Simulate Example: AU2 (Importance Sampling)

This page documents a simulate run with importance sampling enabled (-i).

Note: Table values are rounded to 4 significant figures for readability. Very small/large values use scientific notation. Refer to the Excel/JSON result files for full precision.

Run Context

Problem module:

• problems/AU2Problem.py

Recorded result set:

• results/2026-03-16/11-06-54/AU2-42d54.xlsx
• results/2026-03-16/11-06-54/AU2-42d54.json
• results/2026-03-16/11-06-54/AU2-42d54.py
• results/2026-03-16/11-06-54/AU2-42d54.pickle
• results/2026-03-16/11-06-54/AU2-42d54.pdf
• results/2026-03-16/11-06-54/profile-42d54.yaml

Profile and run mode from saved profile:

• Profile used: default
• run_type: simulate
• include_mc: true
• mc_with_is: true

For results-folder and filename conventions, see CLI Result Files.

Equivalent command shape:

reliafy simulate <profile> -i

The -i flag enables importance sampling (IS), which drastically reduces the number of samples needed to estimate very small failure probabilities by concentrating samples near the failure region.

Profile Customization

This example uses the default profile, but simulate behavior is configurable. See Profile Options Reference.

• Importance sampling controls: run_configuration.mc_with_is (or -i), reliability_options.is_method, is_kde_bandwidth, is_fit_samples, is_mixture_components, is_mcmc_*.
• Monte Carlo controls: reliability_options.mc_n, mc_max_cv, mc_seed, mc_remove_oob.
• Report toggles: reporting_options.save_plots_to_pdf, save_plots_to_pickle, save_excel_summary.

Problem File Used

Source: Au, S. K. and Beck, J. L., "A new adaptive importance sampling scheme for reliability calculations," Structural Safety, vol. 21, no. 2, 1999, pp. 135–158. →

AU2Problem.py defines:

• Stochastic variables: T1, T2 — both standard Normal (μ = 0, σ = 1)
• Deterministic variables: c = 5
• LSFreturnsGradient: False, LSFreturnsHessian: False, LSFreturnsLandR: False
• Series limit state (in-series components): The compound limit state is:
• g = numpy.minimum(g1, g2)

This represents two components in series — the system fails when either component fails (i.e., when g1 < 0 OR g2 < 0). The numpy.minimum() function selects the element-wise minimum, which is essential for handling arrays of samples during Monte Carlo simulation.

• Individual limit state functions:
• Component 1: g1 = c - 1 - T2 + numpy.exp(-T1**2/10) + (T1/5)**4
• Component 2: g2 = c**2/2 - T1*T2

Limit states in series and parallel

The simulate command handles multiple limit states combined using numpy.minimum() (series) and numpy.maximum() (parallel) and any combination of both. This example demonstrates the series pattern with numpy.minimum(). Use numpy.maximum() for parallel (redundant) components where the system fails only when all components fail.

Important: Use the element-wise functions numpy.minimum() and numpy.maximum(). Do not use Python's built-in min() and max(), and do not use NumPy's reduction functions numpy.min() and numpy.max(), because during Monte Carlo simulation the component limit-state values are arrays of sample values and this operation requires element-wise comparison.

• ISplot key defined — required for IS diagnostics plots

ISplot key

The ISplot key works like LSFplot and RFADplot: it specifies the axis variables and plot extents used to render the IS proposal distribution diagnostics. When running with -i, the CLI uses ISplot to produce the IS Proposal Diagnostics figures (both u-space and x-space views). If ISplot is absent, the diagnostics plots will not be generated.

Important: If samples from the proposal distribution fall outside the specified plot limits, the axis ranges will be automatically expanded to include all samples. This ensures the entire sampled region is visible in the diagnostics plots.

"ISplot": {
    "x_var": "T1",
    "x_lim": [-5.0, 5.0],
    "y_var": "T2",
    "y_lim": [-5.0, 5.0],
},

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in AU2-42d54.xlsx.

Header Information

Field	Value
Problem	AU2
Request ID	cfcb942355da4179b9b48eb4dfa42d54
Run time	00 min 08.93 sec

Deterministic Variables

var_name	value
c	5

Stochastic Variables Definition

var_name	var_type	mean	std	param1	param2
T1	Normal	0	1	0	1
T2	Normal	0	1	0	1

Monte Carlo Results (with Importance Sampling)

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is	method	bandwidth	fit_samples
4.7775	8.873e−7	0.02978	0.05	6,000	8.333%	3	True	True	kde	0.2817	2,000

Key IS parameters:

• method: kde — the IS proposal distribution was fitted as a kernel density estimate.
• bandwidth: 0.2817 — KDE bandwidth used for the proposal.
• fit_samples: 2,000 — number of samples drawn near the failure region to fit the proposal.

Monte Carlo Variable Statistics and Correlations (IS-weighted)

IS-weighted statistics

Because importance sampling re-weights each sample, the tabulated means and standard deviations reflect the proposal distribution, not the original input distributions. This is expected behavior.

Similarly, the histogram plots in Figures 3, 4, and 5 display the re-weighted samples and thus show the characteristics of the proposal distribution, not the original input distributions.

var_name	mean	std	%_oob	cor(T1)	cor(T2)
T1	−0.2182	3.5806	0	1.0000	0.5670
T2	2.3820	4.2807	0	0.5670	1.0000

Notes Reported by Reliafy

1. Validation: Stochastic variables definition and limit state function validation required 4 function calls.
2. Monte Carlo: Creation of importance sampling proposal distribution required 109,980 calls to the limit state function.
3. Monte Carlo: Completed 3 cycles with 2.00e+03 samples per cycle.
4. Monte Carlo: 0.03% of T1 and 0.05% of T2 values were not finite (NaN, inf, complex) out of 6.00e+03 samples and were excluded from the Monte Carlo statistics.
5. Monte Carlo: Load and resistance histograms are not available because LSFreturnsLandR: False, so the limit state function returns empty load/resistance placeholders.

Interpretation Snapshot

• The failure probability is very small (pf ≈ 8.87 × 10⁻⁷, beta ≈ 4.78), making plain Monte Carlo impractical — it would require on the order of 10⁸ samples for reliable estimation. Importance sampling achieves good precision with only 6,000 weighted samples.
• The cv = 0.0298 is below max_cv = 0.05, confirming adequate IS precision for this run.
• The IS proposal construction required ~110,000 LSF evaluations to locate and characterize the failure region before the weighted sampling phase began.
• The IS-weighted variable statistics (T1 mean ≈ −0.22, T2 mean ≈ 2.38) clearly reflect bias toward the failure region, which is expected and correct.
• No load/resistance histogram is generated because LSFreturnsLandR: False, so L and R are returned as empty placeholders.

Generated Figures

The PDF result file for this run is saved as results/2026-03-16/11-06-54/AU2-42d54.pdf.

Figure 1: IS Proposal Diagnostics (u-space)

The KDE proposal distribution in standard-normal space. Contour lines show the proposal PDF; the failure region boundary is visible.

Figure 2: IS Proposal Diagnostics (x-space)

The same proposal distribution plotted in original variable space (T1, T2).

Figure 3: IS Histogram — T1

Figure 4: IS Histogram — T2

Figure 5: Limit State Function Histogram (Importance Sampling)