The purpose of this series of entries on null spaces is orthogonal to these sets of questions. We want to show that there are in fact very many instances of linear systems of equations in engineering and physics where one has to make that choice of constraining these systems to get a solution we fancy. For the past 200 years, that fancy solution was the one which reduced noise. That means that all the engineering curriculum revolve around solving problems with energy regularizers. We need to expand and show that there is a new degree of freedom in town provided by the null space: compressive sensing focuses on the sparse or compressible ones, what other regularization will we fancy next and what type of problems can it be applied to. These entries want to answer the latter point.
Right now this is the first entry, but there should be more. If you have an idea of a specific application which may or may not have seen investigations of compressive sensing (CS is the canary in the mine as far as I am concerned) I'll be glad to feature it on the blog (with your name and the attendant ...errrrr... windfall that goes with it :)
Case 1, the null space of the gravitational map as explained in this paper: On the Gravitational Inverse Problem by Ulvi Yurtsever, Caren Marzban, Marina Meilˇa. The abstract reads:
We discuss some mathematical aspects of the problem of inverting gravitational ﬁeld data to extract the underlying mass distribution. While the forward problem of computing the gravity ﬁeld from a given mass distribution is mathematically straightforward, the inverse of this forward map has some interesting features that make inversion a diﬃcult problem. In particular, the forward map has an inﬁnite-dimensional kernel which makes the inversion fundamentally non-unique. We characterize completely the kernels of two gravitational forward maps, one map 2840 U. Yurtsever, C. Marzban and M. Meilˇa ping mass density to the Newtonian scalar potential, and the other mapping mass density to the gravity gradient tensor, which is the quantity most commonly measured in ﬁeld observations. In addition, we present some results on unique inversion under constrained conditions, and comment on the roles the kernel of the forward map and non-uniqueness play in discretized approaches to the continuum inverse problem
I am sure I am not the first one to think that with theorem 7, earthquakes are good way of probing the whole planet's interior.
Credit photo: NASA
Credit photo: NASA
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