# High-dimensional Example Function

**URL:** <https://uqworld.org/t/high-dimensional-example-function/67>\
**Category:** Benchmarks\
**Tags:** test-function, m-dimension, reliability\
**Created:** [January 29, 2019, 2:21pm UTC](https://uqworld.org/t/high-dimensional-example-function/67 "2019-01-29T14:21:47Z")\
**Posts on this page:** 1\
**Page:** 1

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**Author:** ![damarginal](https://uqworld.org/user_avatar/uqworld.org/damarginal/32/73_2.png) [@damarginal](https://uqworld.org/u/damarginal)\
**Post date:** [January 29, 2019, 2:21pm UTC](https://uqworld.org/t/high-dimensional-example-function/67/1 "2019-01-29T14:21:48Z")

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The high-dimensional limit state function is used to test algorithms in structural reliability analysis, such as Adaptive Kriging Monte Carlo simulation (AK-MCS) and others (Rackwitz, 2001). A prominent feature of this test function is that the failure probability does not change significantly when changing the number of input dimensions.

## Description

The analytic expression of the high-dimensional limit state function is given as:

g(\mathbf{x}) = \left(M + 3 a \sqrt{M}\right) - \sum\_{i=1}^{M} x\_i,

where M is the number of input variables (or dimensions; positive integer); \{x\_m\}\_{m=1}^{M} are the input variables; and a is a parameter. By default, the parameter is set to 0.2.

The failure event is defined as g(\mathbf{x}) \leq 0 and the failure probability P\_f = \mathbb{P}[g(\mathbf{x}) \leq 0]. Figure 1 and 2 show the surface and contour plots of the 2-dimensional high-dimensional function using the default values of the constants, respectively.  
In Figure 2, the limit state function at which g(\mathbf{x}) = 0 is highlighted.

![highDimensionalSurface](https://uqworld.org/uploads/default/original/1X/450626b9fd48870a8496d44891104fde7c91e46f.png)  
**Figure 1** : Surface plot of the 2-dimensional high-dimensional limit state function.

![highDimensionalContour](https://uqworld.org/uploads/default/original/1X/114f8d4a2957030c24bf9e3749490d5028866dba.png)  
**Figure 2** : Contour plot of the 2-dimensional high-dimensional limit state function. The limit state function at which g(\mathbf{x})=0 is also shown.

## Inputs

M independent identically distributed lognormal random variables are considered for the high-dimensional limit state function.

| No | Variable | Distribution | Parameters |
| --- | --- | --- | --- |
| 1 | x\_1 | Lognormal | \mu\_{x\_1} = 1,  
\sigma\_{x\_1} = 0.2 |
| \vdots | \vdots | \vdots | \vdots |
| M | x\_M | Lognormal | \mu\_{x\_M} = 1,  
\sigma\_{x\_M} = 0.2 |

## Parameter

By default, the parameter a is set to 0.2 following Rackwitz (2001).

## Reference values

The asymptotic value of the failure probability P\_f with M \rightarrow \infty is 1.35 \times 10^{-3} (Rackwitz, 2001). Some other values of P\_f calculated using different methods for different values of M are shown in the table below.

| Method | M | N | \hat{P}\_f | \text{CoV}[\hat{P}\_f] | Source |
| --- | --- | --- | --- | --- | --- |
| FORM | 50 | 154 | 1.531 \times 10^{-4} | - | Rackwitz (2001) |
| SORM | 50 | 1'480 | 2.555 \times 10^{-3} | - | Rackwitz (2001) |
| MCS | 50 | 10^{6} | 1.915 \times 10^{-3} | 2.28\% | UQLab v1.2.1 |
| FORM | 100 | 304 | 3.369 \times 10^{-4} | - | Rackwitz (2001) |
| SORM | 100 | 5'455 | 2.98 \times 10^{-3} | - | Rackwitz (2001) |
| MCS | 100 | 10^{6} | 1.685 \times 10^{-3} | 2.43\% | UQLab v1.2.1 |
| FORM | 100 | 603 | 6.212 \times 10^{-6} | - | Rackwitz (2001) |
| SORM | 200 | 20'904 | 4.53 \times 10^{-3} | - | Rackwitz (2001) |
| MCS | 200 | 10^{6} | 1.669 \times 10^{-3} | 2.45\% | UQLab v1.2.1 |

## Resources

The vectorized implementation of the high-dimensional limit state function in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLAB can be downloaded below:

[uq\_highDimensional.zip](https://uqworld.org/uploads/default/original/1X/a4e4adaaebb0e44a524a015d57c2383369158406.zip) (2.4 KB)

The contents of the file are:

| Filename | Description |
| --- | --- |
| `uq_highDimensional.m` | vectorized implementation of the high-dimensional limit state function in MATLAB |
| `uq_Example_highDimensional.m` | definitions for the model and probabilistic inputs in UQLab |
| `LICENSE` | license for the function (BSD 3-Clause) |

## Reference

- R. Rackwitz, “Reliability analysis—a review and some perspectives,” _Structural Safety_, vol. 23, no. 4, pp. 365–395, 2001. [DOI:10.1016/S0167-4730(02)00009-7](https://doi.org/10.1016/S0167-4730(02)00009-7)