arXiv subjects

Angkana Rüland

Publications and source records attributed to Angkana Rüland.

At least 19 recordsLinked to original sources

arXiv2026

Instability estimates for the recovery of absorption in the diffusive regime of radiative transfer

We revisit the instability properties of the recovery of the absorption coefficient for the radiative transfer equation in the diffusive regime. To this end, we develop a rather robust framework building on [Koch-Rüland-Salo, 2021] which allows us to deal with nonlinear critical stability transition phenomena. In particular, this permits us to consider rather general geometries based on the identification of compression properties of the forward operator. Given the albedo operator as the measurement data, we show that in the regime of vanishing Knudsen number there is a transition from Hölder to logarithmic stability in the inverse problem for the radiative transfer equation. As a central ingredient, we rely on suitable a priori estimates for the radiative transfer equation which we deduce by building on the strategy from [Demattè-Velázquez, 2025].

math.AP↗

arXiv2026

On the transfer of stability from the local to the fractional anisotropic Calderón problem with exterior measurements

We study the quantitative transfer of uniqueness from the classical to the fractional Calderón problem with exterior data. This allows us to deduce the first stability estimates for the principal part of the isotropic fractional Calderón problem with exterior data in the absence of Liouville transforms. Our argument relies on careful quantitative unique continuation and Runge approximation estimates. Due to the unbounded geometry and the mismatch of the dimensionalities of the measurement domains (exterior data on an open set vs boundary data on a co-dimension one manifold) novel challenges arise compared to the setting of source-to-solution measurements on closed manifolds.

math.AP↗

arXiv2025

The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates

We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather general lower scaling bounds, both in the setting of prescribed Dirichlet data and for periodic configurations with a mean value constraint. In contrast to the setting without gauge invariances, these lower scaling bounds depend on \emph{two} parameters -- the order of lamination of the boundary data as well as the number of involved (non-)degenerate symmetrized rank-one directions. By discussing upper bounds in specific geometries and for a specific constellation of wells, we give evidence of the sharpness of these lower bound estimates. Hence, it is necessary to keep track of the outlined \emph{two} parameters in deducing scaling laws within the geometrically linearized theory of elasticity.

math.AP↗

arXiv2025

Transfer of Stability from the Classical to the Fractional Anisotropic Calderón Problem

We discuss two spectral fractional anisotropic Calderón problems with source-to-solution measurements and their quantitative relation to the classical Calderón problem. Firstly, we consider the anistropic fractional Calderón problem from [FGKU25]. In this setting, we quantify the relation between the local and nonlocal Calderón problems which had been deduced in [R25] and provide an associated stability estimate. As a consequence, any stability result which holds on the level of the local problem with source-to-solution data has a direct nonlocal analogue (up to a logarithmic loss). Secondly, we introduce and discuss the fractional Calderón problem with source-to-solution measurements for the spectral fractional Dirichlet Laplacian on open, bounded, connected, Lipschitz sets on $\mathbb{R}^n$. Also in this context, we provide a qualitative and quantitative transfer of uniqueness from the local to the nonlocal setting. As a consequence, we infer the first stability results for the principal part for a fractional Calderón type problem for which no reduction of Liouville type is known. Our arguments rely on quantitative unique continuation arguments. As a result of independent interest, we also prove a quantitative relation between source-to-solution and Dirichlet-to-Neumann measurements for the classical Calderón problem.

math.AP↗

arXiv2025

On surface energies in scaling laws for singular perturbation problems for martensitic phase transitions

The objective of this article is to compare different surface energies for multi-well singular perturbation problems associated with martensitic phase transformations involving higher order laminates. We deduce scaling laws in the singular perturbation parameter which are robust in the choice of the surface energy (e.g., diffuse, sharp, an interpolation thereof or discrete). Furthermore, we show that these scaling laws do not require the presence of isotropic surface energies but that generically also highly anisotropic surface energies yield the same scaling results. More precisely, the presence of essentially generic partial directional derivatives in the regularization terms suffices to produce the same scaling behaviour as in the isotropic setting. The only sensitive directional dependences are directly linked to the lamination directions of the well structure -- and even for these only the ``inner-most'' lamination direction is of significance in determining the scaling law. In view of experimental applications, this shows that also for higher-order laminates, the precise structure of the surface energies -- which is often very difficult to determine experimentally -- does not have a crucial impact on the scaling behaviour of the investigated structures but only enters when considering finer properties.

math.AP↗

arXiv2025

On instability mechanisms for inverse problems

In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderón type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings. This is a revised version of the article ``On instability mechanisms for inverse problems'' Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp by the same authors.

math.AP↗

arXiv2025

Boundary Reconstruction for the Anisotropic Fractional Calderón Problem

In this article, we provide a boundary reconstruction result for the anisotropic fractional Calderón problem and its associated degenerate elliptic extension into the upper half plane. More precisely, considering the setting from \cite{FGKU21}, we show that the metric on the measurement set can be reconstructed from the source-to-solution data. To this end, we rely on the approach by Brown \cite{B01} in the framework developed in \cite{NT01} (see also \cite{KY02}) after localizing the problem by considering it through an extension perspective.

math.AP↗

arXiv2025

Fractional anisotropic Calderón problem with external data

In this paper, we solve the fractional anisotropic Calderón problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

math.AP↗

arXiv2024

On a $T_3$-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations

We study the rigidity properties of the $T_3$-structure for the symmetrized gradient from \cite{BFJK94} qualitatively, quantitatively and numerically. More precisely, we complement the flexibility result for approximate solutions of the associated differential inclusion which was deduced in \cite{BFJK94} by a rigidity result on the level of exact solutions and by a quantitative rigidity estimate and scaling result. The $T_3$-structure for the symmetrized gradient from \cite{BFJK94} can hence be regarded as a symmetrized gradient analogue of the Tartar square for the gradient. As such a structure cannot exist in $\mathbb{R}^{2\times 2}_{sym}$ the example from \cite{BFJK94} is in this sense minimal. We complement our theoretical findings with numerical simulations of the resulting microstructure.

math.AP↗

arXiv2024

On Instability Properties of the Fractional Calderón Problem

We prove exponential instability properties for the fractional Calderón problem and the conductivity formulation of the fractional Calderón problem in the regime of fractional powers $s\in (0,1)$. We particularly focus on two settings: First, we discuss instability properties in general domain geometries with scaling critical $L^{\frac{n}{2s}}$ potentials and constant background metrics. Secondly, we investigate instability properties in general geometries with $L^{\frac{n}{2s}}$ potentials and low regularity, variable coefficient, possibly anisotropic background metrics. In both settings we make use of the methods introduced in \cite{KRS21} and we deduce strong compression estimates for the forward problem. In the first setting this is based on analytic smoothing estimates for a suitable comparison operator while in the second setting involving low regularity metrics this is based on an iterated compression gain. We thus generalize the results from \cite{RS18} to generic geometries and variable coefficients and further also discuss the setting of fractional conductivity equations. In particular, this proves that the logarithmic stability estimates for the fractional Calderón problem from \cite{RS20} are optimal.

math.AP↗

arXiv2024

On the Effect of Geometry on Scaling Laws for a Class of Martensitic Phase Transformations

We study scaling laws for singular perturbation problems associated with a class of two-dimensional martensitic phase transformations and deduce a domain dependence of the scaling law in the singular perturbation parameter. In these settings the respective scaling laws give rise to a selection principle for specific, highly symmetric domain geometries for the associated nucleation microstructure. More precisely, firstly, we prove a general lower bound estimate illustrating that in settings in which the domain and well geometry are incompatible in the sense of the Hadamard-jump condition, then necessarily at least logarithmic losses in the singular perturbation parameter occur in the associated scaling laws. Secondly, for specific phase transformations in two-dimensional settings we prove that this gives rise to a dichotomy involving logarithmic losses in the scaling law for generic domains and optimal linear scaling laws for very specific, highly compatible polygonal domains. In these situations the scaling law thus gives important insight into optimal isoperimetric domains. We discuss both the geometrically linearized and nonlinear settings.

math.AP↗

arXiv2024

On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators

We study the scaling behaviour of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions for highly symmetric boundary data (but arbitrary tensor order $m \in \mathbb{N}$) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23} provides an example of this family of new scaling laws.

math.AP↗

arXiv2024

On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in \cite{RT21} we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

math.AP↗

arXiv2023

Revisiting the Anisotropic Fractional Calderón Problem Using the Caffarelli-Silvestre Extension

We revisit the source-to-solution anisotropic fractional Calderón problem introduced and analyzed in [FGKU21] and [F21]. Using the Caffarelli-Silvestre interpretation of the fractional Laplacian, we provide an alternative argument for the recovery of the heat and wave kernels from [FGKU21]. This shows that in the setting of the source-to-solution anisotropic fractional Calderón problem the heat and Caffarelli-Silvestre approach give rise to equivalent perspectives and that each kernel can be recovered from the other. Moreover, we also discuss the Dirichlet-to-Neumann anisotropic source-to-solution problem and provide a direct link between the Dirichlet Poisson kernel and the wave kernel. This illustrates that it is also possible to argue completely on the level of the Poisson kernel, bypassing the recovery of the heat kernel as an additional auxiliary step. Last but not least, as in [CGRU23], we relate the local and nonlocal source-to-solution Calderón problems.

math.AP↗

arXiv2023

A Reduction of the Fractional Calderón Problem to the Local Calderón Problem by Means of the Caffarelli-Silvestre Extension

We relate the (anisotropic) variable coefficient local and nonlocal Calderón problems by means of the Caffarelli-Silvestre extension. In particular, we prove that (partial) Dirichlet-to-Neumann data for the fractional Calderón problem in three and higher dimensions determine the (full) Dirichlet-to-Neumann data for the local Calderón problem. As a consequence, any (variable coefficient) uniqueness result for the local problem also implies a uniqueness result for the nonlocal problem. Moreover, our approach is constructive and associated Tikhonov regularization schemes can be used to recover the data. Finally, we highlight obstructions for reversing this procedure, which essentially consist of two one-dimensional averaging processes.

math.AP↗

arXiv2022

On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of $\mathcal{A}$-free operators and for a singularly perturbed $T_3$-structure for the divergence operator. In particular, in the compatible setting of the two-state problem we prove that all homogeneous, first order, linear operators with affine boundary data which enforce oscillations yield the typical $ε^{\frac{2}{3}}$-lower scaling bounds. As observed in \cite{CC15} for higher order operators this may no longer be the case. Revisiting the example from \cite{CC15}, we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on \cite{RT22, GN04, PP04}, we discuss the scaling behaviour of a $T_3$ structure for the divergence operator. We prove that as in \cite{RT22} this yields a non-algebraic scaling law.

math.AP↗

arXiv2022

On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplified $m$-well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by the \emph{order of lamination of the Dirichlet data}. This follows by deducing (essentially) matching upper and lower scaling bounds. For the \emph{upper} bound we argue by providing iterated branching constructions, and complement this with ansatz-free \emph{lower} bounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated "bootstrap arguments: based on the ideas from \cite{RT21}. Relying on these observations, we study models involving laminates of arbitrary order.

math.AP↗

arXiv2022

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components.

math.AP↗