McCann, Hugh

Ouypornkochagorn, Taweechai

Polydorides, Nicholas

We propose a methodology for speeding up thecomputation of a nonlinear inverse solution, by approximatingefficiently the Jacobian of the forward model. Thishas practical advantages not only in linearising the inverseproblem but also in computing the regularised solution

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     Edinburgh Research Explorer                                      An approximate gradient for fast nonlinear inversion of the EITmodelCitation for published version:Polydorides, N, Ouypornkochagorn, T & McCann, H 2015, An approximate gradient for fast nonlinearinversion of the EIT model. in 16th International Conference on Biomedical Applications of ElectricalImpedance Tomography.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:16th International Conference on Biomedical Applications of Electrical Impedance TomographyGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019An approximate gradient for fast nonlinear inversion of the EIT modelNick Polydorides1, Taweechai Ouypornkochagorn1, Hugh McCann11School of Engineering, University of Edinburgh, UK. n.polydorides@ed.ac.ukAbstract: We propose a methodology for speeding up thecomputation of a nonlinear inverse solution, by approxim-ating efficiently the Jacobian of the forward model. Thishas practical advantages not only in linearising the inverseproblem but also in computing the regularised solution.1 IntroductionWe consider the inverse problem of estimating a conduct-ivity function σ ∈ X from a finite set of electrical potentialmeasurements y ∈ℜm from the nonlinear modely= F(σ)+n, (1)where n is additive random noise, F : X→ℜm is an analyticforward mapping, with continuous first and second weakderivatives and X is the space of bounded positive functionsspanning over a closed domain Ω. For a choice of regular-isation term R : X →ℜ, the inverse problem is to estimateσˆ = argminσ∈X{‖F(σ)− y‖2 +R(σ)}. (2)Using a gradient-based algorithm, such as the Gauss-Newton, a local minimum can be traced iteratively forn= 0,1, . . .σn+1 = σn+hn (3)hn = argmin{‖DFσn(h)−δyn‖2 +R(h)}, (4)where DFσn is the Fréchet derivative of F at a feasibleσn ∈ X , and δyn = yn − F(σn). The need for fast im-age reconstruction in EIT has been addressed in the past,and methods like single-step inexact Newton-based regular-ised reconstructions [1], [2] have been popular in this con-text, and to a lesser extend Newton-Kantorovich algorithmsBroyden methods [3].2 Approximate gradient methodLet |Ω| denote the volume of the domain to be imaged, thenconsider the weighted mean conductivity valueσ¯ =1|Ω|∫Ωdxσ(x)w(x), where∫Ωdxw(x) = 1. (5)As any σ ∈ X admits a decomposition σ = σ¯ + p, then weclaim thatDFσ (h)≈ DFσ¯ (h) = 1σ¯2DF1(h), (6)where 1(x) is the unit conductivity function in Ω. Optionsfor the weights may includew(x) ∝ 1(x) or w(x) ∝d∑i=1∣∣σ(x)∇ui(x)∣∣, (7)with ui(x) = F(σ) denoting the forward solution for the ithboundary current. By the analyticity of F we haveDFσ = DFσ¯ +12D2Fσ¯ p+o(p2), (8)and thus to first-order accuracy‖DFσ −DFσ¯‖ ≤ 12‖D2Fσ¯‖‖p‖, (9)and we can replace DFσn in the Gauss-Newton iterationwith DFσ¯n , such that at each iteration we obtain the ap-proximate Jacobian by scaling accordingly a precomputedJacobian DF1. To see the computational advantage of thisscheme let J1 = ∑mi=1 uisiv′i the Jacobian matrix at discretemodel with k > m elements and unit conductivity, andR(h) = λ‖h‖2, then the iterative solution is given byσn+1 = σn+m∑i=1γni vi, for γni = 〈ui,δyn〉( siσ¯2ns2i +λσ¯4n)(10)3 SimulationsWe implement the approximate gradient scheme using aregularised Gauss-Newton algorithm and compare conver-gence and timings against the respective implementationinvolving the exact Jacobian. These tests were based onsimulations performed with EIDORS 3D [3] using a modelwith 8640 elements and 208 measurements, with a targetedoscillatory conductivity function 0.01 ≤ σ(x) ≤ 110. Thetimings for 6 iterations on a macbook pro running Matlabwere 114 s and 8 s for the exact and approximate schemesrespectively, while the errors in data and image are thoseappearing in fig. 1.iteration index1 2 3 4 5 6 7kF(<i)!VkkVk10 -210 -1100101Exact GNApproximate GNiteration index1 2 3 4 5 6 7k<i!<$kk<$ k10 -210 -1100Exact GNApproximate GNFigure 1: Numerical comparison of the Gauss-Newton methodwith exact and approximate gradients using w(x) ∝ 1(x).4 ConclusionsThe numerical results suggest that the compromise in theconvergence of the nonlinear solver due to the approximateJacobian is very small compared to the profound computa-tional savings.References[1] Lechleiter A., Rieder A., Newton regularizations for impedance tomo-graphy: convergence by local injectivity, Inverse Problems, 24, 065009,2008.[2] Cheney M., Isaacson D., Newell J. C., Goble J., Simske S., Noser:an algorithm for solving the inverse conductivity problem. Internat. J.Imaging Systems and Technology, 2, 66-75, 1990.[3] Holder D. S. (ed) Electrical Impedance Tomography: Methods, his-tory and applications, IoP, 2002.

An approximate gradient for fast nonlinear inversion of the EIT model

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An approximate gradient for fast nonlinear inversion of the EIT model

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