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We propose a volatile static all-optical memory capable of storing phase information of a slowly-varying electric field. The scheme and its realization (a memory circuit) are based on two mutually coupled lasers subject to external optical injection. The proposed circuit has a single optical input for write and hold operations and two opposite-sign outputs for reading the memory. The proposed circuit operates with a single wavelength of light, a single direction of propagation, and without a need to switch the state of polarization. We prove mathematically that the proposed arrangement has equilibrium points that may discreetly quantify and store the phase in a bistable manner. The circuit is studied numerically for solid-state and semiconductor lasers with zero and non-zero linewidth enhancement factors, respectively. Simulations based on a rate equation system confirm the essential findings. Using typical parameters of a semiconductor laser and optimizing for a possibly wide range of operation, the write-read operations were simulated using PRBS-9 at the rate of 1 Gb/s with negligible errors. The proposed circuit will enable integrated memory implementations for future all-optical signal processing and computing systems.

We consider the recovery of a potential associated with a semi-linear wave equation on Rn+1, n > 1. We show that an unknown potential a(x, t) of the wave equation ???u + aum = 0 can be recovered in a H & ouml;lder stable way from the map u|onnx[0,T] ???-> (11, avu|ac >= x[0,T])L2(oc >= x[0,T]). This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function ???. We also prove similar stability result for the recovery of a when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation ???u + aum = 0. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Superscript/Subscript Available

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., to appear in Comm. PDE

Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end -- rather than layerwise -- doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks., 48 pages, 18 figures, submitted to JMLR

An all-optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should utilize nonlinearity that overcomes weaknesses that plague many optical processing schemes. An advantageous nonlinearity provides a complete set of logic operations and allows cascaded operations without changes in wavelength or in signal encoding format. Here we demonstrate an all-optical majority gate based on a vertical-cavity surface-emitting laser (VCSEL). Using emulated signal coupling, the arrangement provides Bit Error Ratio (BER) of 10−6 at the rate of 1 GHz without changes in the wavelength or in the signal encoding format. Cascaded operation of the injection-locked laser majority gate is simulated on a full adder and a 3-bit ripple-carry adder circuits. Finally, utilizing the spin-flip model semiconductor laser rate equations, we prove that injection-locked lasers may perform normalization operations in the steady-state with an arbitrary linear state of polarization.

The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension n + 1 >= 3 and with partial data. We consider the case when the set Omega(in), where the sources are supported, and the set Omega(out), where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point p(in) Omega(in) and the past of the point p(out) Omega(out). In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in Omega(in) and observations in Omega(out), determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.

We study approximation of probability measures supported on $n$-dimensional manifolds embedded in $\mathbb{R}^m$ by injective flows -- neural networks composed of invertible flows and injective layers. We show that in general, injective flows between $\mathbb{R}^n$ and $\mathbb{R}^m$ universally approximate measures supported on images of extendable embeddings, which are a subset of standard embeddings: when the embedding dimension m is small, topological obstructions may preclude certain manifolds as admissible targets. When the embedding dimension is sufficiently large, $m \ge 3n+1$, we use an argument from algebraic topology known as the clean trick to prove that the topological obstructions vanish and injective flows universally approximate any differentiable embedding. Along the way we show that the studied injective flows admit efficient projections on the range, and that their optimality can be established "in reverse," resolving a conjecture made in Brehmer and Cranmer 2020., 26 pages, 5 figures

Publisher Copyright: © 2021 Neural information processing systems foundation. All rights reserved. In this work, we consider the linear inverse problem y = Ax+ε, where A: X → Y is a known linear operator between the separable Hilbert spaces X and Y, x is a random variable in X and ε is a zero-mean random process in Y . This setting covers several inverse problems in imaging including denoising, deblurring and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori, but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator A and depends only on the mean and covariance of x. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both x and y, and one unsupervised, based only on samples of x. In both cases we prove generalization bounds, under some weak assumptions on the distribution of x and ε, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.

An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points, however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed., Final version, to appear in SIAM Journal on Applied Dynamical Systems (SIADS)

We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calder\'on problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions., Comment: 25 pages

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging., Comment: 31 pages, 11 figures, 4 tables

In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem., Comment: 48 pages, minor corrections after refereeing process

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem., Comment: 27 pages

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations., Comment: 48 pages

We consider a restricted Dirichlet-to-Neumann map Λ S , R T associated with the operator ∂ t 2 − Δ g + A + q where Δ g is the Laplace-Beltrami operator of a Riemannian manifold ( M , g ) , and A and q are a vector field and a function on M. The restriction Λ S , R T corresponds to the case where the Dirichlet traces are supported on ( 0 , T ) × S and the Neumann traces are restricted on ( 0 , T ) × R . Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Λ S , R T determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T / 2 . We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

In this paper we consider an inverse problem for the $n$-dimensional random Schr\"{o}dinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential $q$, we uniquely determine the principal symbol of the covariance operator of $q$. Especially, for $n=3$ this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance., Comment: Previous version 48 pages; Current version 51 pages, 3 figures, several references have been added

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let $\{X_1,X_2,\dots,X_N\}$ bea set of $N$ sample points sampled randomly from an unknown Riemannian $M$ manifold. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+\eta_{jk}$, where $j,k\in \{1,2,\dots,N\}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $\eta_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|\eta_{jk}|}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.

Electrical impedance tomography (EIT) is an emerging non-invasive medical imaging modality. It is based on feeding electrical currents into the patient, measuring the resulting voltages at the skin, and recovering the internal conductivity distribution. The mathematical task of EIT image reconstruction is a nonlinear and ill-posed inverse problem. Therefore any EIT image reconstruction method needs to be regularized, typically resulting in blurred images. One promising application is stroke-EIT, or classification of stroke into either ischemic or hemorrhagic. Ischemic stroke involves a blood clot, preventing blood flow to a part of the brain causing a low-conductivity region. Hemorrhagic stroke means bleeding in the brain causing a high-conductivity region. In both cases the symptoms are identical, so a cost-effective and portable classification device is needed. Typical EIT images are not optimal for stroke-EIT because of blurriness. This paper explores the possibilities of machine learning in improving the classification results. Two paradigms are compared: (a) learning from the EIT data, that is Dirichlet-to-Neumann maps and (b) extracting robust features from data and learning from them. The features of choice are virtual hybrid edge detection (VHED) functions (Greenleaf et al 2018 Anal. PDE 11) that have a geometric interpretation and whose computation from EIT data does not involve calculating a full image of the conductivity. We report the measures of accuracy, sensitivity and specificity of the networks trained with EIT data and VHED functions separately. Computational evidence based on simulated noisy EIT data suggests that the regularized grey-box paradigm (b) leads to significantly better classification results than the black-box paradigm (a).

The high complexity of various inverse problems poses a significant challenge to model-based reconstruction schemes, which in such situations often reach their limits. At the same time, we witness an exceptional success of data-based methodologies such as deep learning. However, in the context of inverse problems, deep neural networks mostly act as black box routines, used for instance for a somewhat unspecified removal of artifacts in classical image reconstructions. In this paper, we will focus on the severely ill-posed inverse problem of limited angle computed tomography, in which entire boundary sections are not captured in the measurements. We will develop a hybrid reconstruction framework that fuses model-based sparse regularization with data-driven deep learning. Our method is reliable in the sense that we only learn the part that can provably not be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to the remaining parts. Such a decomposition into visible and invisible segments is achieved by means of the shearlet transform that allows to resolve wavefront sets in the phase space. Furthermore, this split enables us to assign the clear task of inferring unknown shearlet coefficients to the neural network and thereby offering an interpretation of its performance in the context of limited angle computed tomography. Our numerical experiments show that our algorithm significantly surpasses both pure model- and more data-based reconstruction methods.

Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

The paper considers the transmission problems for Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero the fields may blow up. When the speed of the blow up is suitable, this is called the Anomalous Localized Reconance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak Anomalous Localized Reconance (w-ALR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-ALR or ALR may, or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary neither ALR nor w-ALR appears with non-zero frequencies (i.e. in the dynamic range) in dimensions $d\ge 3$. In the case when the boundary of the negative material contains a flat subset we show that the w-ALR always happens with some point sources in dimensions $d\ge 2$. These results, together with the earlier results of Milton et al. ( [22, 23]) and Ammari et al. ([2]) show that for strictly convex bodies ALR may appear only for bodies so small that the quasi-static approximation is realistic. This gives limits for size of the objects for which invisibility cloaking methods based on ALR may be used., Comment: 30 pages, 7 figures

In 1995, Tataru proved a Carleman-type estimate for linear operators with partially analytic coefficients that is generally used to prove the unique continuation of those operators. In this paper, we use this inequality to study the stability of the unique continuation in the case of the wave equation with coefficients independent of time. We prove a logarithmic estimate in a ball whose radius has an explicit dependence on the C (1)-norm of the coefficients and on the other geometric properties of the operator.

An inverse boundary value problem for the 1+1 dimensional wave equation \begin{document}$ (\partial_t^2 - c(x)^2 \partial_x^2)u(x,t) = 0,\quad x\in\mathbb{R}_+ $\end{document} is considered. We give a discrete regularization strategy to recover wave speed \begin{document}$ c(x) $\end{document} when we are given the boundary value of the wave, \begin{document}$ u(0,t) $\end{document} , that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed \begin{document}$ \widetilde c $\end{document} , satisfying a Holder type estimate \begin{document}$ \| \widetilde c-c\|\leq C \epsilon^{\gamma} $\end{document} , where \begin{document}$ \epsilon $\end{document} is the noise level.

Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same data, they have identical laws as function-valued random variables., Comment: 16 pages

Classical tomographic imaging is soundly understood and widely employed in medicine, nondestructive testing and security applications. However, it still offers 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.

We experimentally demonstrate BER reduction and orthogonal modulation using an injection locked single-mode VCSEL. It allows us suppressing an amplitude noise of optical signal and/or double the capacity of an information channel.

It has been shown earlier, that the injection locked semiconductor lasers enable effective amplitude noise suppression [1] and makes possible an extra level of signal multiplexing-orthogonal modulation [2], where DPSK and ASK NRZ channels propagate at the same wavelength [3]. In our work we use an injection-locked 1550 nm VCSEL as a slave laser providing separation of amplitude and phase modulations, carrying independent information flows. To validate the possibility of phase modulation extraction by an injection-locked VCSEL, an experimental setup shown in Fig. 1 has been built.

The ill-posedness of Calder\'on's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. Here we show how, using EIT data alone, to efficiently detect interior jumps and other singularities of the conductivity. Analysis of the complex geometrical optics solutions of Astala and P\"aiv\"arinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes it possible to exploit an underlying complex principal type structure of the problem. We show that the leading term in a Neumann series is an invertible nonlinear generalized Radon transform of the conductivity. The wave front set of all higher-order terms can be characterized, and, under a prior, some are smoother than the leading term. Numerics indicate that this approach effectively detects inclusions within inclusions via EIT., Comment: Revised with additional commentary and references. 54 pages, 10 figures

Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation ∂ t 2 u − Δ g u = χ W , where W is a random variable with the white noise statistics on R 1 + n , n ≥ 3 , χ is a smooth function vanishing for negative times and outside a compact set in space, and Δ g is the Laplace–Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R n . The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set X ⊂ R n , C T ( t 1 , x 1 , t 2 , x 2 ) = 1 T ∫ 0 T u ( t 1 + s , x 1 ) u ( t 2 + s , x 2 ) d s , t 1 , t 2 > 0 , x 1 , x 2 ∈ X , for T > 0 . Supposing that χ is non-zero on X and constant in time after t > 1 , we show that in the limit T → ∞ , the data C T becomes statistically stable, that is, independent of the realization of W. Our main result is that, with probability one, this limit determines the Riemannian manifold ( R n , g ) up to an isometry.

Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\in F=N\setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric $g|_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g|_F$, and the collection $\mathcal D(M)=\{D_x;\ x\in M\}$. Moreover, we consider the embedded image $\mathcal D(M)$ of the manifold $M$, in the vector space $C(F\times F)$, as a representation of manifold $M$. The inverse problem of determining $(M,g)$ from $\mathcal D(M)$ arises e.g. in the study of the wave equation on $\mathbb R\times N$ when we observe in $F$ the waves produced by spontaneous point sources at unknown points $(t,x)\in \mathbb R\times M$. 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., An extended preprint version of this paper is at the page http://wiki.helsinki.fi/display/mathstatHenkilokunta/Teemu+Saksal

We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.

We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space $$(X,d_X)$$ ( X , d X ) . This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold $$S\subset {{\mathbb {R}}}^m$$ S ⊂ R m , $$m>n$$ m > n needs to be constructed to approximate a point cloud in $${{\mathbb {R}}}^m$$ R m . 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. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize 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. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in $${{\mathbb {R}}}^m$$ R m and interpolated to a smooth submanifold.

We consider illumination systems and mathematical algorithms for determination of the anisotropy and topographical features of an illuminated surface from its reflection images. As a particular example we study determination of the fiber orientation of paper surface. We also consider illumination systems with multiple light sources, and introduce optimization algorithms that exploit different spectral bands of these light sources. We show that a system of three light sources, e.g., a blue, green and red LED placed in a regular triangular form, efficiently prevents distortion of the above features. It is also easy to implement in applications, e.g., of the paper industry. We furthermore show that a pentagonal illumination system would be even better. However, in this case a five-colour photographic system and the corresponding light sources would be needed.

We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M,g)$. The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood $V\subset M$ of a time-like geodesic $\mu$. Under natural causality conditions, we reconstruct the conformal type of the unknown open, relatively compact set $W\subset M$, when we are given $V$, the conformal class of $g|_V$, and the light observations sets $P_V(q)$ corresponding to all source points $q$ in $W$. The light observation set $P_V(q)$ is the intersection of $V$ and the light-cone emanating from the point $q$, i.e., the points in the set $V$ where light from a point source at $q$ is observed. 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 geodesic $\mu$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$. 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 $\mu$ and return back to $\mu$., Comment: The earlier version, v1, of the preprint had a different title - "Inverse problems in spacetime II: Reconstruction of a Lorentzian manifold from light observation sets" and it concerned only passive observations. In the new versions v2-v4 we have combined the passive observation results with inverse problems with active measurements

We 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 X-ray 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.

Volume: 2017.4 Host publication title: RIMS Kôkyûroku