The basic formulation of a geophysical estimation problem consists of setting up two goals, one for data fitting, and the other for model shaping. With two goals, preconditioning is somewhat different. The two goals may be written as:
(4) | ||
(5) |
To fix ideas, let us examine a toy example. The data and the first three rows of the matrix below are random numbers truncated to integers. The model roughening operator is a first differencing operator times 100.
d(m) F(m,n) iter Norm --- ------------------------------------------------ ---- ----------- 41. -55. -90. -24. -13. -73. 61. -27. -19. 23. -55. 1 20.00396538 33. 8. -86. 72. 87. -41. -3. -29. 29. -66. 50. 2 12.14780140 -58. 84. -49. 80. 44. -52. -51. 8. 86. 77. 50. 3 8.94393635 0. 100. 0. 0. 0. 0. 0. 0. 0. 0. 0. 4 6.04517126 0. -100. 100. 0. 0. 0. 0. 0. 0. 0. 0. 5 2.64737511 0. 0.-100. 100. 0. 0. 0. 0. 0. 0. 0. 6 0.79238468 0. 0. 0.-100. 100. 0. 0. 0. 0. 0. 0. 7 0.46083349 0. 0. 0. 0.-100. 100. 0. 0. 0. 0. 0. 8 0.08301232 0. 0. 0. 0. 0.-100. 100. 0. 0. 0. 0. 9 0.00542009 0. 0. 0. 0. 0. 0.-100. 100. 0. 0. 0. 10 0.00000565 0. 0. 0. 0. 0. 0. 0.-100. 100. 0. 0. 11 0.00000026 0. 0. 0. 0. 0. 0. 0. 0.-100. 100. 0. 12 0.00000012 0. 0. 0. 0. 0. 0. 0. 0. 0.-100. 100. 13 0.00000000
Notice at the tenth iteration,
the residual suddenly plunges 4 significant digits.
Since there are ten unknowns and the matrix is obviously full-rank,
conjugate-gradient theory tells us to expect
the exact solution at the tenth iteration.
This is the first miracle of conjugate gradients.
(The residual actually does not drop to zero.
What is printed in the Norm column
is the square root of the sum of the squares
of the residual components at the iter-th
iteration minus that at the last interation.)