Fast model updating coupling Bayesian inference and PGD model reduction

Abstract

The paper focuses on a coupled Bayesian-Proper Generalized Decomposition (PGD) approach for the real-time identification and updating of numerical models. The purpose is to use the most general case of Bayesian inference theory in order to address inverse problems and to deal with different sources of uncertainties (measurement and model errors, stochastic parameters). In order to do so with a reasonable CPU cost, the idea is to replace the direct model called for Monte-Carlo sampling by a PGD reduced model, and in some cases directly compute the probability density functions from the obtained analytical formulation. This procedure is first applied to a welding control example with the updating of a deterministic parameter. In the second application, the identification of a stochastic parameter is studied through a glued assembly example.

Appendix

In this section the details of a PGD solution is illustrated on the welding example from Sect. 4.

1.1 Problem

The same problem as in Sect. 4 is considered with the convection-diffusion equation:

$$\begin{aligned} \frac{\partial T}{\partial t} + \underline{v}.\underline{\text {grad}} T - \varDelta T = s \end{aligned}$$

(61)

With:

$$\begin{aligned} s(x,y;\sigma )=\frac{u}{2 \pi \sigma ^2} \text {exp} \left( - \frac{ \left( x-x_c\right) ^2+ \left( y-y_c\right) ^2 }{2 \sigma ^2}\right) \end{aligned}$$

(62)

The purpose is to build a multiparametric reduced order model with separation of space, time and parameter $\sigma$.

1.2 Progressive Galerkin PGD

As presented in Sect. 3 the PGD modes are built recursively thanks to the Galerkin orthogonality. The spaces of variation of each parameter is defined as follows: $I=[0,T_f]$ the time interval and $\varSigma =[\sigma _\text {min},\sigma _\text {max}]$, the space of variation of $\sigma$. The admissible field spaces are defined:

$$\begin{aligned} \mathcal {T}= & {} \{ T \in H^1(\varOmega =]0;5[\times ]0;1[), T=0 \text { on } \varGamma _D \} \end{aligned}$$

(63)

$$\begin{aligned} \mathcal {I}= & {} \{T, \int _I{\Vert T(x,y,.;\sigma ) \Vert ^2_{H^1(\varOmega )}} < \infty , \forall (x,y,\sigma ) \in \varOmega \times \varSigma \} \end{aligned}$$

(64)

$$\begin{aligned} \mathcal {E}= & {} \{T, \int _{\varSigma }{\Vert T(x,y,t;.) \Vert ^2_{H^1(\varOmega )}} < \infty , \forall (x,y,t) \in \varOmega \times I \} \end{aligned}$$

(65)

The weak formulation of Eq. (61) on each space reads:

Find $T \in \mathcal {T} \otimes \mathcal {I} \otimes \mathcal {E}$, such as $\forall T^* \in \mathcal {T} \otimes \mathcal {I} \otimes \mathcal {E}$:

$$\begin{aligned} a(T,T^*)=l(T^*) \end{aligned}$$

(66)

with:

$$\begin{aligned}&a(T,T^*)= \end{aligned}$$

(67)

$$\begin{aligned}&\int _{I \times \varSigma \times \varOmega } \frac{\partial T}{\partial t}.T^* + \underline{v}.\underline{\text {grad}} T .T^*+\underline{\text {grad}} T .\underline{\text {grad}} T^* dt d\sigma d \varOmega \end{aligned}$$

(68)

$$\begin{aligned}&l(T^*) = \int _{I \times \varSigma \times \varOmega } s.T^* dt d\sigma d \varOmega \end{aligned}$$

(69)

The solution is searched in the separated form:

$$\begin{aligned} T_{m}(x,y,t;\sigma )=\sum _{n=1}^{m} \varLambda _n(x,y) \lambda _n(t) \alpha _n(\sigma ) \end{aligned}$$

(70)

The $m-1$ first modes are supposed to be known and the m-th mode is searched. Then the solution reads:

$$\begin{aligned}&\{T_m(x,y,t;\sigma ) \} \nonumber \\&\quad =\sum _{n=1}^{m-1} \lambda _n(t) \alpha _n(\sigma ) \varLambda _n(x,y)+ \lambda (t) \alpha (\sigma ) \varLambda (x,y) \end{aligned}$$

(71)

The unknowns are the functions: $\varLambda$, $\lambda$ et $\alpha$.

The test field $T^* \in \mathcal {T} \otimes \mathcal {I} \otimes \mathcal {E}$ is taken in the separated form:

$$\begin{aligned} T^*=\lambda ^* \alpha \varLambda +\lambda \alpha ^* \varLambda +\lambda \alpha \varLambda ^* \end{aligned}$$

(72)

Using this form, the variational formulation (66) leads to decoupled problems with the applications $S_m$, $T_m$, $P_m$ such as:

$$\begin{aligned}&\varLambda =S_m(\lambda ,\alpha ) \end{aligned}$$

(73)

$$\begin{aligned}&\lambda =T_m(\alpha , \varLambda ) \end{aligned}$$

(74)

$$\begin{aligned}&\alpha =P_m(\lambda , \varLambda ) \end{aligned}$$

(75)

1.2.1 Spatial application $S_m$

The decoupled weak formulation for space problem reads:

$$\begin{aligned} a\left( T_{m-1}+ \lambda \alpha \varLambda ,\lambda \alpha \varLambda ^*\right) =l( \lambda \alpha \varLambda ^*) \end{aligned}$$

(76)

with:

$$\begin{aligned}&a\left( \lambda \alpha \varLambda ,\lambda \alpha \varLambda ^*\right) \nonumber \\&\quad =\int _{I \times \varSigma \times \varOmega } \lambda \dot{\lambda } \alpha ^2 \varLambda \varLambda ^* +\alpha ^2 \lambda ^2 \underline{v}.\underline{\text {grad}} \varLambda \varLambda ^*\nonumber \\&\qquad +\,\alpha ^2 \lambda ^2 \underline{\text {grad}} \varLambda \underline{\text {grad}} \varLambda ^* d \varOmega d \sigma dt \end{aligned}$$

(77)

The use of a P1 discretization on all fields reads:

$$\begin{aligned}&\varLambda =[N_x]\{\varLambda \} \end{aligned}$$

(78)

$$\begin{aligned}&\lambda =[N_t]\{\lambda \} \end{aligned}$$

(79)

$$\begin{aligned}&\alpha =[N_\sigma ]\{\sigma \} \end{aligned}$$

(80)

where $[N_\bullet ]$ represents the shape functions matrix and $\{ \bullet \}$ the nodal values of the fields.

Then:

$$\begin{aligned} a\left( \lambda \alpha \varLambda ,\lambda \alpha \varLambda ^*\right) =\{\varLambda ^*\}^T[A_{\varLambda }] \{\varLambda \} \end{aligned}$$

(81)

with:

$$\begin{aligned}{}[A_{\varLambda }]= & {} \left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha \} \right) . \nonumber \\&\left[ \left( \{ \lambda \}^T [H_t] \{ \lambda \} \right) [M] + \left( \{ \lambda \}^T [M_t] \{ \lambda \} \right) [C_H] \right] \end{aligned}$$

(82)

where the following matrices are defined:

$$\begin{aligned}&[M_\bullet ]=\int _\bullet [N_\bullet ]^T[N_\bullet ] d \bullet \end{aligned}$$

(83)

$$\begin{aligned}&[H_\bullet ]=\int _\bullet [dN_\bullet ]^T[N_\bullet ] d \bullet \end{aligned}$$

(84)

$$\begin{aligned}&[C_\bullet ]=\int _\bullet [dN_\bullet ]^T[dN_\bullet ] d_\bullet \end{aligned}$$

(85)

$$\begin{aligned}&[C_H]=\kappa .[C_x]+[H_x] \end{aligned}$$

(86)

Likewise:

$$\begin{aligned} a\left( T_{m-1}, \lambda \alpha \varLambda ^*\right) =\{ \varLambda ^*\}^T \sum _{n=1}^{m-1} [A_{\varLambda _n}]\{ \varLambda _n\} \end{aligned}$$

(87)

with:

$$\begin{aligned}&[A_{\varLambda _n}]= \left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha _n\} \right) . \nonumber \\&\quad \left[ \left( \{ \lambda \}^T [H_t] \{ \lambda _n\} \right) [M]+ \left( \{ \lambda \}^T [M_t] \{ \lambda _n\} \right) [C_H] \right] \end{aligned}$$

(88)

$$\begin{aligned}&l(\{\lambda \alpha \varLambda ^*\}) \nonumber \\&\quad =\int _{I \times \varSigma \times \varOmega } \lambda (t) \alpha (\sigma ) s(x,y;\sigma ).\varLambda ^*(x,y) dt d\sigma d\varOmega \end{aligned}$$

(89)

The right-hand side of the variational formulation can be written as:

$$\begin{aligned} l(\lambda \alpha \varLambda ^*)=\{\varLambda ^*\} ^T\{ Q_{\varLambda }\} \end{aligned}$$

(90)

However the integrand is not represented in a separated form. In order to do so, an asymptotic expansion at the center $\sigma _0$ of the P1 element at the d order is used.

The volumic load s is approximated as:

$$\begin{aligned} s(x,y;\sigma _0+\delta \sigma ) \approx \sum _{i=0}^{d} \frac{\partial ^i s}{\partial \sigma ^i} (x,y;\sigma _0) \frac{\delta \sigma ^i}{i!} \end{aligned}$$

(91)

which leads to:

$$\begin{aligned}&\int _{\sigma _{\text {min}}}^{\sigma _{{\text {max}}}} s(x,y;\sigma ) \alpha (\sigma ) d\sigma \nonumber \\&\quad = \sum _{k=1}^N \int _{\sigma _k}^{\sigma _{k+1}} s(x,y;\sigma ) \alpha (\sigma ) d\sigma \nonumber \\&\quad = \sum _{k=1}^N \int _{\sigma _k-\sigma _{0k}}^{\sigma _{k+1}-\sigma _{0k}} s(x,y;\sigma _{0k}+\tilde{\sigma }) \alpha (\sigma _{0k}+\tilde{\sigma })d\tilde{\sigma } \nonumber \\&\quad = \sum _{k=1}^N \sum _{i=0}^{d} \int _{\sigma _k-\sigma _{0k}}^{\sigma _{k+1}-\sigma _{0k}} \frac{\tilde{ \sigma }^i}{i!} \alpha (\sigma _{0k}+\tilde{\sigma })d\tilde{\sigma } \frac{\partial ^i s}{\partial \sigma ^i} (x,y;\sigma _{0k}) \nonumber \\&\quad = \sum _{k=1}^N \sum _{i=0}^{d} \int _{\sigma _k}^{\sigma _{k+1}} \frac{ (\sigma -\sigma _{0k})^i}{i!} \alpha ({\sigma })d{\sigma } \frac{\partial ^i s}{\partial \sigma ^i} (x,y;\sigma _{0k}) \end{aligned}$$

(92)

In practice $d=1$ is sufficient to have a good approximation with the P1 discretization of the interval $\varSigma$.

The right-hand side reads now:

$$\begin{aligned}&l(\lambda \alpha \varLambda ^*)=\{1_t\}^T[M_t]\{\lambda \}.\nonumber \\&\quad \sum _{k=1}^N \sum _{i=0}^{d} \int _{\sigma _k}^{\sigma _{k+1}} \frac{ (\sigma -\sigma _{0k})^i}{i!} \alpha ({\sigma })d{\sigma } \{\varLambda ^*\} ^T\{ S_{ik}\} \end{aligned}$$

(93)

Finally the application $S_m$ leads to a linear system at each iteration of the fixed-point algorithm with the unknown $\{ \varLambda \}$.

1.2.2 Time application $T_m$

For the time application a Runge–Kutta algorithm with automatic adjustment of the time step is used to solve the encountered ordinary differential equation:

$$\begin{aligned} a.\dot{\lambda }(t)+b.\lambda (t)=c-\sum _n^{m-1} a_n.\dot{\lambda _n}(t)+b_n.{\lambda _n}(t) \end{aligned}$$

(94)

with:

$$\begin{aligned} a&= \int _{\varSigma \times \varOmega } \alpha ^2 \varLambda ^2 d\sigma d\varOmega \end{aligned}$$

(95)

$$\begin{aligned}&=\left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha \} \right) \left( \{ \varLambda \}^T [M] \{\varLambda \} \right) \end{aligned}$$

(96)

$$\begin{aligned} b&= \int _{ \varSigma \times \varOmega } \alpha ^2 \left( \underline{v}.\underline{\text {grad}} \varLambda \varLambda + \underline{\text {grad}} \varLambda \underline{\text {grad}} \varLambda \right) d\sigma d\varOmega \end{aligned}$$

(97)

$$\begin{aligned}&=\left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha \} \right) \left( \{ \varLambda \}^T [C_H] \{\varLambda \} \right) \end{aligned}$$

(98)

$$\begin{aligned} c&= \int _{\varSigma \times \varOmega } \alpha (\sigma ) s(x,y;\sigma ).\varLambda (x,y) dt d\sigma d\varOmega \end{aligned}$$

(99)

$$\begin{aligned}&=\sum _{k=1}^N \sum _{i=0}^{d} \int _{\sigma _k}^{\sigma _{k+1}} \frac{ (\sigma -\sigma _{0k})^i}{i!} \alpha ({\sigma })d{\sigma } \{\varLambda \} ^T\{ S_{ik}\} \end{aligned}$$

(100)

$$\begin{aligned} a_n&= \int _{\varSigma \times \varOmega } \alpha \alpha _n \varLambda \varLambda _n d\sigma d\varOmega \end{aligned}$$

(101)

$$\begin{aligned}&=\left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha _n\} \right) \left( \{ \varLambda \}^T [M] \{\varLambda _n\} \right) \end{aligned}$$

(102)

$$\begin{aligned} b_n&=\int _{ \varSigma \times \varOmega } \alpha \alpha _n \left( \underline{v}.\underline{\text {grad}} \varLambda \varLambda _n+ \underline{\text {grad}} \varLambda \underline{\text {grad}} \varLambda _n \right) d\sigma d\varOmega \end{aligned}$$

(103)

$$\begin{aligned}&=\left( \{ \alpha \}^T [M_{\sigma }] \{ \alpha _n\} \right) \left( \{ \varLambda \}^T [C_H] \{\varLambda _n\} \right) \end{aligned}$$

(104)

At each iteration of the fixed-point algorithm an ordinary differential equation on the unknown $\lambda$ is solved.

1.2.3 Parametric application $P_m$

Here, the same approach used in the spatial application is used. The decoupled weak form for the parametric problem reads:

$$\begin{aligned}&a\left( T_{m-1}+ \lambda \alpha \varLambda ,\lambda \alpha ^* \varLambda \right) =l( \lambda \alpha ^* \varLambda ) \end{aligned}$$

(105)

$$\begin{aligned}&a\left( \lambda \alpha \varLambda ,\lambda \alpha ^* \varLambda \right) \nonumber \\&\quad =\int _{I \times \varSigma \times \varOmega } \lambda \dot{\lambda } \alpha \alpha ^* \varLambda ^2 +\alpha \alpha ^* \lambda ^2 \underline{v}.\underline{\text {grad}} \varLambda \varLambda \nonumber \\&\qquad +\,\alpha \alpha ^* \lambda ^2 \underline{\text {grad}} \varLambda \underline{\text {grad}} \varLambda dt d\sigma d\varOmega \end{aligned}$$

(106)

A P1 discretization leads to:

$$\begin{aligned} a\left( \lambda \alpha \varLambda ,\lambda \alpha ^* \varLambda \right) =\{\alpha ^*\}^T[A_{\alpha }] \{\alpha \} \end{aligned}$$

(107)

with:

$$\begin{aligned}{}[A_{\alpha }]= & {} \left[ \left( \{ \varLambda \}^T [M] \{ \varLambda \} \right) \left( \{ \lambda \}^T [H_t] \{ \lambda \} \right) \right. \nonumber \\&\left. +\, \left( \{ \varLambda \}^T [C_H] \{ \varLambda \} \right) \left( \{ \lambda \}^T [M_t] \{ \lambda \} \right) \right] \left[ M_{\sigma }\right] \end{aligned}$$

(108)

The contribution of the previous modes reads:

$$\begin{aligned} a\left( T_{m-1}, \lambda \alpha ^* \varLambda \right) =\{ \alpha ^*\}^T \sum _{n=1}^{m-1} [A_{\alpha _n}]\{ \alpha _n\} \end{aligned}$$

(109)

with:

$$\begin{aligned}{}[A_{\alpha _n}]= & {} \left[ \left( \{ \varLambda \}^T [M] \{ \varLambda _n\} \right) \left( \{ \lambda \}^T [H_t] \{ \lambda _n\} \right) \right. \nonumber \\&\left. + \left( \{ \varLambda \}^T [C_H] \{ \varLambda _n\} \right) \left( \{ \lambda \}^T [M_t] \{ \lambda _n\} \right) \right] \left[ M_{\sigma }\right] \nonumber \\ \end{aligned}$$

(110)

Finally the right-hand side reads:

$$\begin{aligned}&l(\lambda \alpha ^* \varLambda )\nonumber \\&\quad =\{ 1_t\}^T[M_t]\{\lambda \}. \nonumber \\&\quad \sum _{k=1}^N \sum _{i=0}^{d} \int _{\sigma _k}^{\sigma _{k+1}} \frac{ (\sigma -\sigma _{0k})^i}{i!} \alpha ^*({\sigma })d{\sigma } \{\varLambda \} ^T\{ S_{ik}\} \end{aligned}$$

(111)

At each iteration of the fixed-point algorithm a linear system is solved with the unknown $\{\alpha \}$.