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Suppose that I have a function of, say, three parameters, $f(p_1,p_2,p_3)$ whose output is a field(s) (e.g. velocity field) and is dependent on some real-valued parameters (e.g. viscosity, density, temperature). This function $f$ is computed by some pde solver and is expensive to compute and store. I want to compute $f$ over a 3D grid of parameters $p_1,p_2,p_3$. Each parameter lies between some minimum and maximum value. How would I sample this parameter space with a minimum number of points? Can someone point me to the important papers dealing with this problem? Design of experiments and Latin Hypercube Sampling come to mind, but I don't know what else to look for.

Edit: I'm trying to build reduced order models. And I want to build ROMs at the sampled points and then interpolate in some way (which is not yet decided). I have 8-10 parameters i.e. $(p_1, p_2 ...,p_{10})$ and I'd like to minimize the number of points at which I sample. To me, what I'm asking for sounds like trying to know the answer before solving the problem (one can't reliably know which parameter is important until one samples exhaustively and looks at the fields of interest computed), but I thought I'd still ask.

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  • $\begingroup$ Can you say something more about what your goal of sampling is? From the title it seems like you are just interested in knowing how "important" parameters are, is there any specific criteria you have for importance or what you will do once you know what parameters are important? $\endgroup$
    – user9794
    Sep 8, 2023 at 9:35
  • $\begingroup$ Is it possible to reduce your output quantity of interest to a scalar or small set of scalars; e.g., integrated quantities or steady-state values? Most methods of global sensitivity analysis are designed for scalar outputs and the sensitivity analysis of infinite-dimensional quantities of interest is a topic of active research. $\endgroup$
    – whpowell96
    Sep 8, 2023 at 15:46

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Optimal sampling will in general depend on one's objective. From the title it seems you are mainly interested in variable importance. This is typically one of the objectives of sensitivity analysis, and a variety of methods exist for which the ideal sampling methods may differ.

The book by Saltelli[1] covers sampling from the perspective of factorial design of experiments and Latin Hyper Cube sampling, however quasi-random sequences (Sobol, Halton, Hammersley, etc ) are a popular alternative for sampling based methods.

The number of samples will also depend on what you want to quantify and to what accuracy.

If you have many inputs of interest screening approaches such as Morris' method [2] are a good approach

However, if you already only care about 3 parameters screen may not be advantageous over a full variance based sensitivity analysis (Saltelli's book also covers this).

There are a number of approaches to this which can be divided in two approaches

  1. Meta-model/surrogate modelling approach where a simplified model for which the variance computation is relatively trivial (linear[1], polynomial [3-5], Fourier[1,6]) is used as an approximation of the model of interest.
  2. Direct computation of variance based sensitivities by integrating the model of interest over the input parameter space. The most popular approach for this is Monte Carlo, but direct numerical quadrature might be feasible in some situations.

For both of these methods the simplest procedures specify a sampling scheme (or quadrature grid) a priori, however, the resulting sensitivity estimates are approximate, so it is good practice to see if the results are consistent either between different sample sizes or by bootstrapping the estimation procedure to quantify how variable the estimates are.

Alternatively some adaptive sampling approaches exist, particularly for meta-modelling approaches [4], and try to choose new sample points based on error estimates or other criteria.

  1. Saltelli, A. (2008). Global sensitivity analysis: The primer. John Wiley,
  2. Morris, M. D. (1991). Factorial Sampling Plans for Preliminary Computational Experiments. Technometrics, 33(2), 161–174. https://www.tandfonline.com/doi/abs/10.1080/00401706.1991.10484804
  3. Buzzard, G. T., & Xiu, D. (2011). Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature. Communications in Computational Physics, 9(3), 542–567. https://doi.org/10.4208/cicp.230909.160310s
  4. Blatman, G., & Sudret, B. (2010). An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics, 25(2), 183–197. https://doi.org/10.1016/j.probengmech.2009.10.003
  5. Crestaux, T., Le Maitre, O., & Martinez, J.-M. (2009). Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 94(7), 1161–1172. https://doi.org/10.1016/j.ress.2008.10.008
  6. Xu, C., & Gertner, G. (2011). Understanding and comparisons of different sampling approaches for the Fourier Amplitudes Sensitivity Test (FAST). Computational Statistics & Data Analysis, 55(1), 184–198. https://doi.org/10.1016/j.csda.2010.06.028
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  • $\begingroup$ Morris screening is an effective method for problems this large, but it is worth noting that most of the literature uses step sizes that are way too large to be useful (Morris 1991 recommends $\Delta > 1/2$). Most contemporary uses take small step sizes to get a true finite-difference approximation and use quasi-random sampling for the seed values to get faster convergence. As you are computing these samples, you can perform dimensionality reduction via active subspaces to reduce the number of sampling dimensions further. $\endgroup$
    – whpowell96
    Sep 8, 2023 at 13:48

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