arXiv subjects

Henrik Garde

Publications and source records attributed to Henrik Garde.

At least 19 recordsLinked to original sources

arXiv2026

Regularity and non-degeneracy of $\Phi^*[I]$ implies regularity of the fixed boundary $\partial\Omega$

For a diffeomorphism $\Phi$ of a domain $\overline{\Omega}$ onto itself, which is the identity on $\partial\Omega$, we prove that local regularity of the push-forward $\Phi^*[I]$ implies local regularity of $\partial\Omega$, provided a certain non-degeneracy condition is satisfied. To be precise, if $\nu$ is a normal to $\partial\Omega$ at a point $P$, then the condition $(\Phi^*[I](P)-I)\nu \neq 0$ and the assumption that $\Phi^*[I]$ is of class $C^{k+1,\alpha}$ near $P$ imply that $\partial\Omega$ is also of class $C^{k+1,\alpha}$ near $P$. This result naturally complements recent regularity results for non-scattering inhomogeneities.

math.AP↗

arXiv2026

Reconstruction of shear modulus inclusions in elastostatics via divergence-free localization

We formulate an improved version of the monotonicity method for shape reconstruction in the inverse problem of linear elastostatics. We show that this method can reconstruct inclusions in the second Lam\'e parameter, i.e., the shear modulus, independently of inhomogeneities in the first Lam\'e parameter. This improves on earlier methods that only recover inclusions in the bulk modulus under certain assumptions on the local interplay between the Lam\'e parameters. Our proofs are based on linearization and divergence-free localization of energy using harmonic vector fields.

math.AP↗

arXiv2026

Reconstruction of non-self-adjoint anisotropic and complex inclusions in the Calder\'on problem

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calder\'on problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension $d \geq 2$. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general $L^\infty$ non-self-adjoint perturbations. We only require unique continuation based on the self-adjoint part of the background conductivity, thus making the methods compatible with generic unique continuation results.

math.AP↗

arXiv2026

On the directional growth of the resolvent norm

Let $A$ be a closed densely defined operator on a separable Hilbert space $\mathcal{H}$. Assume the resolvent set $\rho(A)$ is non-empty. For $z,z'\in\rho(A)$ let $[z,z']$ denote the straight line segment from $z$ to $z'$. For each $z\in\rho(A)$ we classify the behavior of the resolvent norm $\zeta\mapsto\lVert R_A(\zeta) \rVert$ near $z$. Either there are $z'\in\rho(A)$, $z'\neq z$, $[z,z']\subset\rho(A)$, such that $\lVert R_A(\zeta) \rVert \geq \lVert R_A(z) \rVert + C\lvert \zeta-z \rvert^\delta$ for $\zeta\in[z,z']$ with $\delta=1$ or $\delta=2$, or the function $\zeta\mapsto\lVert R_A(\zeta) \rVert$ has a global minimum at $\zeta=z$.

math.SP↗

arXiv2025

Direct reconstruction of anisotropic self-adjoint inclusions in the Calder\'on problem

We extend the monotonicity method for direct exact reconstruction of inclusions in the partial data Calder\'on problem, to the case of general anisotropic conductivities in any spatial dimension $d\geq 2$. From a local Neumann-to-Dirichlet map, we give reconstruction methods of inclusions based on unknown anisotropic self-adjoint perturbations to a known anisotropic conductivity coefficient. This additionally provides new insights into the non-uniqueness issues of the anisotropic Calder\'on problem. The main assumption is a definiteness condition for the perturbations near the outer inclusion boundaries. Beyond this condition, they are $L^\infty$-perturbations that may be indefinite away from the outer inclusion boundaries, and with no boundary regularity requirement for the inclusions. Alternatively, we allow extreme parts that are perfectly insulating or perfectly conducting, in which case we require Lipschitz regularity of the outer inclusion boundaries.

math.AP↗

arXiv2025

Reconstruction in the Calder\'on problem on a fixed partition from finite and partial boundary data

This short note modifies a reconstruction method by the author (Comm. PDE, 45(9):1118-1133, 2020), for reconstructing piecewise constant conductivities in the Calder\'on problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

math.AP↗

arXiv2025

Direct reconstruction of general elastic inclusions

The inverse problem of linear elasticity is to determine the Lam\'e parameters, which characterize the mechanical properties of a domain, from pairs of pressure activations and the resulting displacements on its boundary. This work considers the specific problem of reconstructing inclusions that manifest themselves as deviations from the background Lam\'e parameters. The monotonicity method is a direct reconstruction method that has previously been considered for domains only containing positive (or negative) inclusions with finite contrast. That is, all inclusions have previously been assumed to correspond to a finite increase (or decrease) in both Lam\'e parameters compared to their background values. We prove the general outer approach of the monotonicity method that simultaneously allows positive and negative inclusions, of both finite and extreme contrast; the latter refers to either infinitely stiff or perfectly elastic materials.

math.AP↗

arXiv2025

Infinite-dimensional Lipschitz stability in the Calder\'on problem and general Zernike bases

Calder\'on's inverse conductivity problem has, so far, only been subject to conditional logarithmic stability for infinite-dimensional classes of conductivities and to Lipschitz stability when restricted to finite-dimensional classes. Focusing our attention on the unit ball domain in any spatial dimension $d\geq 2$, we give an elementary proof that there are (infinitely many) infinite-dimensional classes of conductivities for which there is Lipschitz stability. In particular, Lipschitz stability holds for general expansions of conductivities, allowing all angular frequencies but with limited freedom in the radial direction, if the basis coefficients decay fast enough to overcome the growth of the basis functions near the domain boundary. We construct general $d$-dimensional Zernike bases and prove that they provide examples of infinite-dimensional Lipschitz stability.

math.AP↗

arXiv2024

The LuViRA Dataset: Synchronized Vision, Radio, and Audio Sensors for Indoor Localization

We present a synchronized multisensory dataset for accurate and robust indoor localization: the Lund University Vision, Radio, and Audio (LuViRA) Dataset. The dataset includes color images, corresponding depth maps, inertial measurement unit (IMU) readings, channel response between a 5G massive multiple-input and multiple-output (MIMO) testbed and user equipment, audio recorded by 12 microphones, and accurate six degrees of freedom (6DOF) pose ground truth of 0.5 mm. We synchronize these sensors to ensure that all data is recorded simultaneously. A camera, speaker, and transmit antenna are placed on top of a slowly moving service robot, and 89 trajectories are recorded. Each trajectory includes 20 to 50 seconds of recorded sensor data and ground truth labels. Data from different sensors can be used separately or jointly to perform localization tasks, and data from the motion capture (mocap) system is used to verify the results obtained by the localization algorithms. The main aim of this dataset is to enable research on sensor fusion with the most commonly used sensors for localization tasks. Moreover, the full dataset or some parts of it can also be used for other research areas such as channel estimation, image classification, etc. Our dataset is available at: https://github.com/ilaydayaman/LuViRA_Dataset

eess.SP↗

arXiv2024

Linearised Calder\'on problem: Reconstruction of unbounded perturbations in 3D

Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L^2$ perturbation from linearised data in the two-dimensional linearised Calder\'on problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calder\'on problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any $L^3$ perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full $L^2$ basis of current densities.

math.AP↗

arXiv2024

Linearization-based direct reconstruction for EIT using triangular Zernike decompositions

This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was originally designed for a linearized setting, we numerically demonstrate its functionality when the input data is the corresponding change in the current-to-voltage boundary operator. Both idealized continuum model and practical complete electrode model measurements are considered in the numerical studies, with the examined domain being either the unit disk or a convex polygon. Special attention is paid to regularizing the algorithm and its connections to the singular value decomposition of a truncated linearized forward map, as well as to the explicit triangular structures originating from the properties of the employed Zernike polynomial basis for the conductivity.

math.NA↗

arXiv2024

Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations

We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,α}$ domain $Ω$. After extending the linearised problem for $L^2(Ω)$ perturbations, we orthogonally decompose $L^2(Ω) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(Ω)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(Ω)$ perturbation.

math.AP↗

arXiv2023

Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisons

We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calderón's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.

math.AP↗

arXiv2023

Series reversion for electrical impedance tomography with modeling errors

This work extends the results of [Garde and Hyvönen, Math. Comp. 91:1925-1953] on series reversion for Calderón's problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fréchet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.

math.NA↗

arXiv2022

Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

We extend recent results on discrete approximations of the Laplacian in $\mathbf{R}^d$ with norm resolvent convergence to the corresponding results for Dirichlet and Neumann Laplacians on a half-space. The resolvents of the discrete Dirichlet/Neumann Laplacians are embedded into the continuum using natural discretization and embedding operators. Norm resolvent convergence to their continuous counterparts is proven with a quadratic rate in the mesh size. These results generalize with a limited rate to also include operators with a real, bounded, and H\"older continuous potential, as well as certain functions of the Dirichlet/Neumann Laplacians, including any positive real power. Note (Nov 27, 2024): A corrigendum has been added to the end of the PDF.

math.FA↗

arXiv2022

Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

math-ph↗

arXiv2022

Reconstruction of singular and degenerate inclusions in Calderón's problem

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calderón's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

math.AP↗

arXiv2021

Series reversion in Calderón's problem

This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is $\nabla\cdot(A\nabla u)=0$ in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends $A$ to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Fréchet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model.

math.AP↗