Resolution measures the degree by which model parameters can be independently determined (from each other), while model covariance measures the degree by which errors in the data propagate into uncertainty in the model parameters. -- Sambridge and Mosegaard, Reviews of Geophysics, 40, 3, 2002.
A functional is a rule that unambiguously assigns a single real number to an elelment in a real linear vector space V. The subset of V for wihch values are defined is the domain of the functional. The subset of reals where results may line is termed the range. If the rule assigns something other than a real scalar to a given element of the sapce, then the terms mapping, operator, or transformation are used.
A Hilbert space is an inner product space that is complete under the norm < f,f >^{½}.
Complete normed linear vector spaces are called Banach space.
In applied mathematics, data kernels are called representers.
For boundary value problems, consider using Fourier transforms to solve the problem. Remember that the integral is evaluated from -infinity to +infinity.
For initial value problems, consider using Laplace transforms to solve the problem. In this case, the integral is evaluated from 0 to +infinity. Because the integration starts at 0, the Laplace transform is good for problems like time decay, for example.
A good reference is Sen and Stoffa's book Global Optimization Methods in Geophysical Inversion, 1985.
Genetic Algorithms are a good choice for non-linear problems in which many local minima exist. The mutation step allows for large jumps in the model space. However, the cost is a lot of starting models (the population) requiring lots of memory.
Simulated Annealing is a good choice for non-linear problems in which a few local minima exist or in which the minima are relatively small. SA allows for small jumps in the model space, unlike GA. SA is not as good at searching the model space as is GA. The advantage over GA is that you start with one model, then find the global minimum.
From the May 2, 2006 New York Times article about Henry Petroski of Duke University (page D3):
"For Dr. Petroski, acceptance of uncertainty and possible failure - he calls it "coping with the imponderable" - is what separates the "given world" of the scientist from the "built world" of the engineer." My father commmented that "the enineer is constantly worried about failure in his world and work, whereas the scientist's world never fails, although his work might."
Last updated on December 22, 2006 by William P. Clement.