Gradient-free Importance Sampling Scheme for Efficient Reliability Estimation

As discussed earlier, the basic idea is to construct a near-optimal sampling target distribution $\tilde{h}$, placing higher importance on the rare event domain $\mathcal{F}$. In ASTPA, this is achieved by approximating the indicator function $I_{\mathcal{F}}$ in Eq. 4 with a smooth likelihood function $\ell_{g_{\bm{X}}}$, chosen as a logistic cumulative distribution function (CDF) of the negative of a scaled limit state function, $F_{\text{CDF}}\left(\nicefrac{{-g(\bm{X})}}{{g_{c}}}\right)$, with $g_{c}$ being a scaling constant:

where $\mu_{g}$ and $\sigma$ are the mean and dispersion factor of the logistic CDF, respectively. The approximate sampling target is ASTPA is then expressed as:

The dispersion factor $\sigma$ controls the spread of the likelihood function and thus influences the shape of the constructed target $\tilde{h}$. Recommended values for $\sigma$ lie in the range of $[0.1,\,0.6]$, with lower values (typically between $0.1$ and $0.4$) generally working efficiently. Higher values within the recommended range may, however, be needed for high-dimensional cases involving strongly nonlinear effects. The mean parameter, $\mu_{g}$, of the logistic CDF is typically chosen as $\mu_{g}=1.21\sigma$, generally placing $\mu_{g}$ inside the rare event domain, as derived in [4]. In Fig. 2, we illustrate how the dispersion factor, $\sigma$, influences the shape of the target distribution, based on a nonlinear bimodal limit state function, as detailed in Section 6.1. As shown, decreasing $\sigma$ results in a more concentrated target distribution inside the rare event domain. Although pushing the target further inside the rare event domain, i.e., closer to the optimal distribution $(I_{\mathcal{F}}\pi_{\bm{X}})$, can generate more rare event samples, it may also complicate the sampling process and, thus, the computation of the normalizing constant $\hat{C}_{h}$, as discussed in [4].

The scaling $\nicefrac{{g(\bm{X})}}{{g_{c}}}$ accounts for the varying magnitudes of limit-state functions, ensuring that the input to the logistic CDF $\left(\nicefrac{{g(\bm{X})}}{{g_{c}}}\right)$, evaluated at the mean of the original distribution, consistently falls within a predefined range. This process standardizes the input while keeping the recommended values for $\sigma$ unchanged. To ensure the input at $\bm{x}=\bm{0}$ lies within a desired range of $[3,7]$, $g_{c}$ is defined as $g_{c}=\nicefrac{{g(\bm{0})}}{{q}}$, where $\,q\in[3,7]\text{ if }g(\bm{0})\notin[3,7]$, and $g_{c}=1$ otherwise. Our empirical analysis confirms the effectiveness of this scaling technique, demonstrating that the recommended range for $q$ is practical for the studied Gaussian problems.

Fig. 3 demonstrates the effect of the scaling constant $g_{c}$ on constructing the target distribution $\tilde{h}(\bm{X})$. Since the limit state function considered in this problem provides $g(\bm{0})>7$, the scaling constant $g_{c}$ can be defined as $\nicefrac{{g(\bm{0})}}{{4}}$. Fig. 3(b) shows the unnormalized target distribution $\tilde{h}(\bm{X})$ constructed without applying the proper scaling constant. As a result, the input $\left(\nicefrac{{g(\bm{X})}}{{g_{c}}}\right)$ at $\bm{x}=0$ does not lie within the desired range, leading to a highly concentrated target. In contrast, Fig. 3(c) presents the corrected target distribution after applying the appropriate scaling factor $g_{c}=\nicefrac{{g(\mathbf{0})}}{{4}}$. These plots clearly demonstrate how scaling the limit-state function helps maintain an appropriately relaxed target, with the recommended parameter values remaining unchanged, thus enhancing sampling efficiency.

Inverse importance sampling (IIS) is a general technique to compute normalizing constants, demonstrating exceptional performance in the context of rare event probability estimation [29]. Given an unnormalized distribution $\tilde{h}$, and samples $\{\bm{x}_{i}\}_{i=1}^{N}\sim h$, inverse importance sampling first fits an ISD $Q(\bm{x})$ based on the samples $\{\bm{x}_{i}\}_{i=1}^{N}\sim h$. IIS then estimates the normalizing constant $C_{h}$ as:

By drawing $\{\bm{x}^{\prime}\}_{i=1}^{M}$ i.i.d. samples from $Q$, the unbiased IIS estimator can be computed as:

In this work, a Gaussian Mixture Model (GMM), based on the available samples $\{\bm{x}_{i}\}_{i=1}^{N}\sim h$, is employed as the ISD $Q(\bm{X})$. In low-dimensional spaces, where $d<20$, we employ a GMM with a large number $(\sim 10)$ of Gaussian components that have full covariance matrices. This approach aims to closely approximate the target distribution $\tilde{h}$ with an ISD $Q(.)$, thus improving the IIS estimator in eq. 11. To address the challenges posed by high-dimensional spaces, GMMs with diagonal covariance matrices are used in high-dimensional examples in this work. This choice reduces the number of GMM parameters to be estimated, thereby mitigating the scalability issues typically associated with density estimation methods. Notably, IIS has been shown to work effectively even with a crudely fitted $Q$ [4, 29].

The IIS technique is further enhanced using the splitting approach introduced in [4] to improve the robustness of the IIS estimator against the impact of rarely observed outliers or anomalous samples. In this approach, the sample set $\{\bm{x}_{i}^{\prime}\}_{i=1}^{M}\sim Q$ is split into two equal-sized subsets, and the IIS estimate $\hat{C}_{h}$ is computed separately for each subset. If the two estimates are within a reasonable range of each other (specifically, $\frac{1}{3}\leq\nicefrac{{\hat{C}_{h}^{1}}}{{\hat{C}_{h}^{2}}}\leq 3$), the final estimate is taken as their average. Otherwise, the final estimate is conservatively set to the minimum of the two. This method has proven effective across all examples presented in this work.

The ASTPA estimator of the rare event probability can finally be computed as:

where $\{\bm{x}_{i}\}_{i=1}^{N}$ and $\{\bm{x}_{i}^{\prime}\}_{i=1}^{M}$ are samples from $h$ and $Q$, respectively.

As shown in [29, 4], the ASTPA estimator is unbiased, i.e., $\mathop{\mathbb{E}}[\hat{p}_{\mathcal{F}}]=\mathop{\mathbb{E}}[\hat{\tilde{p}}_{\mathcal{F}}\,\hat{C}_{h}]=\tilde{p}_{\mathcal{F}}\,C_{h}$. The analytical Coefficient of Variation (C.o.V) of the ASTPA estimator can also be computed as $\textrm{C.o.V}\approx\nicefrac{{\sqrt{\widehat{\text{Var}}(\hat{p}_{\mathcal{F}})}}}{{\hat{p}_{\mathcal{F}}}}$, where the variance ($\widehat{\text{Var}}(\hat{p}_{\mathcal{F}})$) is estimated by:

Here, $\hat{\tilde{p}}_{\mathcal{F}}$ and $\hat{C}_{h}$ can be computed according to Eqs. 6 and 11, respectively. The variances $\widehat{\text{Var}}(\hat{\tilde{p}}_{\mathcal{F}})$ and $\widehat{\text{Var}}(\hat{C}_{h})$ can thus be estimated as follows:

where $N_{s}$ denotes the number of used Markov chain samples, accounting for the fact that the samples are not independent and identically distributed (i.i.d) in this case. To address this, a thinning process is adopted to reduce the sample size from $N$ to $N_{s}$ based on the effective sample size (ESS) of the sample set $\{\bm{x}_{i}\}_{i=1}^{N}$. We select every $j^{th}$ sample, where $j$, an integer, is determined by:

where $\text{ESS}_{\text{min}}$ represents the minimum effective sample size across all dimensions. To ensure $j$ remains within practical bounds, it is constrained to the set $\{3,4,5,\ldots,30\}$. Any calculated value of $j$ outside this range is adjusted to the nearest bound within it. The analytical C.o.V computed according to this approach showed very good agreement with the sample C.o.V in all our numerical examples.

The preconditioned Crank-Nicolson (pCN) algorithm is a gradient-free Markov Chain Monte Carlo (MCMC) method designed for efficient sampling from high-dimensional probability distributions [36, 37]. To sample from a target distribution $\tilde{h}=\ell_{g_{\bm{X}}}\,\pi_{\bm{X}}$, the pCN sampler proposes an update, $\tilde{\bm{x}}$, at each iteration, $t$, by combining the current sample, $\bm{x}_{t}$, with a perturbation drawn from the original (prior) Gaussian distribution, $\pi_{\bm{X}}=\mathcal{N}(\bm{X};\,\bm{0},\mathbf{C})$, as follows:

where $\beta\in(0,1]$ is a tunable scaling parameter, discussed below, and $\mathbf{C}$ is a covariance matrix. This update is equivalent to using a Gaussian proposal distribution $\mathcal{N}(\sqrt{1-\beta^{2}}\,\bm{x}_{t},\,\beta^{2}\mathbf{C})$, yielding the acceptance probability:

where $\ell_{g_{\bm{X}}}$ is the likelihood function in Eq. 8. This formula illustrates a fundamental characteristic of the pCN algorithm: its invariance to the underlying Gaussian distribution $\mathcal{N}(\bm{X};\,\bm{0},\mathbf{C})$. In essence, this ensures that the proposed sample is always accepted when sampling from $\pi_{\bm{X}}$, making the pCN highly suitable for efficiently exploring high-dimensional Gaussian spaces. This invariance can be understood by examining the Metropolis-Hastings (MH) ratio when using the pCN algorithm to sample $\mathcal{N}(\bm{X};\,\bm{0},\mathbf{C})$, as:

Considring the marginal distribution $\mathcal{N}(\tilde{\bm{X}};\,\bm{0},\mathbf{C})$ and the conditional distribution $\mathcal{N}(\bm{X}_{t}\mid\tilde{\bm{X}};\,\sqrt{1-\beta^{2}}\,\tilde{\bm{x}},\,\beta^{2}\mathbf{C})$, the joint random vector $[\tilde{\bm{X}}\,\,\bm{X}_{t}]^{\text{T}}$ is therefore Gaussian with a mean vector of $[\bm{0}\,\,\bm{0}]^{\text{T}}$ and a covariance matrix of $\begin{bmatrix}\mathbf{C}&\sqrt{1-\beta^{2}}\,\mathbf{C}\\ \sqrt{1-\beta^{2}}\,\mathbf{C}&\mathbf{C}\end{bmatrix}$. The conditional distribution of $\tilde{\bm{X}}$ given $\bm{X}_{t}$ is hence Gaussian with a conditional mean $\bm{\mu}_{\tilde{\bm{X}}\mid\bm{X}_{t}}=\sqrt{1-\beta^{2}}\bm{X}_{t}$ and a covariance $\mathbf{\Sigma}_{\tilde{\bm{X}}\mid\bm{X}_{t}}=\beta^{2}\mathbf{C}$. We therefore get:

Consequently, the MH ratio in Eq. 18 simplifies to one, confirming that the pCN proposal is indeed invariant under the original Gaussian distribution $\pi_{\bm{X}}$. As a result, when applying the pCN algorithm to sample from the target distribution $\tilde{h}=\ell_{g_{\bm{X}}}\,\pi_{\bm{X}}$, the acceptance probability reduces to the ratio between the likelihood function for the proposed and the current samples:

The scaling parameter $\beta$ determines the balance between retaining information from the current sample and exploring new regions of the random variable space. When $\beta$ is close to zero, we have $\sqrt{1-\beta^{2}}\approx 1$, which keeps the proposed sample $\tilde{\bm{x}}$ close to the current one. This proximity leads to high acceptance rates but potentially slows the mixing of the Markov chain due to limited exploration, thus reducing the effective sample size. Conversely, as $\beta$ approaches one, we have $\sqrt{1-\beta^{2}}\approx 0$, making $\tilde{\bm{x}}$ largely independent of $\bm{x}_{t}$. This allows the sampler to explore the random variable space more broadly, but the proposed samples are more likely to land in low-probability regions, leading to an increase in rejections. To enhance the sampling efficiency, it is essential to find an optimal $\beta$ balancing these trade-offs, which is typically achieved by tuning $\beta$ to target a desired average acceptance ratio $\alpha^{*}$. The Robbins–Monro stochastic approximation algorithm provides a method for such adaptation as [51, 52]:

where $\alpha_{t}$ is the acceptance probability at iteration $t$, computed by Eq. 17, and $\gamma_{t}$ is a diminishing step size sequence, commonly chosen as $\gamma_{t}=t^{-0.5}$ [32]. In this work, we target an average acceptance probability in the range of $20$–$40\%$, with the lower values applied to high-dimensional distributions and the higher values to low-dimensional ones. This range is consistent with the values typically employed in the literature [37, 41] and has proven effective across all numerical examples considered in this work. Algorithm 1 describes the pCN algorithm, where $\bm{x}_{0}$ is the initial sample, $\beta_{0}$ is the initial scaling parameter ($\beta_{0}=0.5$ in this work), $\alpha^{*}$ is the desired average acceptance ratio, and $L_{\text{chain}}$ indicates the number of steps in the pCN chain.

In this section, we introduce a computationally efficient discovery approach for rare event domains, designed to enhance sampling efficiency by generating (multimodal) rare event samples that serve as initial seeds for the pCN chains. This approach addresses the limitations the pCN algorithm may encounter in scenarios involving multimodality and target distributions, $\tilde{h}$, that are significantly distant from the original (prior) distribution $\pi_{\bm{X}}$.

This process begins by sampling from the original distribution, $\pi_{\bm{X}}=\mathcal{N}(\bm{X};\,\bm{0},\mathbf{I})$, or a more dispersed version, $\mathcal{N}(\bm{X};\,\bm{0},\epsilon\mathbf{I})$, with the dispersion parameter $\epsilon\geq 1$, to facilitate broader exploration of the random variable space. This step is followed by conditional sampling based on the reciprocal of the conditional distribution $\nicefrac{{1}}{{\pi_{\bm{X}\mid\mathcal{F}_{j}}}}$, where $\mathcal{F}_{j}$ is an intermediate event defined as $\mathcal{F}_{j}\coloneqq\{\bm{x}\,:\,g(\bm{x})\leq\lambda_{j}\}$, with $\mathcal{F}_{1}\supset\mathcal{F}_{2}\supset\dots\supset\mathcal{F}_{M}$, where $\mathcal{F}_{M}=\mathcal{F}$ and $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{M}=0$, chosen adaptively as discussed below. The use of $\nicefrac{{1}}{{\pi_{\bm{X}\mid\mathcal{F}_{j}}}}$ significantly facilitates rapid diffusion toward the rare event domain, compared to sampling from the conditional distribution $\pi_{\bm{X}\mid\mathcal{F}_{j}}$. While the latter is necessary for methods sequentially computing the sought probability, our method focuses solely on generating samples in the rare event domain (without utilizing any sequential information). The procedure stops after generating a specified number of rare event samples from the synthetic distribution $\nicefrac{{1}}{{\pi_{\bm{X}\mid\mathcal{F}}}}$, from which a defined number of seeds, $N_{\text{chain}}$, is randomly selected to initialize the pCN chains, ensuring that the sampler is initialized within the rare event domain.

The detailed steps of this discovery process are outlined in Algorithm 2, where $N_{\text{level}}$ denotes the number of samples of each intermediate event $\mathcal{F}_{j}$, $p_{0}$ is the conditional probability for intermediate events, controlling the selection of threshold levels, $\lambda_{j}$, $\epsilon$ is a dispersion parameter for the initial distribution, $\mathcal{N}(\bm{0},\epsilon\mathbf{I})$, and $N_{\text{chain}}$ represents the number of the desired pCN chains. Fig. 4 outlines the proposed rare event domain discovery stage, showing the evolution of samples and seed selection, while the computational benefits of using a more dispersed initial distribution are demonstrated in Fig. 5. In this work, we use $\epsilon=4$ for all numerical examples to facilitate broader exploration of the random variable space by the initial sample set $\{\bm{x}_{i}^{0}\}_{i=1}^{N_{\text{level}}}$.

Let $\mathbf{X}^{\mathcal{F}}=\{\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{N_{\mathcal{F}}}\}$, with $N_{\mathcal{F}}\geq N_{S}$, be the rare event sample set obtained in Algorithm 2. The probability of selecting $\bm{x}_{i}\in\mathbf{X}^{\mathcal{F}}$ as a pCN seed can be given by:

where $\tilde{h}(\bm{x}_{i})$ is the value of the target distribution in ASTPA at sample $\bm{x}_{i}$. Then, $N_{\text{chain}}$ pCN seeds can be sampled from the set $\mathbf{X}^{\mathcal{F}}$ without replacement, using the selection probability $\text{Pr}(\bm{x}_{i})$. This approach ensures that pCN seeds are more likely to be selected from different rare event modes, based on their statistical weights. Fig. 4(e) and Fig. 5(d) illustrate the selected pCN seeds using this weighted selection method. While this technique is effective for generating multimodal seeds in low-dimensional cases, as demonstrated in the low-dimensional examples in this work, it may not scale well in high-dimensional, multimodal spaces due to the sensitivity of the high-dimensional target distribution values, even for relatively small perturbations. In such cases, we recommend using uniform sampling to select pCN seeds, which ensures a broader exploration of high-dimensional, multimodal rare event domains. This uniform sampling method is successfully adopted in the high-dimensional examples in this work. Two examples of this uniform sampling are shown in Fig. 6(a) and Fig. 6(b), illustrating that the selected seeds are located in the rare event domain but do not follow any relevant target distribution ($\pi_{\bm{X}}$ or $\tilde{h}$). The computational efficiency of the proposed discovery approach, which requires significantly fewer model evaluations, just in order to identify the relevant domains, compared to conventional sequential methods utilized for probability estimation, is highlighted by comparing its performance in Fig. 6(a) and Fig. 6(b) with the sampling evolution of the Subset Simulation (SuS) method, as shown in Fig. 6(c).

The first example involves a bimodal nonlinear limit-state function characterized by two independent standard normal random variables:

The approximate sampling target, $\tilde{h}$, in ASTPA is constructed using a likelihood dispersion factor $\sigma=0.3$ and a scaling constant $g_{c}=1$, as discussed in Section 3.1. The discovery stage then starts using $N_{\text{level}}=300$ and $p_{0}=0.1$, yielding 10 rare event seeds for the pCN sampler ($N_{\text{chain}}=10$), with each chain running for $L_{\text{chain}}=150$ iterations. Subsequently, inverse importance sampling (IIS) is applied using $M=300$ samples. Table 1 compares the performance of the guided pCN-ASTPA with iCE-IS, SIS, and aCS-SuS. As shown in table 1, the guided pCN-based ASTPA provides an $\mathbb{E}[\hat{p}_{\mathcal{F}}]$ estimate, closely matching the reference probability obtained through standard Monte Carlo simulation (MCS). Compared to the other methods considered, the guided pCN-based ASTPA demonstrates improved performance, achieving a lower C.o.V using fewer total model calls, $\mathbb{E}[N_{\text{total}}]$. Notably, iCE-IS also demonstrates strong efficiency, outperforming both SIS and aCS-SuS in this 2D example. The excellent agreement between the sampling C.o.V and analytical C.o.V, as reported in parentheses, confirms the accuracy and effectiveness of the ASTPA analytical C.o.V expression. Since this example was introduced earlier in the paper to illustrate the ASTPA framework and the discovery stage, the steps and performance of the proposed framework are visualized in Figs. 1 and 5.

In the second example, we consider a quartic bimodal limit-state function, defined in the standard Gaussian space as:

The approximate target $\tilde{h}$ is constructed with a likelihood dispersion factor of $\sigma=0.2$ and a scaling constant $g_{c}=1$. The discovery stage is applied using $N_{\text{level}}=300$ and $p_{0}=0.1$, generating 18 rare event seeds for the pCN sampler ($N_{\text{chain}}=18$), with each chain running for $L_{\text{chain}}=120$ steps. As in the previous example, inverse importance sampling (IIS) is used with $M=300$ samples. Fig. 7 shows the evolution of the rare event domain discovery stage, highlighting the effectiveness of the proposed algorithm in accurately identifying and sampling rare event regions.

The results, as summarized in Table 2, show that the guided pCN-ASTPA significantly outperforms the other methods, achieving the lowest C.o.V, while using the fewest model calls ($\mathop{\mathbb{E}}[N_{\text{total}}]$). This demonstrates the robustness and efficiency of the guided pCN-ASTPA framework, even in the presence of increased nonlinearity introduced through higher-order terms in the limit-state function. The guided pCN-based ASTPA also provides $\mathop{\mathbb{E}}[\hat{p}_{\mathcal{F}}]$ estimates in close agreement with the reference probability computed by Monte Carlo simulation (MCS). The very good agreement between the sampling and analytical C.o.V values (reported in parentheses) further confirms the reliability of the analytical C.o.V expression for ASTPA in this more challenging problem.

The Himmelblau function [54] is a commonly used fourth-order polynomial test function in nonlinear optimization. In this example, we modify the Himmelblau function to suit rare event scenarios, particularly those with multiple separated rare event domains. The modified limit-state function is defined in the standard Gaussian space as:

where $X_{1}$ and $X_{2}$ are independent standard normal random variables. The approximate target $\tilde{h}$ is constructed using a likelihood dispersion factor $\sigma=0.3$ and a scaling constant $g_{c}=\nicefrac{{g(\bm{0})}}{{4}}$, as described in Section 3.1 and Fig. 3. The discovery stage is applied using $N_{\text{level}}=300$ and $p_{0}=0.1$, generating 16 rare event seeds for the pCN sampler, with each chain subsequently running for $L_{\text{chain}}=160$ iterations. Similar to the previous examples, inverse importance sampling (IIS) is applied using $M=300$ samples. Fig. 8 illustrates the constructed target distribution, the rare event domain discovery process, and the pCN samples, effectively exploring all rare event modes. The results in Table 3 show that the guided pCN-ASTPA outperforms the other methods, achieving a lower C.o.V and fewer total model evaluations. This demonstrates the robustness and efficiency of the guided pCN-ASTPA framework, particularly in handling complex, multi-modal rare event domains. The $\mathop{\mathbb{E}}[\hat{p}_{\mathcal{F}}]$ estimate produced by the guided pCN-based ASTPA closely matches the reference probability, further confirming the method’s accuracy.

This example examines a limit-state function with significant topological changes, making it a challenging scenario for rare event sampling methods, as described in [28]. The function is defined as:

where $X_{1}$ and $X_{2}$ are independent standard normal random variables. This limit-state surface is particularly challenging due to its abrupt changes in topology, as shown in Fig. 9(a), requiring a robust sampling framework to accurately capture the rare event domain.

The guided pCN-ASTPA algorithm is applied with a likelihood dispersion factor of $\sigma=0.1$ and a scaling constant $g_{c}=\nicefrac{{g(\mathbf{0})}}{{5}}$, as outlined in Section 3.1. The discovery stage uses $N_{\text{level}}=300$ and $p_{0}=0.1$, generating 6 rare event seeds for the pCN sampler, with each chain running for $L_{\text{chain}}=100$ iterations. Inverse importance sampling (IIS) is employed with $M=200$ samples. Fig. 9 visualizes the limit-state surface, the constructed sampling target, and the evolution of samples in the rare event domain discovery process, highlighting the effectiveness of guided pCN-ASTPA in accurately capturing the influential rare event region. In contrast, Fig. 9(e) demonstrates the limitations of sequential methods like SuS, which struggle to efficiently identify the rare event mode with a competitive number of model evaluations. These rare event domain discovery limitations are more pronounced in iCE-IS in this example, which exhibits a significant error in the estimated mean of the rare event probability. As presented in Table 4, the guided pCN-ASTPA achieves an accurate rare event probability estimate, $\mathop{\mathbb{E}}[\hat{p}_{\mathcal{F}}]$, with significantly fewer model evaluations and a lower Coefficient of Variation (C.o.V) compared to the other methods. This underscores the effectiveness of the pCN-ASTPA framework in tackling problems with complex topological changes, where traditional sequential methods often underperform.

This example analyzes a thirty-four-story structure, adapted from [55], as represented in Fig. 10. The structure is subjected to thirty four static loads $F_{i}$, for $i=1,2,\dots,34$. The floor slabs/beams are assumed to be rigid, and all columns have the same height, $H=4$ m, but different flexural stiffnesses $EI_{k}$, for $k=1,2,\dots,68$. Both the loads and stiffnesses are treated as random variables, resulting in a total of $d=102$ random variables. The loads are modeled as normally distributed with a mean of 2 kN and a C.o.V. of 0.4. Similarly, the stiffnesses are normally distributed with a mean of 20 MNm² and a C.o.V. of 0.2. The top story displacement, $u$, can be determined by summing the interstory displacements $u_{i}$, expressed as: