Practical considerations when using sparse grids with Bayesian inference for parameter estimation

Figures & data

Figure 1. Comparison of the pdf () for Problem 1.

Figure 2. Effect of ill-conditioning on Bayesian inference for Problem 1.

Figure 3. History of iterations using β = 0.01 for Problem 1 when ill-conditioned.

Table 1. Effect of β for σ = 1% (true values: k = 1.0,  h = 1.0, ρc = 20.0 Bi = 1.0, α = 0.05).

β	0.1	0.01	0.001
Bi	1.015	1.015	1.018
α	0.0494	0.0494	0.0493
N	14	14	10
k	1.480	1.408	0.0302
h	1.503	1.429	0.0308
ρc	29.96	28.49	0.6135
N	14	>200	Never

Figure 4. Solution trajectories for Problem 1 using exact solutions (solid line) and a sparse grid (dashed line) for two parameters of level 3 (29 sample points) for interpolation.

Figure 5. Schematic of the panel (thermocouples are placed at the numbered points).

Figure 6. Time history of panel temperatures (numbers denote the thermocouples shown in Figure 5).

Figure 7. Contours of L(Θ) showing the solution trajectories for Problem 2 using exact solutions of M(Θ) (–-) and interpolation by a sparse grid of level 3 (69 sample points) (- - -).

Figure 8. Level 1 and level 2 sparse grid points: (a) level 1 points (O) and (b) level 2 additional points (X).

Figure 9. Reduction in maximum errors of T and ∂T/∂k as a function of sparse grid level for Problem 1 (maximum errors evaluated at x/L = [0, 0.25, 0.5, 0.75, 1.0], Fo = [0 : 1 : 41] (‘spinterp’ results evaluated at x/L = 0.5, Fo = 21).

Figure 10. Interpolated values for sparse grids of increasing levels for Problem 1 using the Chebyshev grid (numbers denote the level): (a) T at x/L = 0.5, Fo = 21 and (b) ∂T/∂k at x/L = 0.5, Fo = 21.

Figure 11. Interpolated values for sparse grids of increasing levels for Problem 1 using the Clenshaw–Curtis grid (numbers denote the level): (a) T at x/L = 0.5, Fo = 21 and (b) ∂T/∂k at x/L = 0.5, Fo = 21.

Figure 12. Interpolation error for the temperature at x/L = 0.5, Fo = 21, Problem 1: (a) Clenshaw–Curtis grid and (b) Chebyshev grid. Interpolation error for the ∂T/∂k at x/L = 0.5, Fo = 21, Problem 1: (c) Clenshaw–Curtis grid and (d) Chebyshev grid.

Figure 13. Trajectory for different ways of determining the sensitivities: Exact, solving using the model; SG-a, finite differencing interpolated temperatures; SG-b, differentiating interpolating polynomials representing temperature; SG-c, interpolating sensitivities.

Figure 14. Effect of refining the range of parameters for in Problem 1 (ρc specified): (a) expanded range and (b) centred about LS solution using sparse grid interpolation.

Figure 15. Behaviour of the panel: (a) time history of sensitivity of temperature and (b) surface of L(h_t, h_b) for the panel at TC 9.

Figure 16. Joint probability densities for Problem 1: (a) well-conditioned and (b) ill-conditioned.