼ *<sup>a</sup>* ð Þ *<sup>a</sup>*, 1 *<sup>T</sup>*. The second nonlinear model is built from the observable vector *<sup>η</sup>*ð Þ¼ *<sup>a</sup>* ð Þ *<sup>a</sup>*, *<sup>κ</sup>*ð Þ *<sup>a</sup>* , 1 *<sup>T</sup>* with the recurrent time scheme

$$\boldsymbol{\sigma}\_{\mu}^{n+1} = \boldsymbol{A}\_{\mu} \boldsymbol{\eta} \left( \boldsymbol{\sigma}\_{\mu}^{n} \right) \tag{40}$$

**Figure 1.**

*Example of a microcapsule dynamics within a square-base channel for (Ca*, *a=*ℓ*) = (0.1, 0.90). Shapes and locations of the initially spherical capsule are shown at t* ¼ 0 *(in transparency), 0.4, 2.8, 5.2, 7.6.*

*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*

**Figure 2.**

*SVD: Four first spatial principal components computed by the HOSVD. Each mode has been added on the initial spherical shape and amplified by a factor 2 for better visualization. Higher-order modes show oscillations at the rear of the capsule.*

**Figure 3.**

*SVD: Plot of k*↦1 � RICð Þ*k , where* RICð Þ*k represents the relative information content at truncation rank k.*

(ROM-B). For ROM-B, the Gaussian kernel function (21) is used. The standard deviation parameter *<sup>σ</sup>* in (21) is chosen as *<sup>σ</sup>* <sup>¼</sup> max *<sup>n</sup>*∥*a<sup>n</sup> <sup>μ</sup>*<sup>∥</sup> <sup>¼</sup> <sup>∥</sup>*a<sup>N</sup>*þ<sup>1</sup> *<sup>μ</sup>* <sup>∥</sup>2. In both cases ROM-A and ROM-B, the determination of the matrix *A<sup>μ</sup>* by minimization of the least square problem (37) leads to a very small residual. In **Figure 4**, the logarithm of the relative time residual

$$n \mapsto \log\_{10} \left( \left\| \boldsymbol{\sigma}\_{\mu}^{n+1} - \boldsymbol{A}\_{\mu} \boldsymbol{\eta} \left( \boldsymbol{a}\_{\mu}^{n} \right) \right\|^{2} / \left\| \boldsymbol{a}\_{\mu}^{n+1} \right\|^{2} \right) \tag{41}$$

is plotted for each ROM model. One can observe values between 10�<sup>14</sup> and 10�8. The residual for ROM-B appears to be slightly smaller than that of ROM-A thanks to the added nonlinear terms. A surprising result is that the affine ROM-A model returns rather accurate results whereas the fluid–structure interaction problem is intrinsically nonlinear. In order to study the stability of the model ROM-A, in **Figure 5** we plot the complex eigenvalues of the square matrix *Aμ*. We observe that all the eigenvalues have a modulus less or equal to one. One of the eigenvalues is equal to 1 exactly up to Double Precision roundoff errors, meaning that there is a physical invariant in the system. It is known that the capsule volume is kept constant during time, because of the incompressibility of the fluid. For ROM-A, since *<sup>a</sup><sup>n</sup>*þ<sup>1</sup> *<sup>μ</sup>* <sup>¼</sup> *<sup>A</sup>μa<sup>n</sup> <sup>μ</sup>* <sup>þ</sup> *<sup>b</sup>* and *<sup>a</sup>*<sup>0</sup> *<sup>μ</sup>* ¼ **0**, we have

$$\mathfrak{a}\_{\mu}^{n} = A\_{\mu}^{n} \mathfrak{a}\_{\mu}^{0} + \sum\_{k=0}^{n-1} A\_{\mu}^{k} \mathfrak{b} = \sum\_{k=0}^{n-1} A\_{\mu}^{k} \mathfrak{b}. \tag{42}$$

The matrix *A<sup>μ</sup>* is observed to be diagonalizable in ℂ. There is an invertible matrix *<sup>P</sup><sup>μ</sup>* such that *<sup>A</sup><sup>μ</sup>* <sup>¼</sup> *<sup>P</sup>μ*Λ*μP*�<sup>1</sup> *<sup>μ</sup>* where Λ*<sup>μ</sup>* is the diagonal matrix of the eigenvalues. Since it is observed that *ρ* Λ*<sup>μ</sup>* � � <sup>¼</sup> 1, we have

$$\|\boldsymbol{\mathfrak{a}}\_{\mu}^{n}\| \leq \text{Cond}\left(P\_{\mu}\right)n\|\mathbf{b}\|, \quad \forall n \in \mathbb{N},\tag{43}$$

#### **Figure 4.**

*Matrix identification. Log of the normalized residual n*<sup>↦</sup> log <sup>10</sup> <sup>∥</sup>*a<sup>n</sup>*þ<sup>1</sup> *<sup>μ</sup>* � *<sup>A</sup>μ<sup>η</sup> <sup>a</sup><sup>n</sup> μ* � �∥<sup>2</sup>*=*∥*a<sup>n</sup>*þ<sup>1</sup> *<sup>μ</sup>* <sup>∥</sup><sup>2</sup> � � *for both ROM-A and ROM-B.*

*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*

**Figure 5.**

*Matrix identification. Eigenvalues of the computed matrix A<sup>μ</sup> plotted in the complex plane for the ROM-A model. One of the eigenvalue is* 1 *up to round-off error.*

showing that the coefficients in the PCA space grow at most linearly in time.

In **Figure 6**, we compare the computed capsule shapes and positions in the channel for the computed FOM capsules obtained at different times: *t* ¼ 0, 0*:*4, 2*:*8, 5*:*2 and 7*:*6. What can be observed is that the ROM-B model returns very satisfactory results where the shape solutions fully overlap the FOM ones'at the eye norm'. Finely, we plot in **Figure 7** the time evolution of the errors in the capsule 3D shape of the ROM solutions as compared to the FOM solutions. The difference between the shapes are quantified by *ε*Shapeð Þ*t* , the ratio between the modified Hausdorff distance (MHD) computed between the FOM shape SFOMð Þ*t* and the ROM shape SROMð Þ*t* and the channel characteristic length ℓ:

$$\varepsilon\_{\text{Shape}}(t) = \frac{\text{MHD}(\mathcal{S}\_{\text{FOM}}(t), \mathcal{S}\_{\text{ROM}}(t))}{\ell}. \tag{44}$$

The modified Hausdorff distance measures the maximum value of the mean distance between the two shapes to compare [17]. The ROM-A and ROM-B

#### **Figure 6.**

*Sequence of cross-section capsule shapes and positions in the microchannel from the initial spherical shape shown in light green at the beginning of the channel: Comparison of the FOM solutions (gray dots) and of the solutions computed from the dynamical k-DMD reduced-order model (dark green solid line) at the same instants as in Figure 1: t* ¼ 0, 0*:*4, 2*:*8, 5*:*2, 7*:*6*.*

#### **Figure 7.**

*(a) Comparison of the time evolution of the shape error between the affine DMD model (ROM-A with K* ¼ 10*) and the kernel-based one (ROM-B with K* ¼ 10*, M* ¼ 5*). One can observe a maximum error less than 0.1% in both cases. (b) Sensitivity analysis of the parameter <sup>σ</sup> (*∥*a<sup>N</sup>*þ<sup>1</sup>∥<sup>2</sup> <sup>¼</sup> <sup>667</sup>*:*14*.)*

models return very accurate solutions with maximum 0.1% error. It is also observed that the ROM-B models is slightly more accurate than the affine approximation.

### **6.2 HOSVD on the whole data tensor and error measurements on the whole database**

Now the consider the whole database made of 55 samples in the parameter domain. In **Figure 8** we plot the location of the 55 chosen samples in the plane ð Þ *Ca*, *a=*ℓ . The design zone of interest corresponds to capsule shapes that can reach a steady state after a certain time.

A SVD is first performed on the *μ*-flattening of the data tensor T *<sup>x</sup>*. In **Figure 9**, we plot the four first parameter (normalized) eigenmodes in the parameter domain.

*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*

#### **Figure 8.**

*Samples of the design of experiment in the parameter space Ca* ð Þ , *a=*ℓ . *The zone of interest corresponds to capsule shapes that reach a steady state after a certain time.*

**Figure 9.** *HOSVD: The first four parameter eigenmodes in the parameter domain, computed from the μ-flattening of the data cube.*

These modes give an idea on the dependency of the capsule shapes with respect to the parameters. To complete the analysis, we show in **Figure 10** the spectrum of the singular values for the *μ*-flattening matrix. One can observe a rather fast decay of the singular values especially for the ten first modes.

**Figure 10.** *Spectrum of the singular values of the μ-flattening matrix.*

Next, we perform the SVD of the time *t*-flattening matrix of T *<sup>x</sup>*. The SVD provides us temporal eigenmodes. In **Figure 11**, the four first temporal eigenmodes *ωm*, *m* ¼ 1, … , 4 are plotted. The first one appears to be the linear function while the others add details especially in the transient zone of the capsule dynamics. The spectrum of the singular values again shows a fast decay especially for the six first modes.

To conclude this section, we have tested the accuracy of both ROM-A and ROM-B on the whole database. For each sample, we have derived a ROM model, i.e. a loworder dynamical system formulated in the PCA space. Then we have compared the ROM solution to the FOM solution by calculating *ε*Shape between the two capsule shapes. In **Figure 12**, the heat maps of *ε*Shape are plotted for ROM-A and ROM-B. One can observe a uniform accuracy over the whole parameter domain, with errors less than 0.1%, thus showing the efficiency of the approach. Reported computational speedups are between 5000 and 10,000 using ROM models. A computer with two Intel Xeon GOLD 6130 CPU (2.1 Ghz) has been used for the numerical tests.

*HOSVD. Left: First four temporal eigenmodes computed from the SVD decomposition of the t-flattening data tensor. Right: Spectrum of the singular values.*

*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*

**Figure 12.**

*Heat maps of the modified Hausdorff distance between the FOM solutions and the ROM ones at dimensionless time Vt=*ℓ ¼ 10. *Left: ROM-A, right: ROM-B. Errors are less than 0.1%.*
