We investigate the inverse problem of identifying a piecewise constant anisotropic conductivity A(x) in the elliptic equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-\textrm{div}(A\nabla u)=f \text{ in } \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u=g \text{ on } \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation>, from multiple measurements of the internal current density <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\vec {h}=A\nabla u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>h</mi> <mo stretchy="false">→</mo> </mover> <mo>=</mo> <mi>A</mi> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. Owing to its ill-posedness, we suggest minimizing the output least-squares functional with a total variation penalty for regularization. The well-posedness of this regularized formulation is analyzed, including the existence of minimizers, the optimality system, and its consistency to the original problem. To discretize the continuous optimization problem, we employ the standard finite element method and establish the convergence of the finite element approximations as the mesh size approaches zero. Computationally, we adopt the iteratively reweighted least-squares strategy together with the projected Newton method to solve the discretized problem and present several two-dimensional numerical examples to demonstrate its efficiency.

Imaging piecewise constant anisotropic conductivities with total variation regularization

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发表日期：

2026年1月5日

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Huan Liu,
Xiliang Lu,
Taige Wang

摘要

We investigate the inverse problem of identifying a piecewise constant anisotropic conductivity A(x) in the elliptic equation $-\textrm{div}(A\nabla u)=f \text{ in } \Omega $ $- div (A \nabla u) = f in Ω$ , $u=g \text{ on } \Gamma $ $u = g on Γ$ , from multiple measurements of the internal current density $\vec {h}=A\nabla u$ $\vec{h} = A \nabla u$ . Owing to its ill-posedness, we suggest minimizing the output least-squares functional with a total variation penalty for regularization. The well-posedness of this regularized formulation is analyzed, including the existence of minimizers, the optimality system, and its consistency to the original problem. To discretize the continuous optimization problem, we employ the standard finite element method and establish the convergence of the finite element approximations as the mesh size approaches zero. Computationally, we adopt the iteratively reweighted least-squares strategy together with the projected Newton method to solve the discretized problem and present several two-dimensional numerical examples to demonstrate its efficiency.