120 research outputs found
Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation
No full textAn inverse boundary value problem for the 1+1 dimensional wave equation
(
∂
2
t
−
c
(
x
)
2
∂
2
x
)
u
(
x
,
t
)
=
0
,
x
∈
R
+
is considered. We give a discrete regularization strategy to recover wave speed
c
(
x
)
when we are given the boundary value of the wave,
u
(
0
,
t
)
, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed
˜
c
, satisfying a Hölder type estimate
∥
˜
c
−
c
∥
≤
C
ϵ
γ
, where
ϵ
is the noise lev
Uniqueness for a wave propagation inverse problem in a half space
No full textLassas, Matti; Cheney, M.; Uhlmann, Gunther. (1997). Uniqueness for a wave propagation inverse problem in a half space. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/3152
Determination of a Riemannian manifold from the distance difference functions
No full textLet (N, g) be a Riemannian manifold with the distance function d(x, y) and an open subset M subset of N. For x is an element of M we denote by D-x the distance difference function D-x:F x F -> R, given by D-x(z(1), z(2)) = d(x, z(1)) - d(x, z(2)), z(1), z(2) is an element of F = N \ M. We consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric g vertical bar M on it when we are given the distance difference data, that is, the set F, the metric g vertical bar F, and the collection D(M) = {D-x; x is an element of M}. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F x F), as a representation of manifold M. The inverse problem of determining (M, g) from D(M) arises e.g. in the study of the wave equation on R x N when we observe in F the waves produced by spontaneous point sources at unknown points (t, x) is an element of R x M. Then D-x (z(1), z(2)) is the difference of the times when one observes at points z(1) and z(2) the wave produced by a point source at x that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.Peer reviewe
Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations
No full textWe study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are: / 1. Passive observations in spacetime: consider observations in an open set V⊂M . The light observation set corresponding to a point source at q∈M is the intersection of V and the light-cone emanating from the point q.
Let W⊂M be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points q∈W determine uniquely the conformal type of W. / 2. Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path μ and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from μ and return back to μ
QUANTITATIVE STABILITY OF GEL’FAND’S INVERSE BOUNDARY PROBLEM
No full textIn Gel’fand’s inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold M with boundary from the knowledge of the boundary ∂M, the Neumann eigenvalues λj and the boundary values of the eigenfunctions ϕj |∂M. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov–Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator.Peer reviewe
Deep Neural Networks for Inverse Problems with Pseudodifferential Operators: An Application to Limited-Angle Tomography
No full textWe propose a novel convolutional neural network (CNN), called \Psi DONet, designed for learning pseudodifferential operators (\Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting
minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our \Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle Xray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of \Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are \Psi DOs or Fourier integral operators
Shearlet-based regularization in sparse dynamic tomography
No full textClassical tomographic imaging is soundly understood and widely employed in medicine, nondestructive testing and security applications. However, it still o↵ers many challenges when it comes to dynamic tomography. Indeed, in classical tomography, the target is usually assumed to be stationary during the data acquisition, but this is not a realistic model. Moreover, to ensure a lower X-ray radiation dose, only a sparse collection of measurements per time step is assumed to be available. With such a set up, we deal with a sparse data, dynamic tomography problem, which clearly calls for regularization, due to the loss of information in the data and the ongoing motion. In this paper, we propose a 3D variational formulation based on 3D shearlets, where the third dimension accounts for the motion in time, to reconstruct a moving 2D object. Results are presented for real measured data and compared against a 2D static model, in the case of fan-beam geometry. Results are preliminary but show that better reconstructions can be achieved when motion is taken into account
Geometric Whitney problem: Reconstruction of a manifold from a point cloud
No full text"We study the geometric Whitney problem on how a Riemannian manifold can be constructed to approximate a metric space . This problem is closely related to manifold interpolation (or manifold learning) where a smooth -dimensional surface , needs to be constructed to approximate a point cloud in . These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data. The results are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan.
References: [1] C. Fefferman, S. Ivanov, Y. Kurylev, M. Lassas, H. Narayanan: Reconstruction and interpolation of manifolds I: The geometric Whitney problem. ArXiv:1508.00674"Non UBCUnreviewedAuthor affiliation: University of HelsinkiFacult
Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets
No full textInternational audienceWe consider a restricted Dirichlet-to-Neumann map associated with the operatorwhere is the Laplace-Beltrami operator of a Riemannian manifold , and and are a vector field and a function on .The restriction corresponds to the case where the Dirichlet traces are supported on and the Neumann traces are restricted on .Here and are open sets, which may be disjoint, on the boundary of .We show that determines uniquely, up the natural gauge invariance, the lower order terms and in a neighborhood of the set assuming that is strictly convex and thatthe wave equation is exactly controllable from in time . We give also a global result under a convex foliation condition.The main novelty is the recovery of and when the sets and are disjoint.We allow and to be non-self-adjoint,and in particular, the corresponding physical system may have dissipation of energy
Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography
No full textWe propose a novel convolutional neural network (CNN), called PsiDONet, designed for learning pseudodifferential operators (PsiDOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our PsiDONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of PsiDONet on simulated data from limited angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are PsiDOs or Fourier integral operators
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