Research on Inverse Problems for Differential Equations

The general nature of an inverse problem is to deduce a cause from an effect. Consider a physical system, depending on a collection of parameters, in which one can speak of inputs to the system and outputs from the system. If all of the parameters were known perfectly then for a given input we could predict the output. It may happen, however, that some of the parameters characterizing the system are not known, being inaccessible to direct measurement. If it is important to know what these parameters are, in order that the system be understood as completely as possible, we might try to infer them by observing the outputs from the system corresponding to special inputs. Thus we seek the cause (the system parameters) given the effect (the output of the system for a given input).

An important example is the inverse problem of geophysics, in which we seek to investigate the structure of the interior of the earth. Elastic waves may propagate through the earth in a manner which depends on the material properties of the earth. A concentrated source of energy at the surface of causes waves to penetrate into the earth which are then partially reflected back to the surface. If the material properties of the earth's interior were known completely then we could predict the nature of the reflected wave from knowledge of the source. Since in fact we cannot measure these properties directly we seek to infer them by observing the reflected waves in response to a collection of known sources.

In formulating such problems mathematically, we typically find that the problem amounts to that of determining one or more coefficients in a differential equation, or system of differential equations, given partial knowledge of certain special solutions of the equation(s). In the seismology problem just discussed, the propagation of waves in the earth is governed by the equations of elasticity, a system of partial differential equations in which the material properties of the earth manifest themselves as coefficient functions in the equations. The measurements we can make amount to the knowledge of special solutions of the equations at special points, e.g. those points on the surface of the earth in this example.

Inverse problems in differential equations have this general character. One has a certain definite kind of differential equation (or system of equations) containing one or more unknown (or partially known) coefficient functions. From some limited knowledge about certain special solutions of the equations we seek to determine the unknown coefficient functions. Problems of this type arise in a variety of important applications areas, such as geophysics, optics, quantum mechanics, astronomy, medical imaging and materials testing.

Research activities of P. Sacks on inverse problems

• Inverse problems for acoustic media in N-dimensional space. (Items 9,10,16,21,25,27,28,37,44,50,51 in publication list) The governing differential equation here is the acoustic wave equation, a single hyperbolic equation. This work includes uniqueness theorems and analysis of various operators relating coefficients to data.
• Inverse problems for layered acoustic media. (Items 14,20, 43) Here it is assumed that the material properties characterizing the acoustic medium vary in only one direction.
• Inverse problems for layered elastic media. (Items 11,13,15,29,49) The governing equations are the equations of linear elasticity, a hyperbolic system. In comparison to the acoustic case, there is more than one wave speed. Current projects include extension of this work to some cases of anisotropic elasticity.
• Computational methods for classical inverse scattering problem of quantum mechanics and the inverse Sturm-Liouville problem. (Items 17,32,33,35,36,40,54,55,56)  `Time domain' methods are applied to develop new numerical methods for `frequency domain' problems.
• Phase identification problems. (Items 34,38,41,42,45,46,48,52,53) This is an auxiliary problem arising in a number of inverse scattering problems: The scattering data is naturally described in terms of a complex valued function, but only the amplitude is available for measurement. Thus we need to infer phase information in some indirect way, and/or analyze to what extent the inverse problem can be solved using only the amplitude information.

Inverse problems at Iowa State University

Research of a more applied nature on inverse problems may be found in several other places at ISU. See for example