Optimization NLFFF Model J.McTiernan SSL/UCB HMI/SDO 27-Jan-2005

Optimization method (cont): The first step is to set up the initial conditions. The method requires that the B field on all boundaries be unchanged. For the test case of the Low-Lou field, the B field on all boundaries is known. For real data, only the lower boundary is known; the bottom boundary is the magnetogram, the upper and side boundaries are the initial field. The initial field can be a potential field or a linear force-free field that is extrapolated from the magnetogram. In the test case with all known boundaries, this results initially in discontinuities at the outer boundaries. These smooth out during iterations.

Optimization method (cont): Iterative process, starts with initial B field. Calculate F, set new B = B + F*dt (typical dt =1.0e-5) “Objective function”, L, is guaranteed to decrease, because dL/dt is guaranteed to be negative since dB/dt is proportional to F. The change in L becomes smaller as the number of iterations increases, L asymptotically approaches zero, but never quite gets there. Keep iterating until L stops decreasing, or until some preset iteration limit (currently set to ).

Optimization method (cont): For the test case, the quantity F, that drives the minimization, is non zero only on the boundaries. Differences between the calculated and initial field propagate into the volume as we iterate. For the case with only the lower boundary, F is only non-zero at the lower boundary, the difference between the calculated and initial fields propagates upwards from the lower boundary. The final extrapolation is dependent on the boundary conditions. If only the lower boundary condition is specified the solution also depends on the initial field.

Tests of NLFFF model: 4 different tests, based on the Low-Lou model discussed in previous presentation: 1 st test, 64x64x64 cube, all boundary fields are specified by LL model. On 2.4 GHz Linux machine, this took 4300 minutes, (max) iterations, and L was still noticeably decreasing at the end of the calculation. (Note that the code is written in IDL. Run time scales with the total number of grid points, e.g., a 32x32x32 cube will take 1/8 th the time.) A typical value for real data, for a 64x64x64 cube, is 200 minutes. This is because real data has noise.

Tests of NLFFF model: 2 nd Test, added random noise to the initial field; the maximum noise level on the lower boundary was set to be 2% of the max total field strength on that boundary, approximately 6 Gauss. (How realistic is this?) This “converged”-- the objective function, L, stopped decreasing after about 4140 iterations, or about 370 minutes. If the noise level is doubled, the iterations stop after 211 minutes.

Tests of NLFFF model: 3 rd Test, ignored the outer boundaries of the LL field, and set the outer boundaries to the initial potential field. This is still running, (as of 26-Jan pm) after iterations, and about 1790 minutes… 4 th test, same as 3 rd test, but with noise added to lower boundary as in 2 nd test. Converged after 8600 iterations and 500 minutes. (For some reason, the tests with only the lower BC’s are about 60% faster, per iteration, about 3 sec per iteration, instead of 5 sec per iteration for the tests with all BC’s specified.)

Tests of NLFFF model: We can conclude that noise matters, and the level of noise will restrict the eventual use of the field extrapolation, in the sense that the field will not converge to as force-free a state. Also the field lines can be different. This could be an issue when interpreting model results. Future work will include more realistic noise estimates and Monte Carlo tests for simulated and real magnetogram data, spherical coordinates version is being debugged, will have to redo in Fortran…