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"""Borehole function."""
import numpy as np
[docs]
def borehole83_lofi(rw, r, tu, hu, tl, hl, l, kw):
r"""Low-fidelity version of Borehole benchmark function.
Very simple and quick to evaluate eight dimensional function that models
water flow through a borehole. Frequently used function for testing a wide
variety of methods in computer experiments.
The low-fidelity version is defined as in [1] as:
:math:`f_{lofi}({\bf x}) = \frac{5 T_u (H_u-H_l)}{\ln(\frac{r}{r_w})(1.5)
+ \frac{2 L T_u}{\ln(\frac{r}{r_w})r_w^2 K_w} + \frac{T_u}{T_l}}`
For the purposes of uncertainty quantification, the distributions of the
input random variables are often chosen as:
| rw ~ N(0.10, 0.0161812)
| r ~ Lognormal(7.71, 1.0056)
| tu ~ Uniform[63070, 115600]
| hu ~ Uniform[990, 1110]
| tl ~ Uniform[63.1, 116]
| hl ~ Uniform[700, 820]
| l ~ Uniform[1120, 1680]
| kw ~ Uniform[9855, 12045]
Args:
rw (float): Radius of borehole :math:`(m)` [0.05, 0.15]
r (float): Radius of influence :math:`(m)` [100, 50000]
tu (float): Transmissivity of upper aquifer :math:`(\frac{m^2}{yr})` [63070, 115600]
hu (float): Potentiometric head of upper aquifer :math:`(m)` [990, 1110]
tl (float): Transmissivity of lower aquifer :math:`(\frac{m^2}{yr})` [63.1, 116]
hl (float): Potentiometric head of lower aquifer :math:`(m)` [700, 820]
l (float): Length of borehole :math:`(m)` [1120, 1680]
kw (float): Hydraulic conductivity of borehole :math:`(\frac{m}{yr})` [9855, 12045]
Returns:
float: The response is water flow rate, in :math:`(\frac{m^3}{yr})`
References:
[1] Xiong, S., Qian, P. Z., & Wu, C. J. (2013). Sequential design and
analysis of high-accuracy and low-accuracy computer codes.
Technometrics, 55(1), 37-46.
"""
frac1 = 5 * tu * (hu - hl)
frac2a = 2 * l * tu / (np.log(r / rw) * rw**2 * kw)
frac2b = tu / tl
frac2 = np.log(r / rw) * (1.5 + frac2a + frac2b)
y = frac1 / frac2
return y
[docs]
def borehole83_hifi(rw, r, tu, hu, tl, hl, l, kw):
r"""High-fidelity version of Borehole benchmark function.
Very simple and quick to evaluate eight dimensional function, that models
water flow through a borehole. Frequently used function for testing a wide
variety of methods in computer experiments, see, e.g., [1]-[10].
The high-fidelity version is defined as:
:math:`f_{hifi}({\\bf x}) = \frac{2 \pi T_u(H_u-H_l)}{\ln(\frac{r}{r_w})
\left(1 + \frac{2LT_u}{\ln(\frac{r}{r_w})r_w^2K_w} \right)+ \frac{T_u}{T_l}}`
For the purpose of multi-fidelity simulation, Xiong et al. (2013) [8] use
the following function for the lower fidelity code:
:math:`f_{lofi}({\bf x}) = \frac{5 T_u (H_u-H_l)}{\ln(\frac{r}{r_w})(1.5)
+ \frac{2 L T_u}{\ln(\frac{r}{r_w})r_w^2 K_w} + \frac{T_u}{T_l}}`
For the purposes of uncertainty quantification, the distributions of the
input random variables are often chosen as:
| rw ~ N(0.10,0.0161812)
| r ~ Lognormal(7.71,1.0056)
| tu ~ Uniform[63070, 115600]
| hu ~ Uniform[990, 1110]
| tl ~ Uniform[63.1, 116]
| hl ~ Uniform[700, 820]
| l ~ Uniform[1120, 1680]
| kw ~ Uniform[9855, 12045]
Args:
rw (float): Radius of borehole :math:`(m)` [0.05, 0.15]
r (float): Radius of influence :math:`(m)` [100, 50000]
tu (float): Transmissivity of upper aquifer :math:`(\frac{m^2}{yr})` [63070, 115600]
hu (float): Potentiometric head of upper aquifer :math:`(m)` [990, 1110]
tl (float): Transmissivity of lower aquifer :math:`(\frac{m^2}{yr})` [63.1, 116]
hl (float): Potentiometric head of lower aquifer :math:`(m)` [700, 820]
l (float): Length of borehole :math:`(m)` [1120, 1680]
kw (float): Hydraulic conductivity of borehole :math:`(\frac{m}{yr})` [9855, 12045]
Returns:
float: The response is water flow rate, in :math:`\frac{m^3}{yr}`
References:
[1] An, J., & Owen, A. (2001). Quasi-regression. Journal of Complexity,
17(4), 588-607.
[2] Gramacy, R. B., & Lian, H. (2012). Gaussian process single-index models
as emulators for computer experiments. Technometrics, 54(1), 30-41.
[3] Harper, W. V., & Gupta, S. K. (1983). Sensitivity/uncertainty analysis
of a borehole scenario comparing Latin Hypercube Sampling and
deterministic sensitivity approaches (No. BMI/ONWI-516).
Battelle Memorial Inst., Columbus, OH (USA). Office of Nuclear
Waste Isolation.
[4] Joseph, V. R., Hung, Y., & Sudjianto, A. (2008). Blind kriging:
A new method for developing metamodels. Journal of mechanical design,
130, 031102.
[5] Moon, H. (2010). Design and Analysis of Computer Experiments for
Screening Input Variables (Doctoral dissertation, Ohio State University).
[6] Moon, H., Dean, A. M., & Santner, T. J. (2012). Two-stage
sensitivity-based group screening in computer experiments.
Technometrics, 54(4), 376-387.
[7] Morris, M. D., Mitchell, T. J., & Ylvisaker, D. (1993).
Bayesian design and analysis of computer experiments: use of derivatives
in surface prediction. Technometrics, 35(3), 243-255.
[8] Xiong, S., Qian, P. Z., & Wu, C. J. (2013). Sequential design and
analysis of high-accuracy and low-accuracy computer codes.
Technometrics, 55(1), 37-46.
[9] Worley, B. A. (1987). Deterministic uncertainty analysis
(No. CONF-871101-30). Oak Ridge National Lab., TN (USA).
[10] Zhou, Q., Qian, P. Z., & Zhou, S. (2011). A simple approach to
emulation for computer models with qualitative and quantitative
factors. Technometrics, 53(3).
For further information, see also http://www.sfu.ca/~ssurjano/borehole.html
"""
frac1 = 2 * np.pi * tu * (hu - hl)
frac2a = 2 * l * tu / (np.log(r / rw) * rw**2 * kw)
frac2b = tu / tl
frac2 = np.log(r / rw) * (1 + frac2a + frac2b)
y = frac1 / frac2
return y
