arXiv subjects

Tony Liimatainen

Publications and source records attributed to Tony Liimatainen.

At least 19 recordsLinked to original sources

arXiv2026

Improved stability of low Fourier modes in inverse problems for potentials

We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-Δ- λ^2 + q)u=0 \quad \text{in} \quad Ω\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $λ$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-Δ+ q)u=0 \quad \text{in} \quad (-π,π)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

math.AP↗

arXiv2026

Reconstruction for an inverse scattering problem with a Kerr type nonlinearity

We study the inverse scattering problem for the Kerr-nonlinear Helmholtz equation \[ Δu + k^2(1+q(x)|u|^2)u = 0 \quad \text{in }\mathbb{R}^n,\; n\geq 2, \] where the aim is to recover the unknown potential $q$ from the scattering amplitude. We obtain uniqueness for full data and partial data cases of backscattering, fixed angle scattering, and fixed energy scattering. For the linear Helmholtz equation, uniqueness in backscattering and fixed angle cases are classical and largely open problems. We are able to explicitly reconstruct individual Fourier modes of the potential, and if the measured directions and energies cover an open subset, we recover $q$. The simplicity of the approach leads to an efficient numerical method, and numerical experiments show accurate reconstructions, even in the presence of noise.

math.AP↗

arXiv2026

An inverse source problem for a quasilinear elliptic equation

We initiate the study of inverse source problems for quasilinear elliptic equations of the form \[ \left\{ \begin{array}{ll} \nabla \cdot (γ(x,u,\nabla u) \nabla u) = F & \text{in } Ω, \\ u = f & \text{on } \partialΩ, \end{array} \right. \] where $Ω\subset \mathbb{R}^n$, $n \geq 2$, is a simply connected bounded domain. We consider the specific nonlinearity $γ(x,u,\nabla u) = σ(x) + q(x) u$, with $q$ assumed to be known. By exploiting the nonlinearity to break the gauge invariance of the problem, we establish unique recovery of both $σ$ and $F$ from the associated Dirichlet-to-Neumann (DN) map under the structural conditions $q$ and $\nabla(σ/q)$ are nowhere vanishing in $\overlineΩ$. In the absence of these conditions, in particular in the linear case, we demonstrate that the inverse problem admits a gauge obstructing the uniqueness. We use higher order linearizations to obtain a complicated coupled system for the unknowns. The complexity of this system arises in part from the gauge freedom of the linearized equation, which is new in this context. We solve the system by constructing suitable complex geometric optics solutions and applying the unique continuation principle for nonlinear elliptic systems. We anticipate that the solution method developed here will prove useful in other inverse problems as well.

math.AP↗

arXiv2026

An Inverse Problem for the Prescribed Mean Curvature

We extend the recent study of inverse problems for minimal surfaces by considering the inverse source problem for the prescribed mean curvature equation \begin{equation*} \nabla \cdot \left[ \frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}} \right] = H(x). \end{equation*} This work also represents the first treatment of inverse source problems for quasilinear equations. We prove that in two dimensions, the source function $H$ is uniquely determined by the associated Dirichlet-to-Neumann map. A notable feature of this problem is that although the equation is posed on an Euclidean domain, its linearization yields an anisotropic conductivity equation where the coefficient matrix corresponds to a Riemannian metric $g$ depending on the background solution. The main methodological contribution is the derivation of a coupled nonlinear system of algebraic and geometric partial differential equations from boundary measurements. Similar systems will naturally appear in other inverse problems for quasilinear equations. We solve the system using a Liouville type uniqueness result for conformal mappings, which recovers the source function uniquely.

math.AP↗

arXiv2025

Bulk metric reconstruction from entanglement data via minimal surface area variations

We investigate the reconstruction of asymptotically anti-de Sitter (AdS) bulk geometries from boundary entanglement entropy data for ball-shaped entangling regions. By deriving an explicit inversion formula, we relate variations in entanglement entropy to deviations of the bulk metric about a fixed background. Applying this formula, we recover the Schwarzschild-AdS spacetime in the low-temperature regime to first order. We further extend our analysis to include deformations of the bulk geometry with nontrivial dependence on boundary directions, and propose an iterative reconstruction scheme aimed at recovering the full spacetime starting close to a conformal fixed point. We do this by building on recent advances in the mathematics of inverse problems by introducing the higher-order linearization method as a new tool in the context of holographic bulk reconstruction.

hep-th↗

arXiv2025

Generalized boundary rigidity and minimal surface transform

We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric in dimension $n+1$ ($n \geq 2$), the metric is determined by these volumes under an ampleness condition. Furthermore, we establish Hölder stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calderón type problems and complex geometrical optics solutions, we study the linearized forward operator that gives rise to the minimal surface transform, a generalization of the X-ray/Radon transform. We demonstrate that this transform fits into the framework of double fibration transforms and satisfies the Bolker condition in the sense of Guillemin. Under certain assumptions, including a foliation condition, we prove invertibility of this transform on an analytic manifold as well as recovery of the analytic wave front set. The methods developed in this paper offer new tools for addressing the generalized boundary rigidity problem and expand the scope of applications of double fibration transforms. We anticipate that these techniques will also be applicable to other geometric inverse problems. Beyond mathematics, our results have implications for the AdS/CFT correspondence in physics.

math.AP↗

arXiv2025

Gaussian beam interactions and inverse source problems for nonlinear wave equations

We study the inverse source problem for the semilinear wave equation \[ (\Box_g + q_1)u + q_2 u^2 = F, \] on a globally hyperbolic Lorentzian manifold. We demonstrate that the coefficients $q_1$ and $q_2$, as well as the source term $F$, can be recovered up to a natural gauge symmetry inherent in the problem from local measurements. Furthermore, if $q_1$ is known, we establish the unique recovery of the source $F$, which is in a striking contrast to inverse source problems for linear equations where unique recovery is not possible. Our results also generalize previous works by eliminating the assumption that $u= 0$ is a solution, and by accommodating quadratic nonlinearities. A key contribution is the development of a calculus for nonlinear interactions of Gaussian beams. This framework provides an explicit representation for waves that correspond to sources involving products of two or more Gaussian beams. We anticipate this calculus will serve as a versatile tool in related problems, offering a concrete alternative to Fourier integral operator methods.

math.AP↗

arXiv2025

An inverse problem for the Monge-Ampère equation

We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Ampère equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function $F$. The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric $g$. Interestingly, although the equation is posed on a Euclidean domain, the inverse problem becomes anisotropic since the metric $g$ appears as a coefficient matrix in the linearized equation. As an intermediate step, we prove that the DN map of the non-divergence form equation \[ g^{ab} \partial_{ab} v = 0 \] uniquely determines the conformal class of the metric $g$ on a simply connected planar domain, without the usual diffeomorphism invariance. To address the challenges of full nonlinearity, we develop asymptotic expansions for complex geometric optics solutions in the planar setting and solve a resulting nonlocal $\overline{\partial}$-equation by proving a unique continuation principle for it. These techniques are expected to be applicable to a wide range of inverse problems for nonlinear equations.

math.AP↗

arXiv2024

Calderón problem for the quasilinear conductivity equation in dimension $2$

In this paper we prove a uniqueness result for the Calderón problem for the quasilinear conductivity equation on a bounded domain $\R^2$. The proof of the result is based on the higher order linearization method, which reduces the problem to showing density of products of solutions to the linearized equation and their gradients. In contrast to the higher dimensional case, the proof involves delicate analysis of the correction terms of Bukhgeim type complex geometric solutions (CGOs), which have only limited decay. To prove our results, we construct suitable families of CGOs whose phase functions have and do not have critical points. We also combine stationary phase analysis with $L^p$ estimates for the correction terms of the CGOs.

math.AP↗

arXiv2024

The Calderón problem on Riemannian surfaces and of minimal surfaces

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $Δ_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a priori that the surface is a planar domain [36] or that the geometry is a priori known [29]. We will then apply this result to study a geometric inverse problem for determining minimal surfaces embedded in $3$-dimensional Riemannian manifolds. In particular we will show that knowledge of the volumes of embedded minimal surfaces determine not only their topological and differential structure but also their Riemannian structure as an embedded hypersurface. Such geometric inverse problems are partially inspired by the physical models proposed by the AdS/CFT correspondence. The crucial ingredient in removing the planar domain assumption is the determination of the boundary trace of holomorphic functions from knowledge of the Dirichlet-Neumann map of $Δ_g +q$. This requires a new type of argument involving Carleman estimates and construction of CGO whose phase functions are not Morse as in the case of [29]. We anticipate that these techniques could be of use for studying other inverse problems in geometry and PDE.

math.AP↗

arXiv2024

Applications of the Stone-Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone-Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds, and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone-Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

math.CV↗

arXiv2023

An inverse problem for general minimal surfaces

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $Σ$ is a $2$-dimensional embedded minimal surface, then the knowledge of the Dirichlet-to-Neumann map associated to the minimal surface equation determines $Σ$ up to an isometry. Without the topological condition, we show that a conformal factor of a general minimal surface $Σ$ can be recovered. We develop a semiclassical nonlinear calculus for complex geometric optics solutions, which allows an efficient error analysis for multiplication of the correction terms of the solutions. The calculus is independent of the application to the minimal surface equation and we expect it to have applications in various inverse problems for nonlinear equations in dimension $2$, in both $\mathbb{R}^2$ and geometric settings. Other applications of the results include generalized boundary rigidity problem and the AdS/CFT correspondence in physics.

math.AP↗

arXiv2023

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on $\mathbb{R}^n$. Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

math.AP↗

arXiv2023

Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations

We study inverse source problems associated to semilinear elliptic equations of the form \[ Δu(x)+a(x,u)=F(x), \] on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both $a(x,u)$ and $F(x)$ uniquely from the associated DN map. Moreover, for general nonlinearities $a(x,u)$, we show that we can recover the derivatives $\partial_u^ka(x,u)$ and the source $F(x)$ up to a gauge. Especially, we recover general polynomial nonlinearities up to a gauge and generalize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$ is a solution.

math.AP↗

arXiv2022

An Inverse Problem for the Relativistic Boltzmann Equation

We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $μ$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of the point $x^+$, and hence is the maximal set to where causal signals sent from $x^-$ can propagate and return to the point $x^+$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

math.AP↗

arXiv2022

An inverse problem for the Riemannian minimal surface equation

In this paper we consider determining a minimal surface embedded in a Riemannian manifold $Σ\times \mathbb{R}$. We show that if $Σ$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $Σ$ up to an isometry.

math.AP↗

arXiv2022

An inverse problem for a semi-linear wave equation: a numerical study

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We also present an approach to numerically estimating two-dimensional derivatives of noisy data via Tikhonov regularization. The methods are tested using synthetic noisy measurements of the Dirichlet-to-Neumann map. Various examples of reconstructions of the potential functions are given.

math.AP↗

arXiv2021

An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds

Given a conformally transversally anisotropic manifold $(M,g)$, we consider the semilinear elliptic equation $$(-Δ_{g}+V)u+qu^2=0\quad \text{on $M$}.$$ We show that an a priori unknown smooth function $q$ can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.

math.AP↗