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Introduction to inverse problems for non-linear partial differential equations

Matti Lassas

TL;DR

The paper surveys inverse problems for nonlinear partial differential equations, focusing on how nonlinearities enable solutions that are intractable in the linear setting through the method of multiple linearization and artificial sources. It develops passive and active measurement frameworks on Lorentzian and Riemannian manifolds, introducing light-observation sets and the Dirichlet-to-Neumann map as key data objects. Core contributions include injectivity and reconstruction results for geometric data from passive measurements, stability estimates for nonlinear inverse problems with explicit exponents, and reconstruction strategies based on nonlinear wave interactions that generate artificial sources. The work emphasizes both theoretical uniqueness/conformality results and practical considerations for numerical reconstruction, highlighting the significance for geometric inverse problems and potential applications in imaging and seismology.

Abstract

We consider inverse problems for non-linear hyperbolic and elliptic equations and give an introduction to the method based on the multiple linearization, or on the construction of artificial sources, to solve these problems. The method is based on self-interaction of linearized waves or other solutions in the presence of non-linearities. Multiple linearization has successfully been used to solve inverse problems for non-linear equation which are still unsolved for the corresponding linear equations.

Introduction to inverse problems for non-linear partial differential equations

TL;DR

The paper surveys inverse problems for nonlinear partial differential equations, focusing on how nonlinearities enable solutions that are intractable in the linear setting through the method of multiple linearization and artificial sources. It develops passive and active measurement frameworks on Lorentzian and Riemannian manifolds, introducing light-observation sets and the Dirichlet-to-Neumann map as key data objects. Core contributions include injectivity and reconstruction results for geometric data from passive measurements, stability estimates for nonlinear inverse problems with explicit exponents, and reconstruction strategies based on nonlinear wave interactions that generate artificial sources. The work emphasizes both theoretical uniqueness/conformality results and practical considerations for numerical reconstruction, highlighting the significance for geometric inverse problems and potential applications in imaging and seismology.

Abstract

We consider inverse problems for non-linear hyperbolic and elliptic equations and give an introduction to the method based on the multiple linearization, or on the construction of artificial sources, to solve these problems. The method is based on self-interaction of linearized waves or other solutions in the presence of non-linearities. Multiple linearization has successfully been used to solve inverse problems for non-linear equation which are still unsolved for the corresponding linear equations.

Paper Structure

This paper contains 13 sections, 11 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Inverse problems for waves
  3. Inverse problems with passive measurements
  4. Lorentzian light observation sets and waves on a Riemannian manifold
  5. Inverse problems on Lorentzian manifolds
  6. Notations in a Lorentzian space-time
  7. Passive measurements on a Lorentzian manifold
  8. The set of earliest light observations
  9. Active measurements: Inverse problems for non-linear wave equations
  10. Stability results and numerical reconstruction
  11. Ideas for proofs and reconstruction methods
  12. Light-observation sets and the inverse problem for passive observations
  13. Nonlinear interaction of waves and inverse problems for active measurements

Key Result

Theorem 1.1

Let $n\geq 2$, and let $\Omega\subset {\mathbb R}^n$ be a bounded domain with $C^\infty$ boundary $\partial \Omega$. Let $q_1,q_2\in C^\infty(\overline{\Omega})$. Assume the Dirichlet-to-Neumann maps $\Lambda_{q_j}$ for the equation Main semilinear equation with potentials $q_j$, $j=1,2$ satisfy for all $f\in C^s(\partial \Omega)$ with $\lVert f \rVert_{C^s(\partial M)}<\delta$, where $\delta>0$

Figures (4)

  • Figure 1: Left.: When there are no cut points, the earliest light observation set $\mathcal{E}_{{U}}(q)$ is the intersection of the cone and the open set ${{U}}$. The cone is the union of future-pointing light-like geodesics from $q$, and the ellipsoid depicts ${{U}}$. Right: The unknown set $W \subset I^-(\mu(1)) \setminus I^-(\mu(-1))$ contains source points. Light cones emanating from these points (red segment) are observed in the set $U$.
  • Figure 2: Four plane waves propagate in space. When the planes intersect, the non-linearity of the hyperbolic system produces new waves. Left: Plane waves before interacting. Middle left: The two-wave interactions (red line segments) appear but do not cause singularities propagating to new directions. Middle right and Right: All plane waves have intersected and new waves have appeared. The three-wave interactions cause conic waves ( the black surface). Only one such wave is shown in the figure. The interaction of four waves causes a microlocal point source that sends a spherical wave in all future light-like directions.
  • Figure 3: Reconstruction of the coefficient $q(x,t)$ in equation $\partial_t^2u(x,t)-\partial_x^2u(x,t) + q(x,t) u(x,t)^{2} =0$, $x\in [0,1],$$t\in [0,T],$ from the data $f\to \langle \psi_0,\Lambda_q f\rangle_{L^2({\partial \Omega\times [0,T]})}$, where $\Lambda_q$ is the Dirichlet-to-Neumann map $\psi_0$ is a suitable chosen 'device function'. Above, data has additive Gaussian noise and the signal-to-noise ratio is 12 dB. Left figure: Ground truth of the potential $q(x,t)$. Center figure: Numerical reconstruction. Right figure: Error in reconstruction. For numerical results, see Tyninumerical and for mathematical analysis, see TyniStabilityTyniLorentzian.
  • Figure 4: Left: Picture on proving that $q_1=q_2$ by contradiction. Right: The function $X_{\vec{a}}:q\to (F_q({a_j})_{j=1}^{n+1}=(f^+_{a_j}(q))_{j=1}^{n+1}\in {\mathbb R}^{n+1}$ defines coordinates near a point $q$ on $M$.

Theorems & Definitions (12)

  • Theorem 1.1: Reconstruction of non-linear term lassas2019nonlinear
  • Theorem 1.2: Simultaneous recovery of metric and potential lassas2019nonlinear
  • Theorem 1.3: Simultaneous recovery: Cavity and potential LLYS-obstacles
  • Theorem 2.1: BelKurTataru1
  • Theorem 2.2: Focusing of waves Kirpichnikova-Korpela
  • Theorem 2.3: KKLM
  • Theorem 2.4: Distance difference functions determine the manifold LSaksala
  • Definition 2.5
  • Theorem 2.6: kurylev2018inverse
  • Theorem 2.7: kurylev2018inverse
  • ...and 2 more