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Publication Radial basis function interpolation in the limit of increasingly flat basis functions(Elsevier, 2016-02-15) Kindelan Segura, Manuel; Moscoso, Miguel; González Rodríguez, Pedro; Ministerio de Ciencia e Innovación (España)We propose a new approach to study Radial Basis Function (RBF) interpolation in the limit of increasingly flat functions. The new approach is based on the semi-analytical computation of the Laurent series of the inverse of the RBF interpolation matrix described in a previous paper [3]. Once the Laurent series is obtained, it can be used to compute the limiting polynomial interpolant, the optimal shape parameter of the RBFs used for interpolation, and the weights of RBF finite difference formulas, among other things.Publication Quantitative signal subspace imaging(IOP Science, 2021-12) González Rodríguez, Pedro; Kim, Arnold D; Tsogka, Chrysoula; Ministerio de Ciencia e Innovación (España)We develop and analyze a quantitative signal subspace imaging method for single-frequency array imaging. This method is an extension to multiple signal classification which uses (i) the noise subspace to determine the location and support of targets, and (ii) the signal subspace to recover quantitative information about the targets. For point targets, we are able to recover the complex reflectivity and for an extended target under the Born approximation, we are able to recover a scalar quantity that is related to the product of the volume and relative dielectric permittivity of the target. Our resolution analysis for a point target demonstrates this method is capable of achieving exact recovery of the complex reflectivity at subwavelength resolution. Additionally, this resolution analysis shows that noise in the data effectively acts as a regularization to the imaging functional resulting in a method that is surprisingly more robust and effective with noise than without noise.Publication Quantitative phase and absorption contrast imaging(IEEE, 2022-09-12) Moscoso, Miguel; Novikov, Alexei; Papanicolau, George; Tsogka, Chrysoula; Ministerio de Ciencia e Innovación (España)We present an algorithm for coherent diffractive imaging with phaseless measurements. It treats the forward model as a combination of coherent and incoherent waves. The algorithm reconstructs absorption and phase contrast that quantifies the attenuation and the refraction of the waves propagating through an object. It requires coherent or partially coherent illuminations, and several detectors to record the intensity of the distorted wave that passes through the object under inspection. The diversity of illuminations, obtained by putting masks between the source and the object, provides enough information for imaging. The computational cost of our algorithm is linear in the number of pixels of the image. Therefore, it is efficient for high-resolution imaging. Our algorithm guarantees exact recovery if the image is sparse for a given basis. Numerical experiments in the setting of phaseless diffraction imaging of sparse objects validate the efficiency and the precision of the suggested algorithm.Publication Fast signal recovery from quadratic measurements(IEEE, 2021-05-19) Moscoso, Miguel; Novikov, Alexei; Papanicolau, George; Tsogka, Chrysoula; Ministerio de Ciencia e Innovación (España)We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix formed by the cross correlations of the sought signal. This creates a bottleneck for the inversion since the number of unknowns grows quadratically with the dimension of the signal. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose dimension grows linearly with the size of the signal, and use an efficient Noise Collector to absorb the cross-correlated data that come from the off-diagonal elements of this matrix. These elements do not carry extra information about the support of the signal, but significantly contribute to these data. With this strategy, we recover the unknown matrix by solving a convex linear problem whose cost is similar to the one that uses linear measurements. Our theory shows that the proposed approach provides exact support recovery when the data is not too noisy, and that there are no false positives for any level of noise. It also demonstrates that the level of sparsity that can be recovered scales almost linearly with the number of data. The numerical experiments presented in the paper corroborate these findings.Publication An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations(Elsevier, 2022-11-15) Tominec, Igor; Villard, Pierre-Frédéric; Larsson, Elisabeth; Bayona Revilla, Víctor; Cacciani, NicolaThe thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model.Publication One-way radiative transfer(Elsevier, 2016-06-01) González Rodríguez, Pedro; Ilan, Boaz; Kim, Arnold DongkyoonWe introduce the one-way radiative transfer equation (RTE) for modeling the transmission of a light beam incident normally on a slab composed of a uniform forward-peaked scattering medium. Unlike the RTE, which is formulated as a boundary value problem, the one-way RTE is formulated as an initial value problem. Consequently, the one-way RTE is much easier to solve. We discuss the relation of the one-way RTE to the Fokker–Planck, small-angle, and Fermi pencil beam approximations. Then, we validate the one-way RTE through systematic comparisons with RTE simulations for both the Henyey–Greenstein and screened Rutherford scattering phase functions over a broad range of albedo, anisotropy factor, optical thickness, and refractive index values. We find that the one-way RTE gives very good approximations for a broad range of optical property values for thin to moderately thick media that have moderately to sharply forward-peaked scattering. Specifically, we show that the error made by the one-way RTE decreases monotonically as the anisotropic factor increases and as the albedo increases. On the other hand, the error increases monotonically as the optical thickness increases and the refractive index mismatch at the boundary increases.Publication Three-dimensional imaging from single-element holographic data(Optical Society of America (OSA), 2021-02-01) Moscoso, Miguel; Novikov, Alexei; Papanicolaou, George; Tsogka, Chrysoula; Ministerio de Economía y Competitividad (España)We present a holographic imaging approach for the case in which a single source-detector pair is used to scan a sample. The source-detector pair collects intensity-only data at different frequencies and positions. By using an appropriate illumination strategy, we recover field cross correlations over different frequencies for each scan location. The problem is that these field cross correlations are asynchronized, so they have to be aligned first in order to image coherently. This is the main result of the paper: a simple algorithm to synchronize field cross correlations at different locations. Thus, one can recover full field data up to a global phase that is common to all scan locations. The recovered data are, then, coherent over space and frequency so they can be used to form high-resolution three-dimensional images. Imaging with intensity-only data is therefore as good as coherent imaging with full data. In addition, we use an ℓ1-norm minimization algorithm that promotes the low dimensional structure of the images, allowing for deep high-resolution imaging.Publication Imaging with highly incomplete and corrupted data(IOP Publishing, 2020-03) Moscoso, Miguel; Novikov, Alexei; Papanicolau, George; Tsogka, Chrysoula; Ministerio de Economía y Competitividad (España)We consider the problem of imaging sparse scenes from a few noisy data using an L1-minimization approach. This problem can be cast as a linear system of the form Ap = b, where A is an N x K measurement matrix. We assume that the dimension of the unknown sparse vector p E Ck is much larger than the dimension of the data vector b E Cn, i.e. K >>N. We provide a theoretical framework that allows us to examine under what conditions the L1-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that L1-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of L1-minimization we propose to solve instead the augmented linear system [A|C]p = b, where the N = Σ matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension N, can be well approximated. Theoretically, the dimension Σ of the noise collector should be eN which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns Σ~10K.Publication The noise collector for sparse recovery in high dimensions(National Academy of Sciences, 2020-05-26) Moscoso, Miguel; Novikov, Alexei; Papanicolaou, George; Tsogka, Chrysoula; Ministerio de Ciencia e Innovación (España)The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. It is well known that a sparse solution of the linear system Alpharho=b0 can be found efficiently with an l1-norm minimization approach if the data are noiseless. However, detection of the signal from data corrupted by noise is still a challenging problem as the solution depends, in general, on a regularization parameter with optimal value that is not easy to choose. We propose an efficient approach that does not require any parameter estimation. We introduce a no-phantom weight tau and the Noise Collector matrix C and solve an augmented system Alpharho+Ceta=b0+e, where e is the noise. We show that the l1-norm minimal solution of this system has zero false discovery rate for any level of noise, with probability that tends to one as the dimension of b0 increases to infinity. We obtain exact support recovery if the noise is not too large and develop a fast Noise Collector algorithm, which makes the computational cost of solving the augmented system comparable with that of the original one. We demonstrate the effectiveness of the method in applications to passive array imaging.Publication Micro-combustion modelling with RBF-FD: A high-order meshfree method for reactive flows in complex geometries(Elsevier, 2021-06-01) Bayona Revilla, Víctor; Sánchez Sanz, Mario; Fernández Tarrazo, Eduardo Antonio; Kindelan Segura, Manuel; Ministerio de Economía y Competitividad (España)New micro-devices, such as unmanned aerial vehicles or micro-robots, have increased the demand of a new generation of small-scale combustion power system that go beyond the energy-density limitations of batteries or fuel cells. The characteristics short residence times and intense heat losses reduce the efficiency of combustion-based devices, a key factor that requires of an acute modelling effort to understand the competing physicochemical phenomena that hamper their efficient operation. With this objective in mind, this paper is devoted to the development of a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences (RBF-FD) based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly). In our model, the combustion chamber of a micro-rotary engine is simulated by a system of unsteady reaction-diffusion equations coupled with a steady flow passing a bidimensional stenotic channel of great slenderness. The conversion efficiency is characterized by identifying the different combustion regimes that emerged as a function of the ignition point. We show that PHS+poly based RBF-FD is able to achieve high-order algebraic convergence on scattered node distributions, enabling for node refinement in key regions of the fluid domain. This feature makes it specially well adapted to integrate problems in irregular geometries with front-like solutions, such as reactive fronts or shock waves. Several numerical tests are carried out to demonstrate the accuracy and effectiveness of our approach.Publication Synthetic Aperture Imaging With Intensity-Only Data(IEEE, 2020) Moscoso, Miguel; Novikov, Alexei; Papanicolaou, George; Tsogka, Chrysoula; Ministerio de Ciencia e Innovación (España)In this paper, we consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.Publication Multifrequency interferometric imaging with intensity-only measurements(Society for Industrial and Applied Mathematics (SIAM), 2017-07-26) Moscoso, Miguel; Novikov, Alexei; Papanicolaou, George; Tsogka, Chrysoula; Ministerio de Economía y Competitividad (España)We propose an illumination strategy for interferometric imaging that allows for robust depth recovery from intensity-only measurements. For an array with colocated sources and receivers, we show that all the possible interferometric data for multiple sources, receivers, and frequencies can be recovered from intensity-only measurements provided that we have sufficient source location and frequency illumination diversity. There is no need for phase reconstruction in this approach. Using interferometric imaging methods we show that in homogeneous media there is no loss of resolution when imaging with intensities only. If in these imaging methods we reduce incoherence by restricting the multifrequency interferometric data to nearby array elements and nearby frequencies we obtain robust images in weakly inhomogeneous background media with a somewhat reduced resolution.Publication Array imaging of localized objects in homogeneous and heterogeneous media(IOP Science, 2016-10) Chai, Anwei; Moscoso, Miguel; Papanicolaou, GeorgeWe present a comprehensive study of the resolution and stability properties of sparse promoting optimization theories applied to narrow band array imaging of localized scatterers. We consider homogeneous and heterogeneous media, and multiple and single scattering situations. When the media is homogeneous with strong multiple scattering between scatterers, we give a non-iterative formulation to find the locations and reflectivities of the scatterers from a nonlinear inverse problem in two steps, using either single or multiple illuminations. We further introduce an approach that uses the top singular vectors of the response matrix as optimal illuminations, which improves the robustness of sparse promoting optimization with respect to additive noise. When multiple scattering is negligible, the optimization problem becomes linear and can be reduced to a hybrid-ℓ1 method when optimal illuminations are used. When the media is random, and the interaction with the unknown inhomogeneities can be primarily modeled by wavefront distortions, we address the statistical stability of these methods. We analyze the fluctuations of the images obtained with the hybrid-ℓ1 method, and we show that it is stable with respect to different realizations of the random medium provided the imaging array is large enough. We compare the performance of the hybrid-ℓ1 method in random media to the widely used Kirchhoff migration and the multiple signal classification methods.Publication Coherent Imaging without Phases(SIAM (Society for Industrial and Applied Mathematics), 2016-01-01) Moscoso, Miguel; Novikov, Alexei; Papanicolaou, GeorgeIn this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with N transducers. We consider that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper [21] where we showed that using linear combinations of intensity-only measurements, obtained from N2 illuminations, imaging of localized scatterers can be carried out e_ciently using imaging methods based on the singular value decomposition of the time-reversal matrix. Here we show the same strategy can be accomplished with only 3N¬¬-2 illuminations, herefore reducing enormously the data acquisition process. Furthermore, we show that in the paraxial regime one can form the images by using six illuminations only. In particular, this paraxial regime includes Fresnel and Fraunhofer di_raction. The key point of this work is that if one controls the illuminations, imaging with intensity-only can be easily reduced to a imaging with phases and, therefore, one can apply standard imaging techniques. Detailed numerical simulations illustrate the performance of the proposed imaging strategy with and without data noise.Publication Optimized Finite Difference Formulas for Accurate High Frequency Components(Hindawi, 2016-12-14) Kindelan Segura, Manuel; Moscoso, Miguel; González Rodríguez, Pedro; Ministerio de Economía y Competitividad (España)We present a method to obtain optimal finite difference formulas which maximize their frequency range of validity. The optimization is based on the idea of keeping an error of interest (dispersion, phase, or group velocities errors) below a given threshold for a wavenumber interval as large as possible. To find the weights of these optimal finite difference formulas we solve a system of nonlinear equations. Furthermore, we give compact formulas for the optimal weights as function of the error bound. Several numerical experiments illustrate the performance of the obtained finite difference formulas compared to the standard ones.Publication Synthetic Aperture Imaging of Direction- and Frequency-Dependent Reflectivities(Society for Industrial and Applied Mathematics, 2016-01-01) Borcea, Liliana; Moscoso, Miguel; Papanicolaou, George; Tsogka, ChrysoulaWe introduce a synthetic aperture imaging framework that takes into consideration directional dependence of the reflectivity that is to be imaged, as well as its frequency dependence. We use an l(1) minimization approach that is coordinated with data segmentation so as to fuse information from multiple subapertures and frequency subbands. We analyze this approach from first principles and assess its performance with numerical simulations in an X-band radar regime.Publication Optimal shape parameter for the solution of elastostatic problems with the RBF method(Springer, 2014-04-01) Simonenko, Stanislav; Bayona Revilla, Víctor; Kindelan Segura, Manuel; Comunidad de Madrid; Ministerio de Economía y Competitividad (España)Radial basis functions (RBFs) have become a popular method for the solution of partial differential equations. In this paper we analyze the applicability of both the global and the local versions of the method for elastostatic problems. We use multiquadrics as RBFs and describe how to select an optimal value of the shape parameter to minimize approximation errors. The selection of the optimal shape parameter is based on analytical approximations to the local error using either the same shape parameter at all nodes or a node-dependent shape parameter. We show through several examples using both equispaced and nonequispaced nodes that significant gains in accuracy result from a proper selection of the shape parameter.Publication Analysis of the diurnal variation of the global electric circuit obtained from different numerical models(AGU Journals, 2017-12-16) Jansky, Jaroslav; Lucas, Greg M.; Kalb, Christina; Bayona Revilla, Víctor; Peterson, Michael J.; Deierling, Wiebke; Flyer, Natasha; Pasko, Victor P.This work analyzes different current source and conductivity parameterizations and their influence on the diurnal variation of the global electric circuit (GEC). The diurnal variations of the current source parameterizations obtained using electric field and conductivity measurements from plane overflights combined with global Tropical Rainfall Measuring Mission satellite data give generally good agreement with measured diurnal variation of the electric field at Vostok, Antarctica, where reference experimental measurements are performed. An approach employing 85 GHz passive microwave observations to infer currents within the GEC is compared and shows the best agreement in amplitude and phase with experimental measurements. To study the conductivity influence, GEC models solving the continuity equation in 3‐D are used to calculate atmospheric resistance using yearly averaged conductivity obtained from the global circulation model Community Earth System Model (CESM). Then, using current source parameterization combining mean currents and global counts of electrified clouds, if the exponential conductivity is substituted by the conductivity from CESM, the peak to peak diurnal variation of the ionospheric potential of the GEC decreases from 24% to 20%. The main reason for the change is the presence of clouds while effects of 222Rn ionization, aerosols, and topography are less pronounced. The simulated peak to peak diurnal variation of the electric field at Vostok is increased from 15% to 18% from the diurnal variation of the global current in the GEC if conductivity from CESM is used.Publication Comparison of moving least squares and RBF+poly for interpolation and derivative approximation(Springer, 2019-10-01) Bayona Revilla, Víctor; Ministerio de Economía y Competitividad (España)The combination of polyharmonic splines (PHS) with high degree polynomials (PHS+poly) has recently opened new opportunities for radial basis function generated finite difference approximations. The PHS+poly formulation, which relies on a polynomial least squares fitting to enforce the local polynomial reproduction property, resembles somehow the so-called moving least squares (MLS) method. Although these two meshfree approaches are increasingly used nowadays, no direct comparison has been done yet. The present study aims to fill this gap, focusing on scattered data interpolation and derivative approximation. We first review the MLS approach and show that under some mild assumptions PHS+poly can be formulated analogously. Based on heuristic perspectives and numerical demonstrations, we then compare their performances in 1-D and 2-D. One key result is that, as previously found for PHS+poly, MLS can also overcome the edge oscillations (Runge's phenomenon) by simply increasing the stencil size for a fixed polynomial degree. This is, however, controlled by a weighted least squares fitting which fails for high polynomial degrees. Overall, PHS+poly is found to perform superior in terms of accuracy and robustnessPublication An insight into RBF-FD approximations augmented with polynomials(Elsevier, 2019-05-01) Bayona Revilla, Víctor; Ministerio de Ciencia e Innovación (España)Radial basis function-generated finite differences (RBF-FD) based on the combination of polyharmonic splines (PHS) with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts. Among the key features, (i) high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill-conditioning, and (ii) the harmful edge effects associated to the use of high order polynomials (better known as Runge's phenomenon) can be overcome by simply increasing the stencil size for a fixed polynomial degree. The present study complements our previous results, providing an analytical insight into RBF-FD approximations augmented with polynomials. It is based on a closed-form expression for the interpolant, which reveals the mechanisms underlying these features, including the role of polynomials and RBFs in the interpolant, the approximation error, and the behavior of the cardinal functions near boundaries. Numerical examples are included for illustration.