From the above discussion, it can be obviously seen that the uncertain inverse problem as shown in Eq. (2) faces the unbearable computational cost due to the multi-layer nesting solving process. In this section, an effective uncertain inverse method based on PCS and DRD is proposed to realize the inverse uncertainty identification of unknown parameters. Firstly, PCS is employed to quantify the uncertainties and correlation of variables *U* according to the limited sample information. Secondly, an efficient space collocation method based on DRD is proposed to transform the IP-MRU into a few IP-RU. Finally, an interval inverse method based on high dimensional model representation (HDMR) and affine algorithm is further presented to effectively solve the IP-RU.

### 3.1 PCS Model for Quantifying Modeling Uncertainty

In practical engineering problems, the sample distributions of uncertain modeling variables are usually complex and various, and there is a certain correlation. Compared with the traditional convex model with the simple regular shape, PCS model provides a more suitable and reasonable way to quantify uncertainty and correlation with limited sample information by using the irregular boundaries.

Assuming that there are *N* experimental samples \({\varvec{U}}^{N} = \left[ {{\varvec{U}}^{\left( 1 \right)} ,{\varvec{U}}^{\left( 2 \right)} , \cdots ,{\varvec{U}}^{\left( N \right)} } \right]\) of *n*-dimensional uncertain modeling variables *U*, where \({\varvec{U}}^{\left( N \right)} { = }\left[ {U_{1}^{\left( N \right)} ,U_{2}^{\left( N \right)} , \cdots ,U_{n}^{\left( N \right)} } \right]^{{\text{T}}}\). According to the samples, a traditional interval model is firstly established, which is expressed as

$$\Omega_{{\text{I}}} = \left\{ {\left. {\varvec{U}} \right|{\varvec{U}}^{{\text{L}}} \le {\varvec{U}} \le {\varvec{U}}^{{\text{R}}} } \right\},$$

(4)

where \({\varvec{U}}^{{\text{L}}} = \min \left( {{\varvec{U}}^{N} } \right)\) and \({\varvec{U}}^{{\text{R}}} = \max \left( {{\varvec{U}}^{N} } \right)\) represent the left and right boundaries of variables *U*; Ω_{I} denotes the uncertainty domain of interval model. In order to reasonably quantify the correlation in sample data, a principal component analysis (PCA) interval model will be further established. For the known samples \({\varvec{U}}^{N}\), the mean point can be calculated as

$${\kern 1pt} {\varvec{U}}^{{\text{M}}} = \frac{1}{N}\left[ {\sum\limits_{t = 1}^{N} {U_{1}^{\left( t \right)} } ,\sum\limits_{t = 1}^{N} {U_{2}^{\left( t \right)} } , \cdots ,\sum\limits_{t = 1}^{N} {U_{n}^{\left( t \right)} } } \right]^{{\text{T}}} .$$

(5)

The covariance matrix *C* for uncertain samples is defined as

$${\varvec{C}} = \frac{{1}}{n}\left( {{\varvec{U}}^{N} - \overline{{\varvec{U}}}^{{\text{M}}} } \right)\left( {{\varvec{U}}^{N} - \overline{{\varvec{U}}}^{{\text{M}}} } \right)^{{\text{T}}} ,$$

(6)

where \(\overline{{\varvec{U}}}^{{\text{M}}} { = }\left[ {{\varvec{U}}^{{\text{M}}} ,{\varvec{U}}^{{\text{M}}} , \cdots, {\varvec{U}}^{{\text{M}}} } \right]_{n \times N}\) is the mean matrix composed of *N* mean points \({\varvec{U}}^{{\text{M}}}\). Through PCA, the orthogonal eigenvectors with respect to the covariance matrix \({\varvec{p}}_{i} = \left( {p_{1i} ,p_{2i} , \cdots ,p_{ni} } \right)^{{\text{T}}} ,i = 1,2, \cdots ,n\) can be obtained, which are rewritten as a matrix \({\varvec{P}} = \left( {{\varvec{p}}_{1} ,{\varvec{p}}_{2} , \cdots ,{\varvec{p}}_{n} } \right)\) by the decreasing order. Thus, a new coordinate system can be constructed based on these orthogonal eigenvector directions, and the uncertain modeling variables *U* and the samples \({\varvec{U}}^{N}\) can be projected to the new coordinate system through matrix *P*

$${\varvec{Z}} = {\varvec{P}}^{{\text{T}}} \left( {{\varvec{U}} - {\varvec{U}}^{{\text{M}}} } \right).$$

(7)

In the new coordinate system, the correlation coefficient between any two of variables \({\varvec{Z}}\) are zero. According to the transformed samples \({\varvec{Z}}^{N}\), a new interval model based on PCA can be established as

$$\Omega_{{{\text{IP}}}} = \left\{ {\left. {\varvec{U}} \right|{\varvec{Z}}^{{\text{L}}} \le {\varvec{P}}^{{\text{T}}} \left( {{\varvec{U}} - {\varvec{U}}^{{\text{M}}} } \right) \le {\varvec{Z}}^{{\text{R}}} } \right\},$$

(8)

where \({\varvec{Z}}^{{\text{L}}} = \min \left( {{\varvec{Z}}^{N} } \right)\) and \({\varvec{Z}}^{{\text{R}}} = \max \left( {{\varvec{Z}}^{N} } \right)\) represent the left and right boundaries of variables \({\varvec{Z}}\). Ω_{IP} denotes the uncertainty domain of PCA interval model. In view of that, the PCS model is established by combining the intersecting region between Ω_{I} and Ω_{IP}.

$$\Omega_{{\text{P}}} = \left\{ {\left. {\varvec{U}} \right|{\varvec{U}}^{{\text{L}}} \le {\varvec{U}} \le {\varvec{U}}^{{\text{R}}} \cap {\varvec{Z}}^{{\text{L}}} \le {\varvec{P}}^{{\text{T}}} \left( {{\varvec{U}} - {\varvec{U}}^{{\text{M}}} } \right) \le {\varvec{Z}}^{{\text{R}}} } \right\},$$

(9)

where Ω_{P} denotes the uncertainty domain of PCS model.

The examples of the two-dimensional PCS model is shown in Figure 3, it can be found that PCS model envelops all samples through the irregular minimum area. Therefore, the PCS model effectively quantifies the uncertainty represented by the given limited sample information. The more properties of PCS model can be seen in Ref. [28].

### 3.2 Space Collocation Method Based on DRD for Decoupling IP-MRU

As described in Section 2, MCS can realize the decoupling of response and modeling uncertainties, and then transform the IP-MRU into the IP-RU under each sampling point. However, the corresponding solving is unacceptable involving the random sampling. In this section, an efficient space collocation method based on DRD is proposed to replace MCS process, and then transforms the IP-MRU into a small amount IP-RU under collocation points (CPs).

The DRD method [33, 34] provides an efficient analysis framework for uncertainty propagation. This method effectively relieves the computational complexity of uncertainty propagation through transforming the structural performance function into the linear combination of univariate sub functions. Similarly, through the DRD for uncertain modeling variables *U*, the inverse function \(\overleftarrow {{\varvec{G}}}\) can be represented as

$${\varvec{X}}{ = }\overleftarrow {{\varvec{G}}} \left( {{\varvec{U}},{\varvec{Y}}} \right) \simeq \sum\limits_{i = 1}^{n} {\overleftarrow {{{\varvec{G}}_{i} }} \left( {{\varvec{U}}_{ - i} ,{\varvec{Y}}} \right)} - \left( {n - 1} \right)\overleftarrow {{\varvec{G}}} \left( {{\varvec{U}}^{{\text{C}}} ,{\varvec{Y}}} \right),$$

(10)

$$\overleftarrow {{{\varvec{G}}_{i} }} \left( {{\varvec{U}}_{ - i} ,{\varvec{Y}}} \right) = \overleftarrow {{\varvec{G}}} \left( {U_{1}^{{\text{C}}} , \cdot \cdot \cdot ,U_{i - 1}^{{\text{C}}} ,U_{i} ,U_{i + 1}^{{\text{C}}} , \cdot \cdot \cdot ,U_{n}^{{\text{C}}} ,{\varvec{Y}}} \right),$$

(11)

where \({\varvec{U}}^{{\text{C}}}\) denotes the decomposition midpoint of function.

It can be found from Eq. (10) that the responses of inverse function can be efficiently predicted through the linear combination of the responses of each sub inverse function. In this paper, the mean point of PCS model is selected to conduct DRD, namely, \({\varvec{U}}^{{\text{C}}} { = }{\varvec{U}}^{{\text{M}}}\). Figure 4 is schematic diagram of DRD for 3-dimensional problem. Assuming that *k*_{i} marginal CPs are taken for the sub inverse function \(\overleftarrow {{{\varvec{G}}_{i} }}\), and the *l*_{i}th marginal CP on the *i*th expansion axis is \({\varvec{U}}_{ - i}^{{l_{i} }} = \left[ {U_{1}^{{\text{M}}} , \cdots ,U_{i - 1}^{{\text{M}}} ,U_{i}^{{l_{i} }} ,U_{i + 1}^{{\text{M}}} , \cdots ,U_{n}^{{\text{M}}} } \right]\). Because the measured responses are intervals \({\overline{\varvec{Y}}}^{{\text{I}}}\), the solving of sub inverse function under the marginal CP \({\varvec{U}}_{ - i}^{{i_{k} }}\) is an interval-based IP-RU. For the convenience of expression, the corresponding IP-RU is expressed as

$$\begin{gathered} {\varvec{X}}_{i}^{{\text{I}}} \left( {l_{i} } \right){ = }\overleftarrow {{{\varvec{G}}_{i} }} \left( {{\varvec{U}}_{ - i}^{{l_{i} }} ,{\overline{\varvec{Y}}}^{{\text{I}}} } \right) \\ = \overleftarrow {{\varvec{G}}} \left( {U_{1}^{{\text{M}}} , \cdot \cdot \cdot ,U_{i - 1}^{{\text{M}}} ,U_{ - i}^{{l_{i} }} ,U_{i + 1}^{{\text{M}}} , \cdot \cdot \cdot ,U_{n}^{{\text{M}}} ,{\overline{\varvec{Y}}}^{{\text{I}}} } \right), \\ \end{gathered}$$

(12)

where \(i = 1,2, \cdots ,m\), and \(l_{i} = 1,2, \cdots ,k_{i}\).

The solving of IP-RU will be discussed in detail in Section 3.3. Here, it only needs to understand that the intervals of unknown parameter *X* can be obtained through Eq. (12). Because the inverse function is the linear combination of univariate sub inverse functions, the response intervals of inverse function *X*^{I} at any joint CP \(\left[ {U_{1}^{{l_{1} }} , \cdot \cdot \cdot ,U_{i}^{{l_{i} }} , \cdot \cdot \cdot ,U_{n}^{{l_{n} }} } \right]\) marked with triangle can be efficiently predicted through the linear combination of the response intervals of sub inverse function corresponding to marginal CPs marked with dot.

$$\begin{gathered} {\varvec{X}}^{{\text{I}}} \left( {l_{1} , \cdots l_{i} , \cdots ,l_{n} } \right) = \overleftarrow {{\varvec{G}}} \left( {U_{1}^{{l_{1} }} , \cdot \cdot \cdot ,U_{i}^{{l_{i} }} , \cdot \cdot \cdot ,U_{n}^{{l_{n} }} ,{\overline{\varvec{Y}}}^{{\text{I}}} } \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \simeq \sum\limits_{i = 1}^{n} {\overleftarrow {{{\varvec{G}}_{i} }} \left( {{\varvec{U}}_{ - i}^{{l_{i} }} ,{\overline{\varvec{Y}}}^{{\text{I}}} } \right)} - \left( {n - 1} \right)\overleftarrow {{\varvec{G}}} \left( {{\varvec{U}}^{{\text{M}}} ,{\overline{\varvec{Y}}}^{{\text{I}}} } \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \sum\limits_{i = 1}^{n} {{\varvec{X}}_{{\text{i}}}^{{\text{I}}} \left( {l_{i} } \right)} - K^{{\text{I}}} . \hfill \\ \end{gathered}$$

(13)

In order to ensure the propagation of correlation, it is necessary to ensure that all CPs are located in the uncertainty domain of PCS model, which can be easily realized by eliminating samples outside the uncertainty domain using Eq. (9). As shown in Figure 4, by comparing the all inverse intervals of IP-RU corresponding to the CPs marked with blue in the PCS model, the intervals of unknown parameters *X* with respect to the IP-MRU can be effectively obtained.

In summary, the proposed space collocation method based on DRD realizes the efficient decoupling of IP-MRU, and transforms the IP-MRU into a small number of IP-RU under marginal CPs. Compared with MCS, the proposed method effectively avoids the solving of a large number of IP-RU under random samples, and relieves the complexity and efficiency of solving calculation.

### 3.3 Interval Inverse Method Based on Affine Algorithm for IP-RU

Through the above discussion, it can be known that the IP-MRU is transformed into a small number of IP-RU under marginal CPs of variables *U*. In this paper, the uncertainty propagation-based inverse method is adopted to realize the solving of IP-RU under each marginal CP.

Assuming that \({\mathbf{U}}^{{\text{P}}}\) is a known marginal CP, the corresponding IP-RU can be expressed as

$$\begin{gathered} {\varvec{X}}{ = }\overleftarrow {{\varvec{G}}} \left( {{\varvec{U}}^{{\text{P}}} ,{\varvec{Y}}} \right),X_{i} = \overleftarrow {{G_{i} }} \left( {{\varvec{U}}^{{\text{P}}} ,{\varvec{Y}}} \right),i = 1,2, \cdots, m, \hfill \\ {\text{s.t.}}{\varvec{Y}} \in {\overline{\varvec{Y}}}^{{\text{I}}} . \hfill \\ \end{gathered}$$

(14)

Through establishing the interval matching model of the measured and calculated responses, this kind of IP-RU can be effectively solved [35].

$$\min \left\| {{\varvec{Y}}^{{\text{I}}} \left( {{\varvec{X}}^{{\text{L}}} ,{\varvec{X}}^{{\text{R}}} } \right) - {\overline{\varvec{Y}}}^{{\text{I}}} } \right\|,$$

(15)

where \({\varvec{X}}^{{\text{L}}} = \left[ {X_{1}^{{\text{L}}} ,X_{2}^{{\text{L}}} , \cdots ,X_{m}^{{\text{L}}} } \right]\) and \({\varvec{X}}^{{\text{R}}} = \left[ {X_{1}^{{\text{R}}} ,X_{2}^{{\text{R}}} , \cdots ,X_{m}^{{\text{R}}} } \right]\) are the left and right boundary vectors of unknown parameters, which are the variables to be inversed. \({\varvec{Y}}^{{\text{I}}}\) denotes the interval vector of calculated responses corresponding to the given \({\varvec{X}}^{{\text{L}}}\) and \({\varvec{X}}^{{\text{R}}}\). In this paper, the genetic algorithm (GA) is adopted to solve this optimization model in Eq. (15). In out layer, \({\varvec{X}}^{{\text{L}}}\) and \({\varvec{X}}^{{\text{R}}}\) will be updated using GA. In inner layer, the \({\varvec{Y}}^{{\text{I}}}\) with respect to the given \({\varvec{X}}^{{\text{L}}}\) and \({\varvec{X}}^{{\text{R}}}\) is calculated by the interval propagation analysis.

Involving the interval propagation analysis in inner layer, the solving of Eq. (15) will consume a larger number of forward model. In order to obtain the calculated responses \({\varvec{Y}}^{{\text{I}}}\) effectively, the first-order high dimensional model representation (HDMR) of forward model is further expressed as

$${\varvec{Y}} = {\varvec{G}}\left( {{\varvec{X}},{\varvec{U}}^{{\text{P}}} } \right)\; \simeq \;\sum\limits_{j = 1}^{m} {{\varvec{G}}_{j} \left( {{\varvec{X}}_{ - j} ,{\varvec{U}}^{{\text{P}}} } \right)} - \left( {m - 1} \right){\varvec{G}}\left( {{\varvec{X}}^{{\text{C}}} ,{\varvec{U}}^{{\text{P}}} } \right),$$

(16)

$$\user2{G}_{j} \left( {\user2{X}_{{ - j}} ,\user2{U}^{{\text{P}}} } \right) = \user2{G}_{j} \left( {X_{1}^{{\text{C}}} , \cdots, X_{{j - 1}}^{C} ,X_{j} ,X_{{j + 1}}^{C} , \cdots, X_{m}^{{\text{C}}} ,\user2{U}^{{\text{P}}} } \right),$$

(17)

where \({\varvec{X}}^{{\text{C}}} = {{\left( {{\varvec{X}}^{{\text{L}}} + {\varvec{X}}^{{\text{R}}} } \right)} \varvec{\left/ {\vphantom {{\left( {{\varvec{X}}^{{\text{L}}} + {\varvec{X}}^{{\text{R}}} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}\) is the decomposition midpoint of the forward model.

For the *j*th univariate sub function *G*_{ij} of the *i*th response *Y*_{i}, the polynomial-based response surface model is adopted to establish the corresponding surrogate model

$$G_{ij} \left( {{\varvec{X}}_{ - j} ,{\varvec{U}}^{{\text{P}}} } \right) = \sum\limits_{s = 0}^{h} {a_{i\left( s \right)} X_{j}^{s} } ,$$

(18)

where *h* denotes the highest order of polynomial surrogate model, *a*_{i(s)} is the coefficient of *s*-order sub term. The order number *h* is usually determined according to the nonlinear degree of system, while the 3-order is usually enough for the univariate sub function. In view of that, the polynomial surrogate model for univariate sub function can be easily established by taking a small number of samples. Due to the explicit polynomial model, the interval \({\varvec{X}}^{{\text{I}}}\) can be directly substituted into Eqs. (16) and (18) to obtain the interval \(Y_{i}^{{\text{I}}}\). However, the calculation results have the interval expansion problem because of the interval operation. In order to the left and right boundaries of each sub function, the affine algorithm is employed to conduct the interval propagation.

Firstly, the variable *X*_{j} is rewritten by the affine form

$$\hat{X}_{j} { = }X_{j}^{{\text{C}}} + \xi X_{j}^{{\text{W}}} {, }\,\xi \in \left[ { - 1,1} \right],$$

(19)

where \(X_{j}^{{\text{C}}} = \frac{{X_{j}^{{\text{U}}} + X_{j}^{{\text{L}}} }}{2}\), \(X_{j}^{{\text{W}}} = \frac{{X_{j}^{{\text{U}}} - X_{j}^{{\text{L}}} }}{2}\).

Substituting Eq. (19) into Eq. (18), the sub function can be expressed in Eq. (20) through the binomial theorem.

$$\begin{gathered} G_{ij} \left( {{\varvec{X}}_{ - j} ,{\varvec{U}}^{{\text{P}}} } \right) = \sum\limits_{s = 0}^{h} {a_{j\left( s \right)} \left( {X_{j}^{{\text{C}}} + \xi X_{j}^{{\text{W}}} } \right)^{s} } \\ = \sum\limits_{s = 0}^{h} {\sum\limits_{t = s}^{h} {a_{j\left( t \right)} \left( {\begin{array}{*{20}c} t \\ s \\ \end{array} } \right)\left( {X_{j}^{{\text{C}}} } \right)^{t - s} \left( {X_{j}^{{\text{W}}} } \right)^{s} \xi^{s} } } . \\ \end{gathered}$$

(20)

Because \(\xi \in \left[ { - 1,1} \right]\), if *s* is even, \(\xi^{s} \in \left[ {0,1} \right]\), if *s* is odd, \(\xi^{s} \in \left[ { - 1,1} \right]\). Thus, the interval expansion problem is avoided effectively. Under these circumstances, the interval [-1, 1] can be directly substituted into Eq. (20) to obtain the left and right boundary vectors of sub function *G*_{ij}. For the convenience of expression, there is

$$b_{j\left( t \right)} = \sum\limits_{t = s}^{h} {a_{j\left( t \right)} \left( {\begin{array}{*{20}c} t \\ s \\ \end{array} } \right)\left( {X_{j}^{C} } \right)^{t - s} \left( {X_{j}^{W} } \right)^{s} } .$$

(21)

The left and right boundary vectors of sub function *G*_{ij} can be expressed as

$$\left\{ \begin{gathered} G_{ij}^{{\text{R}}} \left( {{\varvec{X}}_{ - j} ,{\varvec{U}}^{{\text{P}}} } \right) = b_{j\left( 0 \right)} + \sum\limits_{s = 1}^{h} {\left\{ \begin{gathered} \max \left( {0,b_{j\left( s \right)} } \right) \, s = 2A,\; \hfill \\ \, \left| {b_{j\left( s \right)} } \right| \, s\; = 2A + 1, \hfill \\ \end{gathered} \right.} \hfill \\ G_{ij}^{{\text{L}}} \left( {{\varvec{X}}_{ - j} ,{\varvec{U}}^{{\text{P}}} } \right) = b_{j\left( 0 \right)} + \sum\limits_{s = 1}^{h} {\left\{ \begin{gathered} \min \left( {0,b_{j\left( s \right)} } \right) \, s\; = 2A, \hfill \\ \, - \left| {b_{j\left( s \right)} } \right| \, s = 2A + 1, \hfill \\ \end{gathered} \right.} \hfill \\ \end{gathered} \right.$$

(22)

where *A* denotes arbitrary integer. Similar with Eq. (13), substituting Eq. (22) into Eq. (13), the intervals \({\varvec{Y}}^{{\text{I}}}\) can be calculated.

Overall, the above interval inverse method based on HDMR and affine algorithm efficiently improves the inverse efficiency IP-RU, which can obtain the left and right boundaries of unknown parameter *X* only calling a few forward problem functions. In view of that, the intervals \({\varvec{X}}^{{\text{I}}}\) under each marginal CP all can be effectively obtained.

### 3.4 Solving Procedure

To summarize, the solving process of the proposed uncertain method can be divided into three parts. The first part is the uncertainty modeling for the variables *U* and the measured responses *Y*, in which the variables *U* are quantified by PCS model, the measured responses *Y* are modeled by the interval model. The second part is the decoupling of IP-MRU, the complex nesting solving process is transformed into a small amount of IP-RU calculation through the proposed space collocation method based DRD. The third part is the solving of IP-RU under CPs, in which the inverse efficiency is efficiently improved through affine algorithm and HDMR based on the polynomial response surface. The solving procedure is illustrated in Figure 5, and the inverse solving steps are described as follows.

Step 1. Establish PCS model and interval model to quantify the modeling parameters and the measured responses, respectively.

Step 2. Take the mean point of PCS model \({\varvec{U}}^{{\text{M}}}\) as the decomposition midpoint \({\varvec{U}}^{{\text{C}}}\), and assign the marginal CPs \({\varvec{U}}_{ - i}^{{i_{k} }}\).

Step 3. Construct the optimization matching model at marginal CP according to Eq. (15).

Step 4. Expand the performance function according to Eqs. (16) and (17).

Step 5. Establish the polynomial-based response surface model of each sub function according Eq. (18).

Step 6. Calculate the response intervals of each sub function by using affine algorithm according to Eq. (22).

Step 7. Calculate the response intervals of the forward function according to Eq. (16).

Step 8. Inverse the interval of unknown parameters combing GA and steps 4–7.

Step 9. Calculate the intervals of unknown parameters at all joint CPs according to Eq. (13).

Step 10. Obtain the intervals of unknown parameters *X* by comparing the all inverse intervals corresponding to the CPs in the PCS model.