Abstract
We study inverse problems for the Poisson equation with source term the divergence of an \({{\mathbb {R}}}^3\)-valued measure, that is, the potential \(\varPhi \) satisfies
and \({{\varvec{\mu }}}\) is to be reconstructed knowing (a component of) the field \(\, {\mathrm{grad}}\,\varPhi \) on a set disjoint from the support of \({{\varvec{\mu }}}\). Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We investigate methods for recovering \({{\varvec{\mu }}}\) by penalizing the measure theoretic total variation norm \(\Vert {{\varvec{\mu }}}\Vert _{\mathrm{TV}}\). We provide sufficient conditions for the unique recovery of \({{\varvec{\mu }}}\), asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.
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Communicated by Endre Süli.
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Jordan–Brouwer Separation Theorem in the Non-Compact Case
Jordan–Brouwer Separation Theorem in the Non-Compact Case
In this section, we record a proof of the Jordan–Brouwer separation theorem for smooth and connected, complete but not necessarily compact surfaces in \({{\mathbb {R}}}^3\). The argument applies in any dimension. We are confident the result is known, but we could not find a published reference. More general proofs, valid for non-smooth manifolds as well, could be given using deeper facts from algebraic topology. For instance, one based on Alexander–Lefschetz duality can be modeled after Theorem 14.13 in www.seas.upenn.edu/~jean/sheaves-cohomology.pdf (which deals with compact topological manifolds). More precisely, using Alexander–Spanier cohomology with compact support and appealing to [43, ch. 6, sec. 6 cor. 12 and sec. 9 thm. 10], one can generalize the proof just mentioned to handle the case of non-compact manifolds. Hereafter, we merely deal with smooth surfaces and rely on basic notions from differential topology, namely intersection theory modulo 2.
Recall that a smooth manifold X of dimension k embedded in \({{\mathbb {R}}}^n\) is a subset of the latter, each point of which has a neighborhood V such that \(V\cap X=\phi (U)\) where U is an open subset of \({{\mathbb {R}}}^k\) and \(\phi :U\rightarrow {{\mathbb {R}}}^n\) a \(C^\infty \)-smooth injective map with injective derivative at every point. The map \(\phi \) is called a parametrization of V with domain U, and the image of its derivative \(D\phi (u)\) at u is the tangent space to X at \(\phi (u)\), hereafter denoted by \(T_{\phi (u)}X\). Then, by the constant rank theorem, there is an open set \(W\subset {{\mathbb {R}}}^n\) with \(W\cap X=V\) and a \(C^\infty \)-smooth map \(\psi :W\rightarrow U\) such that \(\psi \circ \phi =\text {id}\), the identity map of U. The restriction \(\psi _{|V}\) is called a chart with domain V. This allows one to carry over to X local tools from differential calculus, see [30, ch. 1]. We say that X is closed if it is a closed subset of \({{\mathbb {R}}}^n\).
If X, Y are smooth manifolds embedded in \({{\mathbb {R}}}^m\) and \({{\mathbb {R}}}^n\), respectively, and if \(Z\subset Y\) is a smooth embedded submanifold, a smooth map \(f:X\rightarrow Y\) is said to be transversal to Z if \(\text {Im} Df(x)+T_{f(x)}Z=T_{f(x)}Y\) at every \(x\in X\) such that \(f(x)\in Z\). If f is transversal to Z, then \(f^{-1}(Z)\) is an embedded submanifold of X whose codimension is the same as the codimension of Z in Y. In particular, if X is compact and \(\text {dim} X+\text {dim} Z= \text {dim} Y\), then \(f^{-1}(Z)\) consists of finitely many points. The residue class modulo 2 of the cardinality of such points is the intersection number of f with Z modulo 2, denoted by \(I_2(f,Z)\). If in addition Z is closed in Y, then \(I_2(f,Z)\) is invariant under small homotopic deformations of f, and this allows one to define \(I_2(f,Z)\) even when f is not transversal to Z, because a suitable but arbitrary small homotopic deformation of f will guarantee transversality, see [30, ch. 2].
Theorem A.1
If \({\mathcal {A}}\) is a \(C^\infty \)-smooth complete and connected surface embedded in \({{\mathbb {R}}}^3\), then \({{\mathbb {R}}}^3{\setminus }{\mathcal {A}}\) has two connected components.
Proof
Let W be a tubular neighbourhood of \({\mathcal {A}}\) in \({{\mathbb {R}}}^3\) [30, Ch. 2, Sec. 3, ex. 3 & 16]. That is, W is an open neighborhood of \({\mathcal {A}}\) in \({{\mathbb {R}}}^3\) comprised of points y having a unique closest point from X, say x, such that \(|y-x|<\varepsilon (x)\) where \(\varepsilon \) is a suitable smooth and strictly positive function on \({\mathcal {A}}\). Thus, we can write \(W=\{x+tn(x),\,x\in {\mathcal {A}},\,|t|<\varepsilon (x)\}\), where n(x) is a normal vector to \({\mathcal {A}}\) at x of unit length. Note that, for each \(x\in {\mathcal {A}}\), there are two possible (opposite) choices of n(x), but the definition of W makes it irrelevant which one we make. Moreover, if we fix n(x) and \(\eta \in (0,1)\), we can find a neighborhood V of x in \({\mathcal {A}}\) such that, to each \(y\in V\), there is a unique choice of n(y) with \(|n(y)-n(x)|<\eta \). Indeed, if \(\phi :U\rightarrow V\) is a parameterization with inverse \(\psi \) such that \(x\in V\), and if we set \(n_y:=\partial _{x_1}\phi (\psi (y))\wedge \partial _{x_2}\phi (\psi (y))/\Vert \partial _{x_1}\phi (\psi (y))\wedge \partial _{x_2}\phi (\psi (y))\Vert \) where \(x_1\), \(x_2\) are Euclidean coordinates on \(U\subset {{\mathbb {R}}}^2\), while \(\partial _{x_j}\) denotes the partial derivative with respect to \(x_j\) and the wedge indicates the vector product, then the two possible choices for n(y) when \(y\in V\) are \(\pm n_y\). Thus, if we select for instance \(n(x)=n_x\) and subsequently set \(n(y)=n_y\), we get upon shrinking V if necessary that \(|n(x)-n(y)|<\eta \) and \(|n(x)+n(y)|>2-\eta \) for \(y\in V\). As a consequence, if \(\varUpsilon :[a,b]\rightarrow {\mathcal {A}}\) is a continuous path, and if \(n_b\) is a unit normal vector to \({\mathcal {A}}\) at \(\varUpsilon (b)\), there is a continuous choice of \(n(\varUpsilon (\tau ))\) for \(\tau \in [a,b]\) such that \(n(\varUpsilon (b))=n_b\).
Fix \(x_0\in {\mathcal {A}}\) and let \(n_0\) be an arbitrary choice for \(n(x_0)\). Pick \(t_0\) with \(0<t_0<\varepsilon (x_0)\), and define two points in W by \(x_0^\pm =x_0\pm t_0 n_0\). We claim that each \(y\in {{\mathbb {R}}}^3{\setminus } {\mathcal {A}}\) can be joined either to \(x_0^+\) or to \(x_0^-\) by a continuous arc contained in \({{\mathbb {R}}}^3{\setminus } {\mathcal {A}}\). Indeed, let \(\gamma :[0,1]\rightarrow {{\mathbb {R}}}^3\) be a continuous path with \(\gamma (0)=y\) and \(\gamma (1)=x_0\). Let \(\tau _0\in (0,1]\) be smallest such that \(\gamma (\tau _0)\in {\mathcal {A}}\); such a \(\tau _0\) exists since \({\mathcal {A}}\) is closed. Pick \(0<\tau _1<\tau _0\) close enough to \(\tau _0\) that \(\gamma (\tau _1)\in W\), say \(\gamma (\tau _1)=x_1+t_1 n_1\) where \(x_1\in {\mathcal {A}}\), \(|t_1|<\varepsilon (x_1)\), and \(n_1\) is a unit vector normal to \({\mathcal {A}}\) at \(x_1\). Since \({\mathcal {A}}\) is connected, there is a continuous path \(\varUpsilon :[\tau _1,1]\rightarrow {\mathcal {A}}\) such that \(\varUpsilon (\tau _1)=x_1\) and \(\varUpsilon (1)=x_0\). Along the path \(\varUpsilon \), there is a continuous choice of \(\tau \rightarrow n(\varUpsilon (\tau ))\) such that \(n(x_0)=n_0\); this follows from a previous remark. Changing the sign of \(t_1\) if necessary, we may assume that \(n_1=n(x_1)\). Let \(\eta :[\tau _1,1]\rightarrow {{\mathbb {R}}}_+\) be a continuous function such that \(0<|\eta (\tau )|<\varepsilon (\varUpsilon (\tau ))\) with \(\eta (\tau _1)=t_1\) and \(\eta (1)=\text {sgn}\,t_1|t_0|\). Such an \(\eta \) exists, since \(\varepsilon \) is continuous and strictly positive while \(|t_1|<\varepsilon (x_1)\) and \(|t_0|<\varepsilon (x_0)\). Now, the concatenation of \(\gamma \) restricted to \([0,\tau _1]\) and \(\gamma _1:[\tau _1,1]\rightarrow {{\mathbb {R}}}^3\) given by \(\gamma _1(\tau )=\varUpsilon (\tau )+\eta (\tau )n(\varUpsilon (\tau ))\) is a continuous path from y to either \(x_0^+\) or \(x_0^-\) (depending on the sign of \(t_1\)) which is entirely contained in \({{\mathbb {R}}}^3{\setminus }{\mathcal {A}}\). This proves the claim, showing that \({{\mathbb {R}}}^3{\setminus } {\mathcal {A}}\) has at most two components. To see that it has at least two, it is enough to know that any smooth cycle \(\varphi :{\mathbb {S}}^1\rightarrow {{\mathbb {R}}}^3\) has intersection number \(I_2(\varphi ,{\mathcal {A}})=0\) modulo 2. Indeed, if this is the case and if \(x_0^+\) and \(x_0^-\) could be joined by a continuous arc \(\gamma :[0,1]\rightarrow {{\mathbb {R}}}^3\) not intersecting \({\mathcal {A}}\), then \(\gamma \) could be chosen \(C^\infty \)-smooth (see [30, Ch.1, Sec. 6, Ex. 3]) and we could complete it into a cycle \(\varphi :{\mathbb {S}}^1\rightarrow {{\mathbb {R}}}^3\) by concatenation with the segment \([x_0^-,x_0^+]\) which intersects \({\mathcal {A}}\) exactly once (at \(x_0\)), in a transversal manner. Elementary modifications at \(x^-_0\) and \(x^+_0\) will arrange things so that \(\varphi \) becomes \(C^\infty \)-smooth, and this would contradict the fact that the number of intersection points with \({\mathcal {A}}\) must be even. Now, if \({\mathbb {D}}\) is the unit disk, any smooth map \(\varphi :{\mathbb {S}}^1\rightarrow {{\mathbb {R}}}^3\) extends to a smooth map \(f:\overline{{\mathbb {D}}}\rightarrow {{\mathbb {R}}}^3\) (take for example \(f(re^{i\theta })=e^{1-1/r}\varphi (e^{i\theta })\)). Thus, by the boundary theorem [30, p. 80], the intersection number modulo 2 of \(\varphi \) with any smooth and complete embedded submanifold of dimension 2 in \({{\mathbb {R}}}^3\) (in particular with \({\mathcal {A}}\)) must be zero. This achieves the proof. \(\square \)
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Baratchart, L., Villalobos Guillén, C., Hardin, D.P. et al. Inverse Potential Problems for Divergence of Measures with Total Variation Regularization. Found Comput Math 20, 1273–1307 (2020). https://doi.org/10.1007/s10208-019-09443-x
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DOI: https://doi.org/10.1007/s10208-019-09443-x
Keywords
- Divergence free
- Distributions
- Solenoidal
- Total variation of measures
- Magnetization
- Inverse problems
- Purely 1-unrectifiable
- Sparse recovery
- Total variation regularization