Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design
<p>Comparing number of unknown coefficients between the PCE and LRA methods for high dimensional problems.</p> "> Figure 2
<p>Comparison of sampling methods [<a href="#B47-algorithms-15-00250" class="html-bibr">47</a>].</p> "> Figure 3
<p>Surrogate model error metrics for Park (1991) function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>).</p> "> Figure 4
<p>Comparison of probability density function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>h</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>).</p> "> Figure 5
<p>Surrogate model error metrics for Borehole function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>).</p> "> Figure 6
<p>Comparison of probability density function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>h</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>).</p> "> Figure 7
<p>Surrogate model error metrics for Sobol’-Levitan function (<math display="inline"><semantics> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> </semantics></math> = 10,000).</p> "> Figure 8
<p>Comparison of probability density function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>h</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>2500</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> </semantics></math> = 10,000).</p> "> Figure 9
<p>Aeroacoustic propagation in sonic boom prediction. Reproduced with permission from Ref. [<a href="#B51-algorithms-15-00250" class="html-bibr">51</a>].</p> "> Figure 10
<p>The JAXA Wing-Body geometry.</p> "> Figure 11
<p>Pressure coefficient distribution comparison (<b>a</b>) on the upper surface (<b>b</b>) on the lower surface.</p> "> Figure 12
<p>Pressure signature comparison: (<b>a</b>) Near-field pressure signature (<b>b</b>); ground signature for JWB.</p> "> Figure 13
<p>Surrogate model error metrics for sonic boom problem 1 (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>70</mn> </mrow> </semantics></math>).</p> "> Figure 14
<p>Comparison of probability density function (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>70</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>h</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>).</p> "> Figure 15
<p>Surrogate model error metrics for sonic boom problem 2 (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>).</p> "> Figure 16
<p>Comparison of probability density function (<b>a</b>) Results of all surrogate models (<b>b</b>) Results of LRA and MF-LRA (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>h</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>l</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>).</p> ">
Abstract
:1. Introduction
2. Methods
2.1. Polynomial Chaos Expansion
2.2. The Low-Rank Approximation Method
2.3. Multi-Fidelity Extension of PCE and LRA
3. Computational Experiments
- Design of Experiments (DoE): Halton SamplingThe key component of the surrogate modelling process is the design of experiments for the simulations. As already known from the associated studies, surrogate model predictions strictly depend on the applied design of experiments (DoE) set. In this study, several widely used DoE sampling types are demonstrated for a reliable performance assessment of the considered surrogates. To incorporate all the characteristics of the data over the sampling space, samples must be distributed homogeneously as much as possible. If this is achieved, the performance of the surrogate can be evaluated without the biased effect of the selected sampling strategy. With this in mind, Monte Carlo (MC) [43], Latin Hypercube (LHS) [44], Optimized Latin Hypercube (OLHS) [45] and Halton sequence (HS) sampling [46] strategies are compared in Figure 2 to visualise behavioural differences for several conventional sampling types.As indicated in the two-dimensional sampling strategy visualisations, Latin Hypercube sampling has the ability to fill the design space successfully. In contrast, Monte Carlo sampling may have some accumulated data areas that may result in learning deficits for the implemented surrogate model. The Halton sequence distributes the samples homogeneously over the design space. As the number of samples is increased, HS keeps the previous sampling set the same, while adding the new samples homogeneously distributed to the vacant regions of the design space. In the present work, Halton sequence is therefore selected as a DoE strategy in an effort to prevent potential bias effects from the sampling strategy.
- Local/Global accuracy metrics:In order to compare the accuracy of surrogate models, error metrics are calculated using the predicted values of the surrogate models and their corresponding analytical results at the same test points. The proposed metrics are selected to offer a comprehensive assessment of the surrogates both in terms of local and global characteristics of the modelling ability of the real functions. These metrics include coefficient of determination (), root-mean square error (), and maximum absolute error (); their formulas and normalised versions are later given in Equations (22)–(24).The coefficient of determination is a statistical measure of how well the regression estimate converges to the actual data points. takes a value between 0 and 1, with a high value indicating that the model fits well with the data used. Using the test data-set, the formulation is expressed as follows, where is the mean of test data-set, is the exact value of the function and represents the surrogate model prediction.The metric expresses the standard deviation of the prediction error and measures how well the predicted values match the actual values in absolute terms. The value ranges from 0 to ∞, with smaller values indicating that the method makes a more accurate prediction. In this respect, is a global accuracy metric that represents the difference between the true value and the surrogate model value. It is used in the study to measure the success of the methods in capturing global behaviour. Using a test data-set, the formulation is expressed as in Equation (23).The maximum absolute error value represents the greatest difference in the design space between the surrogate model prediction and the actual value. is frequently used in the literature to measure the prediction success of the surrogate model locally, and in this study, it is used to quantify how accurately the surrogate models capture the local features. The formulation is expressed using a test data-set as follows,The proposed assessment metrics might show different characteristics, as global accuracy may differ from local accuracy. In order to present a comprehensive performance evaluation of the surrogates, two different type of metrics are considered. A surrogate model’s ability to capture both local and global characteristics needs to be evaluated. For example, when considering a search for an optimum or any other local response of the surrogate, the has crucial importance to ensure that the model has no local deficits. At the same time, to facilitate an efficient evaluation of the surrogate performance, some kind of a compromise has to be observed between local and global accuracy of the models.
- Uncertainty quantification (UQ) metrics:Evaluation of the uncertainty quantification results is performed using the probability density function. In engineering problems, the terms mean, standard deviation, skewness and kurtosis, which express the behaviour of the density function, are used to numerically compare the results of the uncertainty quantification. The mean () is the general tendency of the response based on the variation of the uncertain variables, and the standard deviation() refers to the variation in the analysis program’s response based on the distribution of the uncertain variable. The mean and standard deviation formulas are given in Equation (25), where N represents the number of responses.Skewness is a measure of the asymmetry of the probability density of response relative to the mean and is calculated as in Equation (26). Kurtosis, on the other hand, expresses a measure of whether the data contain an abundance of outliers or lack of outliers relative to a normal distribution and is calculated as in Equation (26).
- Fidelity cost assignment criteria:Multi-fidelity surrogate assessments are usually depicted by using a constant number of low-fidelity analyses and adding high-fidelity data as necessary. Since design problems have to be solved with limited computational budget to complete the process within a reasonable amount of time, dominance of low fidelity data is more convenient. Preliminary cost assignment is determined by using single-fidelity performance results, which means the data are fixed for low-fidelity cases when an initial convergence occurs. Then, the number of low-fidelity runs are kept constant and high-fidelity runs are increased to a level when an acceptable convergence is reached by the multi-fidelity surrogate.
4. Numerical Results
4.1. Analytical Test Cases
4.1.1. Problem 1: Park (1991) Function (4-D)
4.1.2. Problem 2: Borehole Function (8-D)
4.1.3. Problem 3: Sobol’-Levitan (1999) Function (30-D)
4.2. Application for Real-World Engineering Problems
4.2.1. Problem 1: Sonic Boom Uncertainty Quantification
4.2.2. Problem 2: Sonic Boom Uncertainty Quantification
5. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Model Name | Model Shape |
---|---|
Polynomial Chaos Expansion | |
Low-Rank Approximation |
Variable | Statistics |
---|---|
r | |
L | |
Loudness (dB) | |||
---|---|---|---|
SU2 | 0.0770 | 0.0069 | 80.0247 |
PANAIR | 0.0772 | 0.0064 | 87.0811 |
Error (%) | 6.1656 | 7.5190 | 10.0635 |
Variable | Statistics |
---|---|
Angle of attack, () | |
Mach | |
Reflection factor | |
Ground elevation, (ft) |
Mean (dB) | Standard Dev. (dB) | Skewness | Kurtosis | |
---|---|---|---|---|
Reference | 83.0656 | 0.8435 | −0.4968 | 3.3274 |
PCE | 82.9029 | 0.8354 | 0.0134 | 2.9538 |
LRA | 83.0479 | 0.8493 | −0.4080 | 3.2467 |
MF-PCE | 83.0426 | 0.8060 | −0.4811 | 4.3893 |
MF-LRA | 83.0621 | 0.8478 | −0.5038 | 3.3242 |
Variable | Statistics | Variable | Statistics |
---|---|---|---|
(m) | |||
(m) | (11,000, 1000) | ||
(m) | (20,000, 1000) | ||
(oF) | (m) | ||
(oF) | (−69.7,5) | (m) | |
(oF) | (−69.7,5) | (m) | (20,000, 1000) |
(m) | (m/s) | ||
(m) | (m/s) | ||
(m) | (m/s) | ||
(m) | (m) | ||
(m) | (10,060, 1000) | (m) | |
(m) | (13,720, 1000) | (m) | (20,000, 1000) |
(m/s) | |||
(m/s) | |||
(m/s) | |||
Temperature Profile | : Altitude | : Temperature | |
Humidity Profile | : Altitude | : Relative Humidity | |
X-Wind Profile | : Altitude | : X-wind | |
Y-Wind Profile | : Altitude | Y-wind |
Variable | Statistics |
---|---|
Ground altitude () | |
Reflection factor |
Mean (dB) | Standard Dev. (dB) | Skewness | Kurtosis | |
---|---|---|---|---|
MC | 83.5511 | 0.5062 | −0.1590 | 3.1699 |
PCE | 83.5649 | 0.7311 | −0.0565 | 3.8712 |
LRA | 83.5486 | 0.5000 | −0.0357 | 2.8596 |
MF-PCE | 28.9314 | 40.087 | −0.6869 | 5.0330 |
MF-LRA | 83.5507 | 0.5010 | −0.0710 | 2.9130 |
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Yildiz, S.; Pehlivan Solak, H.; Nikbay, M. Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design. Algorithms 2022, 15, 250. https://doi.org/10.3390/a15070250
Yildiz S, Pehlivan Solak H, Nikbay M. Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design. Algorithms. 2022; 15(7):250. https://doi.org/10.3390/a15070250
Chicago/Turabian StyleYildiz, Sihmehmet, Hayriye Pehlivan Solak, and Melike Nikbay. 2022. "Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design" Algorithms 15, no. 7: 250. https://doi.org/10.3390/a15070250