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The fixed angle inverse scattering problem for a velocity consists in determining a sound speed, or a Riemannian metric up to diffeomorphism, from measurements obtained by probing the medium with a single plane wave. This is a formally determined inverse problem that is open in general. In this article we consider the rigidity question of distinguishing a sound speed or a Riemannian metric from the Euclidean metric. We prove that a general smooth metric that is Euclidean outside a ball can be distinguished from the Euclidean metric. The methods involve distorted plane waves and a combination of geometric, topological and unique continuation arguments.

We study the Lorentzian Calder\'on problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calder\'on problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same hyperbolic Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem. To prove these results we introduce a new method for geometric hyperbolic inverse problems. The method is based on distorted plane wave solutions and on a combination of geometric, topological and unique continuation arguments., Comment: 39 pages, 2 figures

We study the inverse backscattering problem for time-dependent potentials. We prove uniqueness and Lipshitz stability for the recovery of small potentials.

We develop a linearized boundary control method for the inverse boundary value problem of determining a density in the acoustic wave equation. The objective is to reconstruct an unknown perturbation in a known background density from the linearized Neumann-to-Dirichlet map. A key ingredient in the derivation is a linearized Blagovescenskii's identity with a free parameter. When the linearization is at a constant background density, we derive two reconstructive algorithms with stability estimates based on the boundary control method. When the linearization is at a non-constant background density, we establish an increasing stability estimate for the recovery of the density perturbation. The proposed reconstruction algorithms are implemented and validated with several numerical experiments to demonstrate the feasibility., Comment: 22 pages, 6 figures. arXiv admin note: text overlap with arXiv:2112.10976

In this paper, we consider the unique continuation problem for the Schr\"odinger equations. We prove a H\"older type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory.

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.

Recent developments in thin-film fabrication and processing open up interesting possibilities for both established and emerging optics technologies. There, one of the key questions requiring more complete understanding is by how much one can improve the performance of thin-film devices by utilizing resonance effects and surface texturation. In this work, we report on our recent theoretical investigations around two aspects of this question: (1) how much the overall (=angle and energy-integrated) emission of extremely thin ($\sim10$ nm) layers can be enhanced through cavity effects, and (2) how much resonances affect the emission of moderately thin ($>100$ nm) layers in a typical device interacting with free space (in this case an ultra-thin solar cell). Beginning with topic (1), we find that the total emission of active layers with thicknesses $<50$ nm in particular can be boosted through resonant effects by placing them in a cavity. For topic (2), the results indicate that a radiative transfer approach (i.e., one not accounting for resonant effects) can give even quantitatively accurate predictions of the total emission of moderately thin layers in a thin-film device, as long as the reflectances of the device's outer boundaries are known, and the emitting layer is not very close to optical elements supporting direct evanescent coupling (such as metal mirrors). Finally, we demonstrate that extending the self-consistent radiative transfer--drift-diffusion approach for diffusive scattering presents an interesting tool to optimize thin-film devices even with textured surfaces.

We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.

There are two main approaches to solve inverse coefficient determination problems for wave equations: the Boundary Control method and an approach based on geometric optics. These notes focus on the Boundary Control method, but we will have a brief look at the geometric optics as well., Comment: These lecture notes were written for CIRM SMF School Spectral Theory, Control and Inverse Problems, November 2022

We present a campaign designed to train the GRANDMA network and its infrastructure to follow up on transient alerts and detect their early afterglows. In preparation for O4 II campaign, we focused on GRB alerts as they are expected to be an electromagnetic counterpart of gravitational-wave events. Our goal was to improve our response to the alerts and start prompt observations as soon as possible to better prepare the GRANDMA network for the fourth observational run of LIGO-Virgo-Kagra (which started at the end of May 2023), and future missions such as SM. To receive, manage and send out observational plans to our partner telescopes we set up dedicated infrastructure and a rota of follow-up adcates were organized to guarantee round-the-clock assistance to our telescope teams. To ensure a great number of observations, we focused on Swift GRBs whose localization errors were generally smaller than the GRANDMA telescopes' field of view. This allowed us to bypass the transient identification process and focus on the reaction time and efficiency of the network. During 'Ready for O4 II', 11 Swift/INTEGRAL GRB triggers were selected, nine fields had been observed, and three afterglows were detected (GRB 220403B, GRB 220427A, GRB 220514A), with 17 GRANDMA telescopes and 17 amateur astronomers from the citizen science project Kilonova-Catcher. Here we highlight the GRB 220427A analysis where our long-term follow-up of the host galaxy allowed us to obtain a photometric redshift of $z=0.82\pm0.09$, its lightcurve elution, fit the decay slope of the afterglows, and study the properties of the host galaxy.

The CPT theorem originally proven by L\"uders and Pauli ensures the equality of masses, lifetimes, magnetic moments and cross sections of any particle and its antiparticle. We show that in a Lorentz invariant quantum field theory described by its Lagrangian, CPT-violating interaction alone does not split the masses of an elementary particle and its antiparticle but breaks only the equality of lifetimes, magnetic moments and cross sections. However, CPT violation in the mass term of a field in the Lagrangian, which can be attributed to be due to the size of the particle described by a form factor, breaks only the equality of masses. Also it is shown that the two separate effects of CPT violation in the interaction terms or in the mass term do not mix due to higher quantum corrections and remain distinguishable. Thus, we urge the experimentalists to search for such observable effects concerning differences in the masses, magnetic moments, lifetimes and cross sections between the elementary or bound state particles and their antiparticles. In the case of CPT violation only in the mass term, besides the difference in the masses of elementary bound state particles and their antiparticles, there will be also an extremely tiny difference in the lifetimes of bound states due to the difference in their phase spaces. From the details of calculations, it appears that the separate effects of the CPT violation described above are quite general, neither depending on how the nonlocality is achieved, nor depending on what this violation is due to: due to T violation, as considered in the present work, which can be attributed to a cosmological direction of time; to CP or to both T and CP violations. The latter two cases satisfy the Sakharov's conditions for explaining the baryon asymmetry in the Universe., Comment: 12 pages; 1 figure; v3: two references added, one reference removed, minor corrections; to appear in Phys. Lett. B

We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates., Comment: Final version with added details and corrections, to appear in Arch. Ration. Mech. Anal

We consider an inverse problem for a non-linear hyperbolic equation. We show that conformal structure of a Lorentzian manifold can be determined by the source-to-solution map evaluated along a single timelike curve. We use the microlocal analysis of non-linear wave interaction.

We identify a previously undetected periodicity at a frequency of 49.08$\pm$0.01 d$^{-1}$ (period of 29.34$\pm$0.01 minutes) during a super-outburst of V844 Her observed by TESS. V844 Her is an SU UMa type cataclysmic variable with an orbital period of 78.69 minutes, near the period minimum. The frequency of this new signal is constant in contrast to the superhump oscillations commonly seen in SU UMa outbursts. We searched without success for oscillations during quiescence using MDM, TESS, and XMM-Newton data. The lack of a periodic signal in the XMM light curve and the relatively low X-ray luminosity of V844 Her suggests that it is not a typical IP. We consider the possibility that the 29 min signal is the result of super-Nyquist sampling of a Dwarf Nova Oscillation with a period near the 2-minute cadence of the TESS data. Our analysis of archival AAVSO photometry from a 2006 super-outburst supports the existence of a 29 min oscillation, although a published study of an earlier superoutburst did not detect the signal. We compare the X-ray properties of V844 Her with short orbital period intermediate polars (IP), V1025 Cen and DW Cnc. We conclude that the new signal is a real photometric oscillation coming from the V844 Her system and that it is unlikely to be an aliased high-frequency oscillation. The steady frequency of the new signal suggests that its origin is related to an asynchronously rotating white dwarf in V844 Her, although the precise mechanism producing the flux variations remains unclear., Comment: 13 pages, 10 figures, accepted for publication in the Astrophysical Journal

Dust in the agricultural landscape is a significant challenge and influences, for example, the environmental perception of autonomous agricultural machines. Image enhancement algorithms can be used to reduce dust. However, these require dusty and dust-free images of the same environment for validation. In fact, to date, there is no dataset that we are aware of that addresses this issue. Therefore, we present the agriscapes RB-Dust dataset, which is named after its purpose of reference-based dust removal. It is not possible to take pictures from the cabin during tillage, as this would cause shifts in the images. Because of this, we built a setup from which it is possible to take images from a stationary position close to the passing tractor. The test setup was based on a half-sided gate through which the tractor could drive. The field tests were carried out on a farm in Bavaria, Germany, during tillage. During the field tests, other parameters such as soil moisture and wind speed were controlled, as these significantly affect dust development. We validated our dataset with contrast enhancement and image dehazing algorithms and analyzed the generalizability from recordings from the moving tractor. Finally, we demonstrate the application of dust removal based on a high-level vision task, such as person classification. Our empirical study confirms the validity of RB-Dust for vision-based dust removal in agriculture., Comment: Accepted by CVPR Workshop NTIRE 2023. Errata: Caption Figure 6 changed

We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration.

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider., Comment: 42 pages, 3 figures; added numerical examples (appendix A)

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

We consider the inverse problem to determine a smooth compact Riemannian manifold $(M,g)$ from a restriction of the source-to-solution operator, $\Lambda_{\mathcal{S,R}}$, for the wave equation on the manifold. Here, $\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and $\Lambda_{\mathcal{S,R}}$ represents the measurements of waves produced by smooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. We emphasise that $\overline{\mathcal{S}}$ and $\overline{\mathcal{R}}$ could be disjoint. We demonstrate that $\Lambda_{\mathcal{S,R}}$ determines the manifold $(M,g)$ uniquely under the following spectral bound condition for the set $\mathcal{S}$: There exists a constant $C>0$ such that any normalized eigenfunction $\phi_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies \begin{equation*} 1\leq C\|\phi_k\|_{L^2(\mathcal{S})}. \end{equation*} We note that, for the Anosov surface, this spectral bound condition is fulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based on the paper [18] and the spectral bound condition above is an analogue of the Hassell-Tao condition there., Comment: 16 pages

The European Spallation Source (ESS) in Lund, Sweden will become the world's most powerful thermal neutron source. The Macromolecular Diffractometer (NMX) at the ESS requires three 51.2 x 51.2~cm$^{2}$ detectors with reasonable detection efficiency, sub-mm spatial resolution, a narrow point spread function (PSF) and good time resolution. This work presents measurements with the improved version of the NMX detector prototype consisting of a Triple-GEM detector with natural Gd converter and a low material budget readout. The detector was successfully tested at the neutron reactor of the Budapest Neutron Centre (BNC) and at the D16 instrument at the Institut Laue-Langevin (ILL) in Grenoble. The measurements with Cadmium and Gadolinium masks in Budapest demonstrate that the point spread function of the detector lacks long tails that could impede the measurement of diffraction spot intensities. On the D16 instrument at ILL, diffraction spots from Triose phosphate isomerase w/ 2-phosphoglycolate (PGA) inhibitor were measured both in the D16 Helium-3 detector and the Gd-GEM. The comparison between the two detectors show a similar point spread function in both detectors, and the expected efficiency ratio compared to the Helium-3 detector. Both measurements together thus give good indications that the Gd-GEM detector fits the requirements for the NMX instrument at ESS.

The $R+R^2$ model of gravity with the corresponding shallow potential in the Einstein frame is consistent with the observations. Recently, many efforts have been made to generalize the $R+R^2$ (Starobinsky) model of inflation or use other shallow potentials to construct a model for the early Universe. We revise the question about the shallow potential. We propose a model in which the Starobinsky model can emerge through a dynamical mechanism. We show that the absence of ghost modes results to constraints on the parameters of the Starobinsky model. We obtain the scalar spectral index and the tensor-to-scalar ratio of the extended model and study the three-point correlation function of the curvature perturbation to estimate the primordial non-Gaussianities of the proposed model., Comment: 24 pages, 4 figures. v4: corrections to language and references; added affiliation; to appear in Phys. Rev. D. arXiv admin note: substantial text overlap with arXiv:2012.10889

V392 Persei is a known dwarf nova (DN) that underwent a classical nova eruption in 2018. Here we report ground-based optical, Swift UV and X-ray, and Fermi-LAT \gamma-ray observations following the eruption for almost three years. V392 Per is one of the fastest evolving novae yet observed, with a $t_2$ decline time of 2 days. Early spectra present evidence for multiple and interacting mass ejections, with the associated shocks driving both the \gamma-ray and early optical luminosity. V392 Per entered Sun-constraint within days of eruption. Upon exit, the nova had evolved to the nebular phase, and we saw the tail of the super-soft X-ray phase. Subsequent optical emission captured the fading ejecta alongside a persistent narrow line emission spectrum from the accretion disk. Ongoing hard X-ray emission is characteristic of a standing accretion shock in an intermediate polar. Analysis of the optical data reveals an orbital period of 3.230 \pm 0.003 days, but we see no evidence for a white dwarf (WD) spin period. The optical and X-ray data suggest a high mass WD, the pre-nova spectral energy distribution (SED) indicates an evolved donor, and the post-nova SED points to a high mass accretion rate. Following eruption, the system has remained in a nova-like high mass transfer state, rather than returning to the pre-nova DN low mass transfer configuration. We suggest that this high state is driven by irradiation of the donor by the nova eruption. In many ways, V392 Per shows similarity to the well-studied nova and DN GK Persei., Comment: Accepted by MNRAS. 20 pages including references, with 37 pages of supplementary material

We show that we can retrieve a Yang--Mills potential and a Higgs field (up to gauge) from source-to-solution type data associated with the classical Yang--Mills--Higgs equations in Minkowski space $\mathbb{R}^{1+3}$. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits the non-linear interaction of waves generated by sources with values in the centre of the Lie algebra showing that abelian components can be used effectively to recover the Higgs field.

We show uniqueness results for the anisotropic Calder\'{o}n problem stated on transversally anisotropic manifolds. Moreover, we give a convexity result for the range of Dirichlet-to-Neumann maps on general Riemannian manifolds near the zero potential. Finally, we present results for Calder\'{o}n type inverse problems associated to semilinear elliptic equations on general Riemannian manifolds.

We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface., Comment: to appear in Comm. PDE

The massive outburst of the comet 17P/Holmes in October 2007 is the largest known outburst by a comet thus far. We present a new comprehensive model describing the evolution of the dust trail produced in this phenomenon. The model comprises of multiparticle Monte Carlo approach including the solar radiation pressure effects, gravitational disturbance caused by Venus, Earth and Moon, Mars, Jupiter and Saturn, and gravitational interaction of the dust particles with the parent comet itself. Good accuracy of computations is achieved by its implementation in Orekit, which executes Dormad-Prince numerical integration methods with higher precision. We demonstrate performance of the model by simulating particle populations with sizes from 0.001 mm to 1 mm with corresponding spherically symmetric ejection speed distribution, and towards the Sun outburst modelling. The model is supplemented with and validated against the observations of the dust trail in common nodes for 0.5 and 1 revolutions. In all cases, the predicted trail position showed a good match to the observations. Additionally, the hourglass pattern of the trail was observed for the first time within this work. By using variations of the outburst model in our simulations, we determine that the assumption of the spherical symmetry of the ejected particles leads to the scenario compatible with the observed hourglass pattern. Using these data, we make predictions for the two-revolution dust trail behavior near the outburst point that should be detectable by using ground-based telescopes in 2022., Comment: 70 pages, 29 main figures, open access in MNRAS

In this paper we consider determining a minimal surface embedded in a Riemannian manifold $\Sigma\times \mathbb{R}$. We show that if $\Sigma$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $\Sigma$ up to an isometry., Comment: 18 pages

We consider an inverse problem of recovering the unknown coefficients $\beta(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schr\"odinger equation $ (\mathrm{i} \partial_t +\Delta +V)u + \beta u^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $\Omega \subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{\beta, V}$ that maps a source $f$ supported in $ \Omega\times (0,T) $ to the restriction of the solution $u$ in $ \Omega\times (0,T) $. We show that the map $L_{\beta, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schr\"odinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $\beta |u|^2 \, u$, that is encountered in quantum physics., Comment: 31 pages

We develop a linearized boundary control method for the inverse boundary value problem of determining a potential in the acoustic wave equation from the Neumann-to-Dirichlet map. When the linearization is at the zero potential, we derive a reconstruction formula based on the boundary control method and prove that it is of Lipschitz-type stability. When the linearization is at a nonzero potential, we prove that the problem is of H\"{o}lder-type stability. The proposed reconstruction formula is implemented and evaluated using several numerical experiments to validate its feasibility., Comment: 19 pages, 9 figures

We study a Lorentzian version of the well-known Calder\'{o}n problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound as well as spacetime perturbations of such manifolds. This relies on a new optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calder\'{o}n problem near the Minkowski geometry., Comment: Comments are welcome

We report on the masses ($M_\mathrm{WD}$), effective temperatures ($T_\mathrm{eff}$) and secular mean accretion rates ($\langle \dot{M} \rangle$) of 43 cataclysmic variable (CV) white dwarfs, 42 of which were obtained from the combined analysis of their $\mathit{Hubble~Space~Telescope}$ ultraviolet data with the parallaxes provided by the Early Third Data Release of the $\mathit{Gaia}$ space mission, and one from the white dwarf gravitational redshift. Our results double the number of CV white dwarfs with an accurate mass measurement, bringing the total census to 89 systems. From the study of the mass distribution, we derive $\langle M_\mathrm{WD} \rangle = 0.81^{+0.16}_{-0.20}\,\mathrm{M_\odot}$, in perfect agreement with previous results, and find no evidence of any evolution of the mass with orbital period. Moreover, we identify five systems with $M_\mathrm{WD} < 0.5\mathrm{M_\odot}$, which are most likely representative of helium-core white dwarfs, showing that these CVs are present in the overall population. We reveal the presence of an anti-correlation between the average accretion rates and the white dwarf masses for the systems below the $2-3\,$h period gap. Since $\langle \dot{M} \rangle$ reflects the rate of system angular momentum loss, this correlation suggests the presence of an additional mechanism of angular momentum loss that is more efficient at low white dwarf masses. This is the fundamental concept of the recently proposed empirical prescription of consequential angular momentum loss (eCAML) and our results provide observational support for it, although we also highlight how its current recipe needs to be refined to better reproduce the observed scatter in $T_\mathrm{eff}$ and $\langle \dot{M} \rangle$, and the presence of helium-core white dwarfs., Comment: Accepted for publication in MNRAS. 22 pages of main body (6 tables and 14 figures) and 54 pages of appendices. Appendix B includes the best-fitting parameters and models to the HST data

We develop the Seiberg-Witten map using the gauge-covariant star product with the noncommutativity tensor $\theta^{\mu\nu}(x)$. The latter guarantees the Lorentz invariance of the theory. The usual form of this map and its other recent generalizations do not consider such a covariant star product. We construct the Seiberg-Witten map for the gauge parameter, the gauge field and the strength tensor to the first order in the noncommutativity parameter $\theta^{\mu\nu}(x)$. Prescription for the generalization of the map to higher orders is also given. Interestingly, the associativity of the covariant star product both in the first and second orders requires the same constraints, namely, on the $\theta^{\mu\nu}(x)$ and on the space-time connection. This fact suggests that the same constraints could be enough to ensure the associativity in all orders. The resulting Seiberg-Witten map applies both to the internal and space-time gauge theories. Comparisons with the Seiberg-Witten map based on other (non-covariant) star products are given and some characteristic properties are also presented. As an application, we consider the $GL(2, C)$ noncommutative gauge theory of gravitation, in which it is shown that the connection determines a space-time with symplectic structure (as proposed by Zumino et al [AIP Conf. Proc. 1200 (2010), 204, arXiv:0910.0459]). This example shows that the constraints required for the associativity of the gauge-covariant star product can be satisfied. The presented $GL(2, C)$ noncommutative gauge theory of gravitation is also compared to the one (given by Chamseddine [Phys. Rev. D 69 (2004), 024015, hep-th/0309166]) with non-covariant star product., Comment: 32 pages