setup_genz_function

pyapprox.benchmarks.setup_genz_function(nvars, test_name, coeff_type=None, w=0.25, c_factor=1, coeff=None)[source]

Setup one of the six Genz integration benchmarks $f_{d} (x) : R^{D} \to R$ , where $x = [x_{1}, \dots, x_{D}]^{⊤}$ . The number of inputs $D$ and the anisotropy (relative importance of each variable and interactions) of the functions can be adjusted. The definition of each function is in the Notes section.

For example, the two-dimensional oscillatory Genz problem can be defined using

>>> from pyapprox.benchmarks.benchmarks import setup_benchmark
>>> benchmark=setup_benchmark('genz',nvars=2,test_name='oscillatory')
>>> print(benchmark.keys())
dict_keys(['fun', 'mean', 'variable'])

Parameters:

nvarsinteger: The number of variables of the Genz function
test_namestring: The test_name of the specific Genz function. See notes for options the string needed is given in brackets e.g. (‘oscillatory’). Choose from [“oscillatory”, “product_peak”, “corner_peak”, “c0continuous”, “discontinuous”]
coef_typestring: Choose from [“no_decay”, “quadratic_decay”, “quartic_decay”, “exponential_decay”. “squared_exponential_decay”]
wfloat 0<=w<=1: Set $w_{d} = w, d = 1, \dots, D$ .
c_factorfloat c_factor>0: Scale the integrand.
coefftuple (ndarray (nvars, 1), ndarray (nvars, 1)): The coefficients $c_{d}$ and $w_{d}$ If provided it will overwite the coefficients defined by coeff_type, w and c_factor

Returns:

benchmarkpyapprox.benchmarks.Benchmark: Object containing the benchmark attributes
funcallable: The function being analyzed
variableJointVariable: Class containing information about each of the nvars inputs to fun
mean: np.ndarray (nvars): The mean of the function with respect to the PDF of var

Notes

The six Genz test function are:

Oscillatory (‘oscillatory’)

f (z) = \cos (2 π w_{1} + \sum_{d = 1}^{D} c_{d} z_{d})

Product Peak (‘product_peak’)

f (z) = \prod_{d = 1}^{D} {(c_{d}^{- 2} + (z_{d} - w_{d})^{2})}^{- 1}

Corner Peak (‘corner_peak’)

f (z) = {(1 + \sum_{d = 1}^{D} c_{d} z_{d})}^{- (D + 1)}

Gaussian Peak (‘gaussian’)

f (z) = \exp (- \sum_{d = 1}^{D} c_{d}^{2} (z_{d} - w_{d})^{2})

C0 Continuous (‘c0continuous’)

f (z) = \exp (- \sum_{d = 1}^{D} c_{d} | z_{d} - w_{d} |)

Discontinuous (‘discontinuous’)

\begin{array}{r} f (z) = {\begin{cases} 0 & z_{1} > w_{1} or z_{2} > w_{2} \\ \exp (\sum_{d = 1}^{D} c_{d} z_{d}) & otherwise \end{cases} \end{array}

Increasing $‖ c ‖$ will in general make the integrands more difficult.

The $0 \leq w_{d} \leq 1$ parameters do not affect the difficulty of the integration problem. We set $w_{1} = w_{2} = \dots = W_{D}$ .

The coefficient types implement different decay rates for $c_{d}$ . This allows testing of methods that can identify and exploit anisotropy. They are as follows:

No decay (none)

{\hat{c}}_{d} = \frac{d + 0.5}{D}

Quadratic decay (qudratic)

{\hat{c}}_{d} = \frac{1}{(D + 1)^{2}}

Quartic decay (quartic)

{\hat{c}}_{d} = \frac{1}{(D + 1)^{4}}

Exponential decay (exp)

{\hat{c}}_{d} = \exp (\log (c_{\min}) \frac{d + 1}{D})

Squared-exponential decay (sqexp)

{\hat{c}}_{d} = 10^{(\log_{10} (c_{\min}) \frac{(d + 1)^{2}}{D})}

Here $c_{\min}$ is argument that sets the minimum value of $c_{D}$ .

Once the formula are used the coefficients are normalized such that

c_{d} = c_{factor} \frac{{\hat{c}}_{d}}{\sum_{d = 1}^{D} {\hat{c}}_{d}} .

References

[Genz1984]

Genz, A. Testing multidimensional integration routines. In Proc. of international conference on Tools, methods and languages for scientific and engineering computation (pp. 81-94), 1984