. . Adaptive and stochastic algorithms for piecewise constant EIT and DC resistivity problems with many measurements . . . . . Uri Ascher Department of Computer Science University of British Columbia October 2011 . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 1 / 44
Motivation EIT on wikipedia . From the horse’s mouth (wikipedia) . Electrical impedance tomography (EIT) is a medical imaging technique in which an image of the conductivity or permittivity of part of the body is inferred from surface electrical measurements. Typically, conducting electrodes are attached to the skin of the subject and small alternating currents are applied to some or all of the electrodes. The resulting electrical potentials are measured, and the process may be repeated for numerous different configurations of applied current . Proposed applications include monitoring of lung function, detection of cancer in the skin and breast and location of epileptic foci [...] In 2011 the first commercial EIT device for lung function monitoring in intensive care patients was introduced. [...] . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 2 / 44
Motivation EIT on wikipedia . From the horse’s mouth (wikipedia) . Electrical impedance tomography (EIT) is a medical imaging technique in which an image of the conductivity or permittivity of part of the body is inferred from surface electrical measurements. Typically, conducting electrodes are attached to the skin of the subject and small alternating currents are applied to some or all of the electrodes. The resulting electrical potentials are measured, and the process may be repeated for numerous different configurations of applied current . Proposed applications include monitoring of lung function, detection of cancer in the skin and breast and location of epileptic foci [...] In 2011 the first commercial EIT device for lung function monitoring in intensive care patients was introduced. [...] . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 2 / 44
Motivation EIT on wikipedia . Lung imaging . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 3 / 44
Motivation EIT on wikipedia . From the horse’s mouth cont. . Mathematically [...] non-linear inverse problem and is severely ill-posed. The mathematical formulation of the problem is due to [Calderon, 1980] There is extensive mathematical research on the problem of uniqueness of solution and numerical algorithms for this problem [Uhlmann, 1999] . In geophysics a similar technique (called electrical resistivity tomography ) is used using electrodes on the surface of the earth or in bore holes to locate resistivity anomalies, and in industrial process monitoring the arrays of electrodes are used for example to monitor mixtures of conductive fluids in vessels or pipes. [...] broadly similar to the medical case. In geophysics, the idea dates from the 1930s. . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 4 / 44
Motivation EIT on wikipedia . From the horse’s mouth cont. . Mathematically [...] non-linear inverse problem and is severely ill-posed. The mathematical formulation of the problem is due to [Calderon, 1980] There is extensive mathematical research on the problem of uniqueness of solution and numerical algorithms for this problem [Uhlmann, 1999] . In geophysics a similar technique (called electrical resistivity tomography ) is used using electrodes on the surface of the earth or in bore holes to locate resistivity anomalies, and in industrial process monitoring the arrays of electrodes are used for example to monitor mixtures of conductive fluids in vessels or pipes. [...] broadly similar to the medical case. In geophysics, the idea dates from the 1930s. . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 4 / 44
Motivation EIT, DC resistivity, and many measurements . Inverse problem setup . Goal : Recover distributed parameter function m ( x ) given forward operator F ( m ) data b s.t. b = F ( m ) + ϵ ϵ - noise Sensitivity matrix J = ∂ F ∂ m . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 5 / 44
Motivation EIT, DC resistivity, and many measurements . Inverse problem setup . Goal : Recover distributed parameter function m ( x ) given forward operator F ( m ) data b s.t. b = F ( m ) + ϵ ϵ - noise Sensitivity matrix J = ∂ F ∂ m . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 5 / 44
Motivation EIT, DC resistivity, and many measurements . Forward operator . F ( m ) = Qu Field u solves discretized elliptic PDE Au = q , with A depending on m . Multi-experiment version: F i ( m ) = Q i u i , i = 1 , . . . , s data b i . Thus, F = ( F 1 , . . . , F s ), b = ( b 1 , . . . , b s ). Field u i solves discretized elliptic PDE A ( m ) u i = q i . Note same A for different experimental settings i . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 6 / 44
Motivation EIT, DC resistivity, and many measurements . Forward operator . F ( m ) = Qu Field u solves discretized elliptic PDE Au = q , with A depending on m . Multi-experiment version: F i ( m ) = Q i u i , i = 1 , . . . , s data b i . Thus, F = ( F 1 , . . . , F s ), b = ( b 1 , . . . , b s ). Field u i solves discretized elliptic PDE A ( m ) u i = q i . Note same A for different experimental settings i . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 6 / 44
Motivation EIT, DC resistivity, and many measurements . EIT and DC resistivity problem setup . PDE with multiple sources ∇ · ( σ ( x ) ∇ u i ) q i , = i = 1 , . . . , s , ∂ u i ∂ν | ∂ Ω = 0 . Conductivity σ ( x ) is expressed as a pointwise function of m ( x ). The operator A ( m ) is the above PDE discretized on a staggered grid. Source q i and observation locations Q i correspond to different selection of sources and receivers. Need to solve s PDEs just to evaluate forward operator F : proceed with caution . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 7 / 44
Motivation EIT, DC resistivity, and many measurements . EIT and DC resistivity problem setup . PDE with multiple sources ∇ · ( σ ( x ) ∇ u i ) q i , = i = 1 , . . . , s , ∂ u i ∂ν | ∂ Ω = 0 . Conductivity σ ( x ) is expressed as a pointwise function of m ( x ). The operator A ( m ) is the above PDE discretized on a staggered grid. Source q i and observation locations Q i correspond to different selection of sources and receivers. Need to solve s PDEs just to evaluate forward operator F : proceed with caution . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 7 / 44
Motivation EIT, DC resistivity, and many measurements . EIT theory . If the entire Dirichlet-to-Neumann map (equivalently, the boundary data u | ∂ Ω expressed in terms of the source q for any q ( x ) restricted to ∂ Ω) is given then σ ( x ) can be uniquely recovered provided that it is at least differentiable on the domain Ω. Much thoeretical effort spent on reducing continuity requirements on σ [Astala & Paivarinta, 2006] . In practice there is noise, discretization of PDEs and discretization of D2N map. Still, theory correctly predicts that (i) things can go wrong when σ has discontinuities and that (ii) many experiments are useful . The problem is much stabilized if we may add a priori information: assume σ ( x ) can take only one of two (or four) values at each x . Thus, shape optimization. This is a stabilizing assumption [Alessandrini & Vessella, 2005] . ∇ · ( σ ( x ) ∇ u i ) q i , = i = 1 , . . . , s . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 8 / 44
Motivation EIT, DC resistivity, and many measurements . EIT theory . If the entire Dirichlet-to-Neumann map (equivalently, the boundary data u | ∂ Ω expressed in terms of the source q for any q ( x ) restricted to ∂ Ω) is given then σ ( x ) can be uniquely recovered provided that it is at least differentiable on the domain Ω. Much thoeretical effort spent on reducing continuity requirements on σ [Astala & Paivarinta, 2006] . In practice there is noise, discretization of PDEs and discretization of D2N map. Still, theory correctly predicts that (i) things can go wrong when σ has discontinuities and that (ii) many experiments are useful . The problem is much stabilized if we may add a priori information: assume σ ( x ) can take only one of two (or four) values at each x . Thus, shape optimization. This is a stabilizing assumption [Alessandrini & Vessella, 2005] . ∇ · ( σ ( x ) ∇ u i ) q i , = i = 1 , . . . , s . . . . . . . Kees van den Doel & (UBC) Adaptive stochastic algorithms October 2011 8 / 44
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