A subinterval bivariate dimension-reduction method for nonlinear problems with uncertainty parameters

Abstract

A subinterval bivariate dimension-reduction method is proposed to predict the upper and lower bounds of nonlinear problems with uncertain-but-bounded parameters, especially for nonmonotonic problems. The existing interval function decomposition method solves the dimensional curse problem, but is only suitable for the monotone case. To address this limitation, the original structural response function with multidimensional interval parameters is decomposed by the bivariate dimension-reduction method into a set of univariate and bivariate interval response functions, and then the interval parameters are partitioned into some subintervals with low uncertainty. The upper and lower bounds of the structural response are approximately predicted not by analyzing all discrete points in the entire uncertain domain, but by calculating the responses at bivariate points, univariate points, and the midpoint, which can improve the computational efficiency. Finally, the accuracy and efficiency of the proposed method are verified using several numerical examples and engineering applications.

Appendices

Appendix 1: interval analysis using Taylor series expansion

An approximate response function ${f^*}\left( \varvec{x} \right)$ using second-order Taylor series expansion around the midpoint of the interval vector can be described as follows:

$${f^*}\left( {{\varvec{x}^{\rm{I}}}} \right) = f\left( {{\varvec{x}^{\rm{C}}}} \right) + \sum\limits_{i = 1}^n {{f_{,{x_i}}}\left( {{x_i} - x_i^{\rm{C}}} \right)} + \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{f_{,{x_i}{x_j}}}\left( {{x_i} - x_i^{\rm{C}}} \right)} } \left( {{x_j} - x_j^{\rm{C}}} \right)$$

(34)

where $f_{{,x_{i} }}$ and $f_{{,x_{i} x_{j} }}$ denote first- and second-order differentiation $\partial f\left( \varvec{x} \right)/\partial {x_i}{\left| x \right._i} = x_i^{\rm{C}}$ and ${\partial ^2}f\left( \varvec{x} \right)/\partial {x_i}\partial {x_j}{\left| x \right._i} = x_i^{\rm{C}},{\left| x \right._j} = x_j^{\rm{C}}$ of the objective function at the midpoint, respectively.

If only the first order differentiation is considered, the response bounds of the first-order Taylor expansion can be expressed as:

$$\begin{aligned} f^{{\text{L}}} \left( {{\varvec{x}^{\rm{I}}}} \right) & = f\left( {{\varvec{x}^{\rm{C}}}} \right) - \sum\limits_{i = 1}^{n} {\left| {f_{{,x_{i} }} x_{i}^{{\text{R}}} } \right|} \\ f^{{\text{U}}} \left( {{\varvec{x}^{\rm{I}}}} \right) & = f\left( {{\varvec{x}^{\rm{C}}}} \right) + \sum\limits_{i = 1}^{n} {\left| {f_{{,x_{i} }} x_{i}^{{\text{R}}} } \right|} \\ \end{aligned}$$

(35)

The approximation using the first-order Taylor series expansion requires only a few calculations, but its results may contain large errors, especially in the case of nonlinearity and large uncertainties. In order to improve the accuracy of interval analysis, the second-order Taylor series expansion approximation method retains the second-order differentiation. In this case, there is a dependence phenomenon of the interval parameters, and using the interval algorithm can easily lead to overestimation. When n is large, calculating all elements of the Hessian matrix requires significant computational effort. Therefore, a simplified method that ignores the nondiagonal elements of the Hessian matrix is proposed. The response bounds of the second-order Taylor method are expressed as follows:

$$\begin{aligned} f^{{\text{L}}} \left( {{\varvec{x}^{\rm{I}}}} \right) & = f\left( {{\varvec{x}^{\rm{C}}}} \right) - \sum\limits_{i = 1}^{n} {\left| {f_{{,x_{i} }} x_{i}^{{\text{R}}} } \right|} + \frac{1}{2}\sum\limits_{i = 1}^{n} {f_{{,x_{i} x_{i} }} \left( {x_{i}^{{\text{R}}} } \right)^{2} } \\ f^{{\text{U}}} \left( {{\varvec{x}^{\rm{I}}}} \right) & = f\left( {{\varvec{x}^{\rm{C}}}} \right) + \sum\limits_{i = 1}^{n} {\left| {f_{{,x_{i} }} x_{i}^{{\text{R}}} } \right|} + \frac{1}{2}\sum\limits_{i = 1}^{n} {f_{{,x_{i} x_{i} }} \left( {x_{i}^{{\text{R}}} } \right)^{2} } \\ \end{aligned}$$

(36)

Appendix 2: The error estimation of the bivariate dimension-reduction

For the simplicity of exposition, n is given as 3 in the following formulation. For n > 3, the error estimation process is the same, but it has more expression terms and the computation is more time consuming. From Eq. (15), the bivariate dimension-reduction approximation contains the higher order differentials of univariate and bivariate terms. Therefore, it has higher approximation accuracy than the interval analysis method based on the Taylor expansion method.

The truncation error $R_{3}^{{\text{I}}}$ of the bivariate dimension-reduction approximation of $f\left( {{\varvec{x}^{\rm{I}}}} \right)$ is expressed as follows:

$$R_3^{\rm{I}} = \frac{{{\partial ^3}f({\varvec{x}^{\rm{C}}} + \theta \delta \varvec{x})}}{{\partial {x_1}\partial {x_2}\partial {x_3}}}\delta {x_1}\delta {x_2}\delta {x_3},{\mkern 1mu} 0 < \theta < 1$$

(37)

Using

$$g\left( \varvec{a} \right) = \frac{{{\partial ^3}f({\varvec{x}^{\rm{C}}} + \varvec{a})}}{{\partial {a_1}\partial {a_2}\partial {a_3}}},{\mkern 1mu} {\rm{ }} - {\varvec{x}^{\rm{R}}} < \varvec{a} < {\varvec{x}^{\rm{R}}}$$

(38)

Eq. (37) can be rewritten as follows:

$$R_{3}^{{\text{I}}} = g(\theta \delta {\varvec{x}})\delta x_{1} \delta x_{2} \delta x_{3} ,\quad 0 < \theta < 1$$

(39)

Here the method of linear approximation [46, 49] is introduced to deal with $g({\varvec{a}})$, namely for the small interval vector ${\varvec{x}^{\rm{I}}}$. $g({\varvec{a}})$ can be approximately replaced by a 3D space within the volume defined by $- x_{1}^{{\text{R}}} < a_{1} < x_{1}^{{\text{R}}}$, $- x_{2}^{{\text{R}}} < a_{2} < x_{2}^{{\text{R}}}$ and $- x_{3}^{{\text{R}}} < a_{3} < x_{3}^{{\text{R}}}$.The linear approximation of the function $g({\varvec{a}})$ at (0,0,0) is:

$$g({\varvec{a}}) \approx L(a_{1} ,a_{2} ,a_{3} ) = l_{0} + l_{1} a_{1} + l_{2} a_{2} + l_{3} a_{3} , \, - {\varvec{x}}^{{\text{R}}} < {\varvec{a}} < {\varvec{x}}^{{\text{R}}}$$

(40)

where the coefficient vectors $l_{0}$, $l_{1}$, $l_{2}$ and $l_{3}$ can be determined by:

$$\begin{aligned} l_{0} & = g(0,0,0) \\ l_{1} & = \frac{{g(x_{1}^{{\text{R}}} ,0,0) - g( - x_{1}^{{\text{R}}} ,0,0)}}{{2x_{1}^{{\text{R}}} }} \\ l_{2} & = \frac{{g(0,x_{2}^{{\text{R}}} ,0) - g(0, - x_{2}^{{\text{R}}} ,0)}}{{2x_{2}^{{\text{R}}} }} \\ l_{3} & = \frac{{g(0,0,x_{3}^{{\text{R}}} ) - g(0,0, - x_{3}^{{\text{R}}} )}}{{2x_{3}^{{\text{R}}} }} \\ \end{aligned}$$

(41)

Substituting Eqs. (40) and (41) into Eq. (39) we obtain:

$$R_{3}^{{\text{I}}} = \left( {l_{0} + l_{1} \theta \delta x_{1} + l_{2} \theta \delta x_{2} + l_{3} \theta \delta x_{3} } \right)\delta x_{1} \delta x_{2} \delta x_{3}$$

(42)

Since $\delta x_{1}$, $\delta x_{2}$ and $\delta x_{3}$ are intervals, the interval of the error can be calculated by the natural expansion of interval mathematics [18]:

$$\begin{aligned} R_{3}^{{\text{I}}} & = \left( {l_{0} + l_{1} \theta [ - 1,1]x_{1}^{{\text{R}}} + l_{2} \theta [ - 1,1]x_{2}^{{\text{R}}} + l_{3} \theta [ - 1,1]x_{3}^{{\text{R}}} } \right)[ - 1,1]x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} \\ & = \left[ {\left| {l_{0} } \right|[ - 1,1]x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{1} } \right|\theta [ - 1,1]\left( {x_{1}^{R} } \right)^{2} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{2} } \right|\theta [ - 1,1]x_{1}^{{\text{R}}} \left( {x_{2}^{{\text{R}}} } \right)^{2} x_{3}^{{\text{R}}} + \left| {l_{3} } \right|\theta [ - 1,1]x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} \left( {x_{3}^{{\text{R}}} } \right)^{2} } \right] \\ \end{aligned}$$

(43)

where |•| represents the absolute values for each component of the vector. Thus, the bounds of $R_{3}^{I}$ can be obtained as follows:

$$\begin{aligned} R_{3}^{{\text{L}}} & = - \left[ {\left| {l_{0} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{1} } \right|\theta \left( {x_{1}^{R} } \right)^{2} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{2} } \right|\theta x_{1}^{{\text{R}}} \left( {x_{2}^{{\text{R}}} } \right)^{2} x_{3}^{{\text{R}}} + \left| {l_{3} } \right|\theta x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} \left( {x_{3}^{{\text{R}}} } \right)^{2} } \right] \\ R_{3}^{{\text{U}}} & = \left[ {\left| {l_{0} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{1} } \right|\theta \left( {x_{1}^{{\text{R}}} } \right)^{2} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{2} } \right|\theta x_{1}^{{\text{R}}} \left( {x_{2}^{{\text{R}}} } \right)^{2} x_{3}^{{\text{R}}} + \left| {l_{3} } \right|\theta x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} \left( {x_{3}^{{\text{R}}} } \right)^{2} } \right] \\ \end{aligned}$$

(44)

Owing to 0 < θ < 1, the following equations are obtained:

$$\begin{aligned} R_{3}^{{\text{L}}} & \ge - \left[ {\left| {l_{0} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{1} } \right|\left( {x_{1}^{{\text{R}}} } \right)^{2} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{2} } \right|x_{1}^{{\text{R}}} \left( {x_{2}^{{\text{R}}} } \right)^{2} x_{3}^{{\text{R}}} + \left| {l_{3} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} \left( {x_{3}^{{\text{R}}} } \right)^{2} } \right] \\ R_{3}^{{\text{U}}} & \le \left[ {\left| {l_{0} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{1} } \right|\left( {x_{1}^{{\text{R}}} } \right)^{2} x_{2}^{{\text{R}}} x_{3}^{{\text{R}}} + \left| {l_{2} } \right|x_{1}^{{\text{R}}} \left( {x_{2}^{{\text{R}}} } \right)^{2} x_{3}^{{\text{R}}} + \left| {l_{3} } \right|x_{1}^{{\text{R}}} x_{2}^{{\text{R}}} \left( {x_{3}^{{\text{R}}} } \right)^{2} } \right] \\ \end{aligned}$$

(45)

From Eq. (45), it can be seen that $R_{3}^{{\text{L}}}$ and $R_{3}^{{\text{U}}}$ represent the maximum errors of the lower and upper bounds, respectively.