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The Calder\'on problem for the multi-fractional Laplacian Schr\"{o}dinger equation with drift terms
Jin Wen, Yun-Long Liu, Jun-Liang Fu
TL;DR
The paper asks whether the relevant DtN map can simultaneously recover drift coefficient b and potential q when f = 0. The paper combines strong unique continuation, Runge approximation, and a generalized Alessandrini identity with a numerical framework using an RBF surrogate and hierarchical Bayesian inversion. The DtN map uniquely determines (b, q) in the homogeneous case and (b, q, f) in the non-homogeneous case.
Problem
The paper asks whether the relevant DtN map can simultaneously recover drift coefficient b and potential q when f = 0.
Source: §1 · p. 3Method
The paper combines strong unique continuation, Runge approximation, and a generalized Alessandrini identity with a numerical framework using an RBF surrogate and hierarchical Bayesian inversion.
Source: §5.1 · p. 23, §7 · p. 33Results
The DtN map uniquely determines (b, q) in the homogeneous case and (b, q, f) in the non-homogeneous case.
Source: §7 · p. 33Takeaways & Limitations
The paper presents a unified framework for fractional Calderón problems involving drift, potential, and source terms for multi-fractional operators.
Source: §7 · p. 32The forward problem assumes zero is not a Dirichlet eigenvalue of the operator.
Source: §1 · p. 4
Abstract
from arXiv · showhide
This paper studies the Calder\'on inverse problem for the multi-fractional Laplacian Schr\"odinger equation with drift terms, whose leading operator is a sum of fractional Laplacians of different orders, perturbed by a drift vector field and a potential. We consider two inverse settings: in the homogeneous source case \(f=0\), we aim to simultaneously recover the drift coefficient \(b\) and the potential \(q\) from the associated Dirichlet-to-Neumann map; in the non-homogeneous case \(f\neq 0\), we aim to simultaneously determine \(b\), \(q\), and the source term \(f\). Under suitable assumptions, we establish the well-posedness of the forward problem, derive a strong unique continuation principle, prove Runge-type approximation properties in fractional Sobolev spaces and weighted \(C^k\) spaces, and obtain generalized Alessandrini identities. These tools yield global uniqueness results for the inverse problems. In addition, we develop a numerical reconstruction framework combining finite difference discretization, radial basis function surrogate modeling, and hierarchical Bayesian inversion. Numerical examples in one dimension for smooth and discontinuous coefficients and sources illustrate the effectiveness of the proposed method and support the theoretical uniqueness results.
1. Introduction
The paper extends fractional Calderón inverse problems to a multi-fractional Schrödinger equation with drift, addressing simultaneous recovery of coefficients and, in the non-homogeneous case, the source. It establishes uniqueness using approximation and continuation tools and presents numerical experiments to illustrate reconstruction.
1. Introduction
- The study builds on nonlocal Calderón problems motivated by fractional calculus models of nonlocality, memory effects, and anomalous diffusion.The passages also cite applications in quantum mechanics, medical imaging, geophysical exploration, and environmental science.Source: §1 · p. 2
- The inverse problem extends prior work on fractional Schrödinger equations with drift to multi-fractional operators.The cited prior work established simultaneous uniqueness for drift and potential from infinitely many exterior measurements; the paper describes its setting as an extension of work on multi-fractional operators.Source: §1 · p. 2
- The complete DtN map uniquely determines b and q when f = 0, and b, q, and f when f ≠ 0.The paper also establishes Runge approximation and a generalized Alessandrini identity for these inverse problems.Source: §1 · p. 3
- Unique continuation is used to establish Runge approximation, a key tool for fractional Schrödinger-type inverse problems.The introduction contrasts this route with local problems, where the cited text says unique continuation is not a feature of local operators and CGO construction is unnecessary for the fractional case.Source: §1 · pp. 3–4
- The forward problem assumes zero is not a Dirichlet eigenvalue of the multi-fractional operator.The introduction states this assumption as the condition that the homogeneous zero-exterior-data problem has only the trivial solution.Source: §1 · p. 4
- The paper reports four one-dimensional numerical examples covering smooth and discontinuous coefficients in homogeneous and non-homogeneous settings.The examples also consider different noise levels.Source: §1 · pp. 4–5
2. Preliminaries
The preliminaries set up fractional Sobolev spaces and the multi-fractional operator with drift. Under coefficient regularity and a non-eigenvalue condition, they establish forward well-posedness and strong unique continuation.
2. Preliminaries
- The section introduces fractional Sobolev spaces, including Lp-based W^s,p spaces for treating drift terms.The stated range for these spaces is 1 < p < ∞ and 0 < s < 1.Source: §2 · p. 6
2.1. Multi-fractional Laplacian. Let m ∈N, and let
- The multi-fractional drift operator's bilinear forms are bounded on H^sm(R^n) when b ∈ W^(1−s_m,∞)(Ω)^n and q ∈ L∞(Ω).The stated range includes 1/2 < s_m < 1.Source: §2.1 · p. 7
- The Dirichlet problem is well-posed away from a countable exceptional set of Dirichlet eigenvalues; at λ = 0 outside that set, solutions are unique.The same exceptional set applies to the adjoint problem, and the zero-exterior-data homogeneous problem has only the trivial solution under the stated condition.Source: §2.1 · pp. 7–8
- A strong unique continuation principle is stated for homogeneous multi-fractional operators: under its hypotheses, u ≡ 0 in R^n.Source: §2.1 · p. 10
3. Approximation property
The approximation results use unique continuation and duality to approximate interior targets with solutions generated by exterior data. The paper establishes density in fractional Sobolev spaces and higher-order weighted C^k spaces under stronger regularity assumptions.
3.1. Approximation in e Hsm(Ω)
- Unique continuation and duality yield the Runge approximation property for solutions of the multi-fractional Schrödinger equation.Source: §3 · p. 10
- The solution set generated by exterior data supported in an open exterior set is dense in L2(Ω) and in e H^s_m(Ω).The section states these density results for the Poisson operator associated with the homogeneous problem.Source: §3 · p. 11
3.2. Higher-order Runge approximation
- Under stronger coefficient regularity and compactly supported b and q, solutions approximate smooth targets in weighted C^k norms.The stated result allows any h ∈ C∞(Ω), ε > 0, and k ∈ N; the weighted topology uses d(x)^−s_m w.Source: §3 · pp. 10–12
- The higher-order result uses a weighted function space whose elements satisfy the stated d^−s_m regularity condition.The section identifies d(x) as a smooth positive function in Ω tied to distance from the boundary.Source: §3 · p. 12
4. Uniqueness
The section derives Alessandrini-type identities and uses them with Runge approximation to prove uniqueness for both source settings. The DtN map determines (b, q) when f = 0 and (b, q, f) when f ≠ 0.
4. Uniqueness
- Alessandrini-type identities relate differences of DtN maps to differences in unknown coefficients and support both uniqueness proofs.The section treats homogeneous and non-homogeneous sources separately.Source: §4 · p. 14
- When f = 0, the DtN map uniquely determines the drift b and potential q.The proof recovers q first, then b.Source: §4 · pp. 15–16
- The result establishes that b1 = b2, q1 = q2, and f1 = f2 in Ω.Source: §4 · p. 18
- In the non-homogeneous proof, once f is identified, the identity is simplified to recover q and then b.The proof uses Runge approximation for non-homogeneous solutions and homogeneous adjoint solutions.Source: §4 · pp. 19–20
5. Numerical method
The numerical method combines a finite-difference forward solver, an RBF surrogate, and hierarchical Bayesian inversion to reconstruct unknown coefficients and source terms. Its parameterization uses compactly supported basis functions, with priors selected for smooth or discontinuous unknowns.
5. Numerical method
- The framework combines a finite-difference solver, an RBF surrogate, and hierarchical Bayesian inversion to reconstruct b, q, and f while quantifying uncertainty.The surrogate replaces repeated, expensive forward solves during inference.Source: §5 · p. 20, §5.1 · p. 23
5.1. Parameterization. We introduce the following parameterizations,
- The unknown drift, potential, and source are represented by coefficients in compactly supported basis functions.Their coefficient vectors are β, θ, and η, respectively.Source: §5.1 · p. 20
- The RBF surrogate is trained on parameter samples by solving the forward problem and computing the corresponding observations.The method uses an inverse multiquadric RBF and regularizes the fit when its matrix is not well conditioned.Source: §5.1 · p. 22
- The Bayesian model uses an ARD prior for smooth unknowns and a TV-GSM prior for discontinuous ones, then estimates the posterior by IAS and approximates uncertainty with the Laplace approximation.The surrogate approximation error is included in the likelihood.Source: §5.1 · pp. 23–26
6. Numerical examples
Four one-dimensional examples test the reconstruction framework in homogeneous and non-homogeneous settings, with smooth and discontinuous coefficients. The authors report effective hierarchical Bayesian inversion and reduced computational cost from the RBF surrogate.
Numerical examples
- Four one-dimensional examples use finite-difference forward solves and RBF surrogate modeling with hierarchical Bayesian inference.The first two examples are homogeneous; the last two are non-homogeneous.Source: §6 · pp. 26–27
Example 1
- Example 1 uses smooth coefficients with f = 0 and shows reconstructions of b and q at δ = 10^-3, 5 × 10^-4, and 10^-4.The figures assign panels (a)–(c) to b and (d)–(f) to q.Source: §6 · pp. 27–28, Figure 6.1 · p. 28
Example 2
- Example 2 considers a discontinuous potential with f = 0 and presents inversion images of b and q across error levels.The supplied passage does not specify the error-level values.Source: §6 · pp. 28–29, Figure 6.2 · p. 29
Example 3
- Example 3 reconstructs b, q, and f in a non-homogeneous setting, showing inversion images at δ = 10^-3 and δ = 10^-4.Panels (a)–(c) show b, q, and f at 10^-3; panels (d)–(f) show them at 10^-4.Source: §6 · p. 30, Figure 6.3 · p. 31
Example 4
- Example 4 treats a non-homogeneous case with discontinuous potential and displays inversion images of b, q, and f at δ = 10^-4 and δ = 10^-5.The first three panels correspond to 10^-4 and the next three to 10^-5.Source: §6 · pp. 30, 32, Figure 6.4 · p. 32
Numerical examples
- The authors report effective hierarchical Bayesian inversion and significantly reduced computational cost using the RBF surrogate.They also state that the examples support their uniqueness theory for simultaneous inversion.Source: §6 · p. 32
7. Conclusion
The paper presents a unified framework for fractional Calderón problems with drift, potential, and source terms. It establishes uniqueness for simultaneous recovery of (b, q) when f = 0 and (b, q, f) when f ≠ 0.
Conclusion
- The framework addresses fractional Calderón problems involving drift, potential, and source terms for multi-fractional operators.Source: §7 · p. 32
- For f = 0, the paper proves uniqueness for simultaneous recovery of (b, q); for f ≠ 0, it proves uniqueness for (b, q, f).Source: §7 · p. 33
- The uniqueness results use strong unique continuation, Runge approximation, and a generalized Alessandrini identity.Source: §7 · p. 33
Appendix
The appendix supplies arguments for forward-problem estimates and approximation properties used in the paper. Its passages establish norm control, solvability under a spectral condition, and density results.
Forward-problem estimates
- The fractional Poincaré estimate bounds the L2 norm by the fractional Laplacian seminorm on the zero-exterior space, yielding an equivalent H^sm norm.Source: Appendix · p. 34
Forward-problem solvability
- The Fredholm argument gives a unique solution when λ lies outside the exceptional spectrum, and the homogeneous problem is trivial when 0 lies outside it.Source: Appendix · p. 36
Approximation arguments
- The Hahn–Banach argument establishes density of D in the zero-exterior fractional Sobolev space and, consequently, in L2(Ω).Source: Appendix · p. 37
- The weighted C^k-space density proof concludes by applying strong unique continuation to show the associated function vanishes.Source: Appendix · p. 38