This study presents an importance sampling formulation based on adaptively relaxing parameters from the indicator function and/or the probability density function. The formulation embodies the prevalent mathematical concept of relaxing a complex problem into a sequence of progressively easier sub-problems. Due to the flexibility in constructing relaxation parameters, relaxation-based importance sampling provides a unified framework for various existing variance reduction techniques, such as subset simulation, sequential importance sampling, and annealed importance sampling. More crucially, the framework lays the foundation for creating new importance sampling strategies, tailoring to specific applications. To demonstrate this potential, two importance sampling strategies are proposed. The first strategy couples annealed importance sampling with subset simulation, focusing on low-dimensional problems. The second strategy aims to solve high-dimensional problems by leveraging spherical sampling and scaling techniques. Both methods are desirable for fragility analysis in performance-based engineering, as they can produce the entire fragility surface in a single run of the sampling algorithm. Three numerical examples, including a 1000-dimensional stochastic dynamic problem, are studied to demonstrate the proposed methods.

翻译：本研究提出了一种重要性抽样公式，其核心思想是通过自适应地松弛指示函数和/或概率密度函数中的参数。该公式体现了将复杂问题逐步简化为一系列越来越简单的子问题这一普遍的数学概念。由于在构造松弛参数方面具有灵活性，基于松弛的重要性抽样为多种现有的方差缩减技术提供了一个统一框架，例如子集模拟、序贯重要性抽样和退火重要性抽样。更为关键的是，该框架为针对特定应用创建新的重要性抽样策略奠定了基础。为展示这一潜力，本文提出了两种重要性抽样策略。第一种策略将退火重要性抽样与子集模拟相结合，专注于低维问题。第二种策略旨在通过利用球面抽样和缩放技术来解决高维问题。这两种方法对于基于性能的工程中的易损性分析尤为理想，因为它们能够在单次抽样算法运行中生成完整的易损性曲面。通过三个数值算例（包括一个1000维的随机动力问题）对所提方法进行了验证。