Patent ID: 8126648

Claim:
In the problem of inverting a matrix equation that can be expressed in the form [ G ] ⁡ [ m 1 m 2 ⋮ m N ] = [ data ] , where m 1 . . . m N are physical parameters to be solved for and G is a matrix based on a model of a physical system that relates the m i to measured data, expressed as components of vector [data] in the equation, wherein the equation may be non-uniquely inverted by numerical methods yielding an infinite number of possible solutions all of which fit the data substantially equally well and from which a most likely solution can be determined, a computer implemented method for determining the largest and smallest of said possible solutions, said method comprising: (a) finding orthonormal basis vectors that diagonalize the G matrix, and using said vectors to diagonalize G; (b) selecting a threshold below which the values of the elements of the diagonalized G are considered insignificant in terms of their effect on the most likely solution to said matrix equation; (c) identifying from the orthonormal basis vectors those vectors (the “null” vectors) associated with the insignificant diagonal elements; (d) choosing an L P mathematical norm where pε[0, ∞]; (e) determining an upper and lower bound for possible solutions m 1 , m 2 . . . m N by summing corresponding components of said null vectors according to the chosen norm, said lower bound being given by the negative of said sums; (f) finding a linear combination of the null vectors that most closely approaches said upper bound, and repeating this finding for said lower bound; and (g) adding each of said two linear combinations in turn to the most likely solution to yield a largest and a smallest solution, scaling as needed to eliminate any physically unreal results; wherein at least one of (a), (c), (e), and (f) is performed using a computer.