Update of the Low-Rank Response Surface via Computing the Residual
In real-world applications, the deterministic solver is very complicated and it is difficult or even impossible to change it, but one can often print out the norm of the residual. Assume that we already approximated the unknown solution by a response surface. Our response surface is approximation via multivariate Hermite polynomials like in Eq. 14, where coefficients are computed like in Eq. 13 with quadrature points 9i, i = 1..nq. The following algorithm updates the given response surface. Algorithm: (Update of the response surface)
1. Take the next point 9nq+1 and evaluate the response surface Eq. 14 in this point. Let u(x, 9nq+i) be the obtained predicted solution.
2. Compute the norm of the residual ||r|| of the deterministic problem (e. g. evaluate one iteration). If ||r|| is small then there is no need to solve the expensive deterministic problem in 9nq+i, otherwise (if ||r|| is large) solve the deterministic problem and recompute A, Br and cp in Eq. 12.
3. Go to item (1).
In the best case we never solve the deterministic problem again. In the worst case we must solve the deterministic problem for each 9nq+i, i = 1,2,… To test this algorithm we computed the solution in Case 1 with 10000 TAU iterations (is usual number of iterations). Then, first, we computed the solution with the response surface (as described above) and, second, corrected it with 1000 TAU iterations. Then we compared both solutions and observed only a very small difference. Thus, the response surface reduced the number of needed iterations from 10000 to 1000. We note that the solution in Case 1 is smooth and there is no shock.
We tested this Algorithm also in the Case 9 (solution with a shock) and it failed. We pre-computed the solution by two different response surfaces (of order P = 2 and P = 4). Both response surfaces failed to produce a good result. For instance, we observed not only one shock, but many smaller shocks. Then we observed an increasing range of e. g. pressure (range (—6;5) in contrast to (0.5,1.3)). It is similar when one tries to approximate a step function by a polynomial — the amplitude of oscillations grows up. Another negative effect which we observed during further iterating the solution, obtained from the response surface, was that the deterministic solver (TAU) produces “nan” after few iterations. A possible reason is that some important solution values, obtained from the response surface, are out of the physical range (e. g. negative density) and are non-realistic.
Thus, we can come to the conclusion that if the solution is smooth (e. g. as in Case 1) then response surface produces a good starting value. Otherwise, if the solution has a shock, the response surface produces a very poor approximation and further iterations do not help.
The computed solution u(x, 9nq+1) can be used to update the response surface, i. e. to recompute the matrices A, B and […c^…] and PCE coefficients (Eq. 13). Please note that this update works only in the case of the usage of embedded sparse grids or (Q)MC in Eq. 11.