  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.
