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SPIKE DETECTION FROM INACCURATE SAMPLINGS
"... Abstract. This article investigates the superresolution phenomenon using the celebrated statistical estimator LASSO in the complex valued measure framework. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, Stieltjes tra ..."
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Abstract. This article investigates the superresolution phenomenon using the celebrated statistical estimator LASSO in the complex valued measure framework. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, Stieltjes transformation...). In particular, we provide an explicit quantitative localization of the spikes. Moreover, our analysis is based on the Rice method and provide an upper bound on the supremum of white noise perturbation in the measure space. hal00780808, version 1 24 Jan 2013 1.
Optimal control of the undamped linear wave equation with measure value controls
, 2014
"... Abstract. Measure valued optimal control control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc, L2(I)) is chosen as control space and the corresponding total variation norm as control cost functional. The support of the controls (sparsity pattern) is ..."
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Abstract. Measure valued optimal control control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc, L2(I)) is chosen as control space and the corresponding total variation norm as control cost functional. The support of the controls (sparsity pattern) is timeindependent which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the wellposedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.
• Government of Styria • City of GrazAPPROXIMATION OF ELLIPTIC CONTROL PROBLEMS IN MEASURE SPACES WITH SPARSE SOLUTIONS ∗
, 2011
"... Approximation of elliptic control problems in measure spaces with sparse solutions ..."
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Approximation of elliptic control problems in measure spaces with sparse solutions
OPTIMAL CONTROL OF THE UNDAMPED LINEAR WAVE EQUATION WITH MEASURE VALUED CONTROLS
, 2014
"... Abstract. Measure valued optimal control control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc, L2(I)) is chosen as control space and the corresponding total variation norm as control cost functional. The support of the controls (sparsity pattern) is ..."
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Abstract. Measure valued optimal control control problems governed by the linear wave equation are analyzed. The space of vector measures M(Ωc, L2(I)) is chosen as control space and the corresponding total variation norm as control cost functional. The support of the controls (sparsity pattern) is timeindependent which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the wellposedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.
Exact solutions to Super Resolution on semialgebraic domains in higher dimensions
, 2015
"... We investigate the multidimensional Super Resolution problem on closed semialgebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the `1minimization in the space of Radon measures in the multidimensional frame on ..."
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We investigate the multidimensional Super Resolution problem on closed semialgebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the `1minimization in the space of Radon measures in the multidimensional frame on semialgebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal `1minimization exact recovery problem of Super Resolution that unleashes standard techniques (such as momentsumofsquares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the Super Resolution problem in dimension greater than 2 and for a large family of domains described by semialgebraic sets.
Towards a Mathematical Theory of SuperResolution
, 2012
"... This paper develops a mathematical theory of superresolution. Broadly speaking, superresolution is the problem of recovering the fine details of an object—the high end of its spectrum— from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources ..."
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This paper develops a mathematical theory of superresolution. Broadly speaking, superresolution is the problem of recovering the fine details of an object—the high end of its spectrum— from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0, 1] and with unknown complexvalued amplitudes. We only observe Fourier samples of this object up until a frequency cutoff fc. We show that one can superresolve these point sources with infinite precision—i.e. recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the superresolved signal is expected to degrade when both the noise level and the superresolution factor vary.
Sparse Spikes Deconvolution on Thin Grids
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.