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Rakesh

Publications and source records attributed to Rakesh.

18 recordsLinked to original sources
arXiv2026

Semiglobal uniqueness for the Lorentzian Calder\'on problem

We prove uniqueness in the Lorentzian Calder\'on problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted $L^2$-estimates.

math.AP↗
arXiv2026

Fixed angle inverse scattering with non-constant velocity

In this article, we study formally determined inverse problems for wave equations in the presence of a variable sound speed. We prove that by measuring the boundary data of finitely many plane waves and their complementary solutions, one can uniquely recover the unknown coefficients of the highest order terms of a second order hyperbolic operator with time independent coefficients. This improves earlier rigidity results in [10], [11] which compared the wave operator generated by a Riemannian metric with the wave operator generated by the Euclidean metric. We compare two general second order hyperbolic operators with time independent coefficients, with the same lower order terms. However, we require the geometry associated with one of the operators to satisfy a pseudoconvexity condition, a no-caustics condition, and a spanning condition. In particular one of the operators could be a wave operator with the sound speed close to a constant and the other operator could be arbitrary. To prove the results, we introduce the notion of a complementary solution for a generalized plane wave solution generated by an incoming plane wave. The complementary solution extends smoothly, across an interface, the generalized plane wave. The unknown coefficients appear in a transport equation at the interface. We show that the unknown coefficients in the interior can be extracted from this transport equation, from the boundary data, via a sequence of Carleman estimates for the wave operator.

math.AP↗
arXiv2024

Rigidity in fixed angle inverse scattering for Riemannian metrics

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.

math.AP↗
arXiv2024

Rigidity in the Lorentzian Calder\'on problem with formally determined data

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.

math.AP↗
arXiv2022

Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients

We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheim-Klibanov method based on Carleman estimates.

math.AP↗
arXiv2020

Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients

We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.

math.AP↗
arXiv2019

Fixed angle inverse scattering for almost symmetric or controlled perturbations

We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally controlled potentials are uniquely determined by their fixed angle scattering data. This is done by establishing an equivalence between the frequency domain and the time domain formulations of the problem, and by solving the time domain problem by extending the methods of [RS19] which adapts the ideas introduced in [BK81] and [IY01] on the use of Carleman estimates for inverse problems.

math.AP↗
arXiv2019

The fixed angle scattering problem and wave equation inverse problems with two measurements

We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the far field pattern generated by plane waves coming from exactly two opposite directions. This implies that a reflection symmetric potential is uniquely determined by its fixed angle scattering data. We also prove a Lipschitz stability estimate for an associated problem. Motivated by the point source inverse problem in geophysics, we show that a compactly supported potential is uniquely determined from boundary measurements of the waves generated by exactly two sources - a point source and an incoming spherical wave. These results are proved by using Carleman estimates and adapting the ideas introduced by Bukhgeim and Klibanov on the use of Carleman estimates for inverse problems.

math.AP↗
arXiv2018

Recovering initial values from light cone traces of solutions of the wave equation

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R^n.

math.AP↗
arXiv2018

Utilizing Semantic Visual Landmarks for Precise Vehicle Navigation

This paper presents a new approach for integrating semantic information for vision-based vehicle navigation. Although vision-based vehicle navigation systems using pre-mapped visual landmarks are capable of achieving submeter level accuracy in large-scale urban environment, a typical error source in this type of systems comes from the presence of visual landmarks or features from temporal objects in the environment, such as cars and pedestrians. We propose a gated factor graph framework to use semantic information associated with visual features to make decisions on outlier/ inlier computation from three perspectives: the feature tracking process, the geo-referenced map building process, and the navigation system using pre-mapped landmarks. The class category that the visual feature belongs to is extracted from a pre-trained deep learning network trained for semantic segmentation. The feasibility and generality of our approach is demonstrated by our implementations on top of two vision-based navigation systems. Experimental evaluations validate that the injection of semantic information associated with visual landmarks using our approach achieves substantial improvements in accuracy on GPS-denied navigation solutions for large-scale urban scenarios

cs.CV↗
arXiv2017

Input-to-State Stability of Periodic Orbits of Systems with Impulse Effects via Poincar\'e Analysis

In this paper we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincar\'e maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding 0-input fixed point of the associated \emph{forced} Poincar\'e map. This result extends the classical Poincar\'e analysis for asymptotic stability of periodic solutions to establish orbital input-to-state stability of such solutions under external excitation. In our proof, we define the forced Poincar\'e map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the 0-input (unforced) Poincar\'e map and ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincar\'e analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.

eess.SY↗
arXiv2015

Determining the twist in an optical fiber

We determine the twist in a birefringent optical fiber from measurements, at one end of the fiber, of the fiber response to an impulsive source at the same end. This is the inverse problem of determining a non-constant coefficient, of a first order hyperbolic system in one space dimension with two speeds of propagation, from measurements at one end of an interval, of the solution of this system corresponding to an impulsive source at the same end. We prove a stability result for this inverse problem and give a provable reconstruction algorithm for this inverse problem.

math.AP↗
arXiv2014

The point source inverse back-scattering problem

We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering problem. We show that if the angular derivatives of the difference of two potentials having the same data is controlled by the L^2 norm of the difference of the potentials they must be equal. In particular this shows injectivity of the inverse problem for radial potentials.

math.AP↗
arXiv2013

Uniqueness for the inverse backscattering problem for angularly controlled potentials

We consider the problem of recovering a smooth, compactly supported potential on R^3 from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled angular derivatives then the two potentials are identical. In particular, if two potentials differ by a finite linear combination of spherical harmonics with radial coefficinets and have the same backscattering data then the two potentials are identical.

math.AP↗
arXiv2010

Uniqueness for a hyperbolic inverse problem with angular control on the coefficients

Suppose $q_i(x)$, $i=1,2$ are smooth functions on $\R^3$ and $U_i(x,t)$ the solutions of the initial value problem {gather*} \pa_t^2 U_i- \Delta U_i - q_i(x) U_i = \delta(x,t), \qquad (x,t) \in \R^3 \times \R U_i(x,t) =0, \qquad \text{for} ~ t<0. {gather*} Pick $R,T$ so that $0 < R < T$ and let $C$ be the vertical cylinder $\{(x,t) \, : |x|=R, ~ R \leq t \leq T \}$. We show that if $(U_1, U_{1r}) = (U_2, U_{2r})$ on $C$ then $q_1 = q_2$ on the annular region $R \leq |x| \leq (R+T)/2$ provided there is a $\gamma>0$, independent of $r$, so that \[\int_{|x|=r} | \Delta_S (q_1 - q_2)|^2 \, dS_x \leq \gamma \int_{|x|=r} |q_1 - q_2|^2 \, dS_x, \qquad \forall r \in [R, (R+T)/2].\] Here $\Delta_S$ is the spherical Laplacian on $|x|=r$.

math.AP↗
arXiv2010

Spherical means with centers on a hyperplane in even dimensions

Given a real valued function on R^n we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with great simplicity and economy. We apply their method to derive an inversion formula for the even n case. A feature of our inversion formula, for the even n case, is that it does not require the Fourier transform of the mean values or the use of the Hilbert transform, unlike the previously known inversion formulas for the even n case. Along the way, we extend the isometry identity of Bukhgeim and Kardakov for odd n, for solutions of the wave equation, to the even n case.

math.AP↗
arXiv2007

Inversion of spherical means and the wave equation in even dimensions

We establish inversion formulas of the so called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form (f,0) (or (0,g)) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided the initial data has support in the ball.

math.AP↗