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On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation

  • * Corresponding author: William Rundell

    * Corresponding author: William Rundell
The first author is supported by FWF grant P30054; the second author is supported by NFS grant DMS-1620138
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  • This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $ a(x) $ and the nonlinear reaction term $ f(u) $ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $ T $; time-trace profiles from two solutions; or both final time and time-trace measurements from a single forwards solve data run. We prove both uniqueness results and the convergence of iteration schemes designed to recover these coefficients. The last section of the paper shows numerical reconstructions based on these algorithms.

    Mathematics Subject Classification: Primary: 35R30; 65M32; Secondary: 35R11.


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  • Figure 1.  Logarithms of the singular values for $ a(x) $ and $ q(x) $ recovery from time trace data using $ F:\delta c \to u(1,t) $ with $ \{\delta c = \sin(n\pi x)\} $

    Figure 2.  Recovery of $ a(x) $ and $ f(u) $ from final data

    Figure 3.  Recovery of $ \,a(x), $ and $ \,f(u)\, $ from space-and-time data

    Figure 4.  Recovery of a(x); and f(u) from space-and-time data: 1% noise

    Figure 5.  Recovery of $ \,a(x)\, $ and $ \,f(u)\, $ from space-and-time data when $ u_0\not = 0 $

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