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Inverse Problems

are best understood by comparing them to the corresponding direct (or forward) problems. In direct problems we have an input which we run through some known process to get an output. We know what the input is and we also know what the process is. From this information we need to produce the output. For inverse problems we reverse the order. Now we have given an output, we have some understanding about the process, and possibly are given a priori information about the input. Our task is to find an input which resulted in this output.

One might ask if the inverse problem is equivalent to a direct problem if we know the process exactly, so that we could then simply run the process backwards. Unfortunately this is rarely possible since most processes cannot be reversed or they are very sensitive for measurement errors. As a simple example one could consider the process which takes two numbers X and Y, and gives the sum X+Y. But if we are given some number Z and are asked to determine which X and Y sum to this Z, we would quickly find that there are many possible solutions if do not pose any restrictions on the allowed inputs.

An other example is to find missing integers, of the grid given here, such that the sum of each row and column is the integer given in the picture. This toy problem is closely connected to medical imaging modality known the X-ray tomography. Each tissue type attenuates the X-ray beams differently and in principle the intensity loss of a beam, traveling trough a body, is the sum of attenuation coefficients multiplied by the depth of corresponding tissue. Thus to figure out the structure of the body we can try to solve an inverse problem of X-ray tomography, which is to recover the attenuation coefficient at each location, from the total intensity loss of X-ray beams traveling through the body.

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Geometric inverse problems in seismology

Earthquakes produce seismic waves which are strong enough to travel through the Earth. If a dense enough network of seismic sensors is deployed on the surface of the Earth and we measure a large number of arrivals of seismic waves, we can hope to recover the wave speed inside the planet. This is the inverse problem of ‘seeing inside the Earth’. The wave speed depends on the material through which the wave travels. In this way recovering the wave speed from earthquake based measurements provides us indirect information about the deep structures of our planet. Almost all of our knowledge of the deep Earth is obtained by using such data.

By Fermat’s principle a seismic wave takes a path that locally minimizes the travel time. For this reason in the mathematical theory of indirect measurements, it is common to model the Earth as a compact, connected Riemannian manifold with boundary. A classical geometric inverse problem is to determine the wave speed inside an object when the travel times between the boundary points are known. This is the mathematical analog of the aforementioned geophysical problem.