It is of high interest to design a device or circuit that can still perform well under process variations. In other words, the robustness of its performance is of primary concern, because fabrication variations are often unavoidable in reality. The goal is to optimize the quantity of interest (or a function of the quantity of interest) under uncertainties and design constraints. Because the quantity of interest is a random variable, it is reasonable to use its expectation or its associated function as the objective in the optimization problem. A common form of the design optimization problem is the following:

$$\begin{array}{ll}\underset{\overrightarrow{x}}{\text{minimize}}\hfill & {\mathbb{E}}_{\xi}\text{[}u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}\text{]\hspace{0.17em}and/or\hspace{0.17em}}{\mathbb{V}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]\hfill \\ \text{subject\hspace{0.17em}to}\hfill & {a}_{i}\le {x}_{i}\le {b}_{i}\mathrm{,}\text{\hspace{0.17em}}i=\mathrm{1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}m\mathrm{,}\hfill \end{array}$$(2)

where $u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}$ is the quantity of interest. The expectation and variance of *u* over $\overrightarrow{\xi}$ are known to be

$${\mathbb{E}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]={C}_{\overrightarrow{0}}\mathrm{(}\overrightarrow{x}\mathrm{)}$$(3)

and

$${\mathbb{V}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]={\displaystyle \sum _{\overrightarrow{\alpha}\ne \overrightarrow{0}}}{C}_{\overrightarrow{\alpha}}^{2}\mathrm{(}\overrightarrow{x}\mathrm{}\mathrm{)}\mathrm{,}$$(4)

assuming $\overrightarrow{x}$ is a constant vector and that the basis functions ${\Psi}_{\overrightarrow{\alpha}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}$ are orthonormal.

If (3) and (4) are used to solve the optimization problem (2), then $u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}$ has to be reexpanded each time for each design point $\overrightarrow{x}$ inside the optimization loop, which is time consuming and only non-gradient-based optimization algorithm such as evolutionary genetic algorithm can be used to search for the optimum. However, genetic algorithm is a heuristic-based method, and it does not guarantee local and global optima, although it is often used as a tool to try to find a global optimum in practice. Alternatively, if $u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}$ can be expanded both on $\overrightarrow{x}$ and $\overrightarrow{\xi},$ then (3) and (4) will be analytical functions of $\overrightarrow{x};$ hence, gradient-based optimizers can be employed. Notice that the constraints of *x*_{i} in (2) are boxed constraints, which is equal to express *x*_{i} as uniformly distributed variables in the interval [*a*_{i}, *b*_{i}]. Therefore, rewrite $u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}$ as

$\begin{array}{c}u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}\approx {\displaystyle \sum _{\overrightarrow{\alpha}}}{C}_{\overrightarrow{\alpha}}{\Psi}_{\overrightarrow{\alpha}}\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}\\ ={\displaystyle \sum _{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}}{C}_{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}{\mathbb{L}}_{\overrightarrow{\beta}}\mathrm{(}\overrightarrow{x}\mathrm{)}{\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}\end{array}$

where ${\mathbb{L}}_{\overrightarrow{\beta}}$ is a multivariate Legendre polynomial with multi-index $\overrightarrow{\beta},$ Φ is a multivariate polynomial of $\overrightarrow{\xi}$ with multi-index $\overrightarrow{\gamma},$ and $\overrightarrow{\alpha}=\mathrm{(}\overrightarrow{\beta}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\gamma}\mathrm{}\mathrm{)}\mathrm{.}$ $u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}$ is now referred to the combined gPC model. Using the combined gPC model, we have

$\begin{array}{c}{\mathbb{E}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]={\displaystyle \int u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}{\rho}_{\overrightarrow{\xi}}d\overrightarrow{\xi}}\\ ={\displaystyle \int {\displaystyle \sum _{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}}{C}_{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}{\mathbb{L}}_{\overrightarrow{\beta}}\mathrm{(}\overrightarrow{x}\mathrm{)}{\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}{\rho}_{\overrightarrow{\xi}}d\overrightarrow{\xi}}\\ ={\displaystyle \sum _{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}}{C}_{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}{\mathbb{L}}_{\overrightarrow{\beta}}\mathrm{(}\overrightarrow{x}\mathrm{)}{\displaystyle \int {\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}{\rho}_{\overrightarrow{\xi}}d\overrightarrow{\xi}}\end{array}$

Because ${\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}$ are orthogonal and ${\Phi}_{\overrightarrow{0}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}=1,$ we have

$\int {\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}{\rho}_{\overrightarrow{\xi}}d\overrightarrow{\xi}}=\{\begin{array}{l}0\text{,\hspace{0.17em}if\hspace{0.17em}}\overrightarrow{\gamma}\ne \overrightarrow{0}\hfill \\ 1\text{,\hspace{0.17em}if\hspace{0.17em}}\overrightarrow{\gamma}=\overrightarrow{0}\hfill \end{array$

Therefore, we have

$${\mathbb{E}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]={\displaystyle \sum _{\overrightarrow{\beta}}}{C}_{\overrightarrow{\beta}\mathrm{,}\overrightarrow{0}}{\mathbb{L}}_{\overrightarrow{\beta}}\mathrm{(}\overrightarrow{x}\mathrm{}\mathrm{)}\mathrm{.}$$(5)

Similarly, using the orthogonality of basis functions

$\int {\Phi}_{\overrightarrow{\gamma}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}{\Phi}_{{\overrightarrow{\gamma}}^{\prime}}\mathrm{(}\overrightarrow{\xi}\mathrm{)}{\rho}_{\overrightarrow{\xi}}d\overrightarrow{\xi}}=\{\begin{array}{l}0\text{,\hspace{0.17em}if\hspace{0.17em}}\overrightarrow{\gamma}\ne {\overrightarrow{\gamma}}^{\prime}\hfill \\ 1\text{,\hspace{0.17em}if\hspace{0.17em}}\overrightarrow{\gamma}={\overrightarrow{\gamma}}^{\prime}\hfill \end{array},$

we have

$$\begin{array}{c}{\mathbb{V}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]={\mathbb{E}}_{\xi}[{u}^{2}\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]-\mathrm{(}{\mathbb{E}}_{\xi}[u\mathrm{(}\overrightarrow{x}\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\xi}\mathrm{)}]{\mathrm{)}}^{2}\\ ={\displaystyle \sum _{\overrightarrow{\beta}\mathrm{,}{\overrightarrow{\beta}}^{\prime}\mathrm{,}\overrightarrow{\gamma}\ne \overrightarrow{0}}}{C}_{\overrightarrow{\beta}\mathrm{,}\overrightarrow{\gamma}}{C}_{{\overrightarrow{\beta}}^{\prime}\mathrm{,}\overrightarrow{\gamma}}{\mathbb{L}}_{\overrightarrow{\beta}}\mathrm{(}\overrightarrow{x}\mathrm{)}{\mathbb{L}}_{{\overrightarrow{\beta}}^{\prime}}\mathrm{(}\overrightarrow{x}\mathrm{}\mathrm{)}\mathrm{.}\end{array}$$(6)

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