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  • 1.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Learning to solve inverse problems using Wasserstein lossManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We propose using the Wasserstein loss for training in inverse problems. In particular, we consider a learned primal-dual reconstruction scheme for ill-posed inverse problems using the Wasserstein distance as loss function in the learning. This is motivated by miss-alignments in training data, which when using standard mean squared error loss could severely degrade reconstruction quality. We prove that training with the Wasserstein loss gives a reconstruction operator that correctly compensates for miss-alignments in certain cases, whereas training with the mean squared error gives a smeared reconstruction. Moreover, we demonstrate these effects by training a reconstruction algorithm using both mean squared error and optimal transport loss for a problem in computerized tomography.

  • 2.
    Banert, Sebastian
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Adler, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Data-driven nonsmooth optimizationManuskript (preprint) (Övrigt vetenskapligt)
  • 3. Chen, Y.
    et al.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    State Tracking of Linear Ensembles via Optimal Mass Transport2018Ingår i: IEEE Control Systems Letters, ISSN 2475-1456, Vol. 2, nr 2, s. 260-265Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the problems of tracking an ensemble of indistinguishable agents with linear dynamics based only on output measurements. In this setting, the dynamics of the agents can be modeled by distribution flows in the state space and the measurements correspond to distributions in the output space. In this letter, we formulate the corresponding state estimation problem using optimal mass transport theory with prior linear dynamics, and the optimal solution gives an estimate of the state trajectories of the ensemble. For general distributions of systems this can be formulated as a convex optimization problem which is computationally feasible when the number of state dimensions is low. In the case where the marginal distributions are Gaussian, the problem is reformulated as a semidefinite programming problem and can be efficiently solved for tracking systems with a large number of states.

  • 4. Chen, Y.
    et al.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Georgiou, T. T.
    The role of the time-arrow in mean-square estimation of stochastic processes2018Ingår i: IEEE Control Systems Letters, ISSN 2475-1456, Vol. 2, nr 1, s. 85-90Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The purpose of this letter is to point out a certain dichotomy between the information that the past and future values of a multivariate stochastic process carry about the present. More specifically, vector-valued, secondorder stochastic processes may be deterministic in one time-direction but not in the other. This phenomenon, which is absent in scalar-valued processes, is deeply rooted in the geometry of the shift-operator. The exposition and the examples we discuss are based on the work of Douglas, Shapiro, and Shields on cyclic vectors of the backward shift and relate to classical ideas going back to Wiener and Kolmogorov. We focus on rank-one stochastic processes for which we obtain an explicit characterization of all regular processes that are deterministic in the reverse timedirection. This letter builds on examples and the goal is to provide insights to a control engineering audience with interests in estimation theory and modeling of time-series.

  • 5. Elvander, F.
    et al.
    Adalbjörnsson, S. I.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Jakobsson, A.
    Using optimal transport for estimating inharmonic pitch signals2017Ingår i: 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Institute of Electrical and Electronics Engineers (IEEE), 2017, s. 331-335, artikel-id 7952172Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this work, we propose a novel multi-pitch estimation technique that is robust with respect to the inharmonicity commonly occurring in many applications. The method does not require any a priori knowledge of the number of signal sources, the number of harmonics of each source, nor the structure or scope of any possibly occurring inharmonicity. Formulated as a minimum transport distance problem, the proposed method finds an estimate of the present pitches by mapping any found spectral line to the closest harmonic structure. The resulting optimization is a convex and highly tractable linear programming problem. The preferable performance of the proposed method is illustrated using both simulated and real audio signals.

  • 6. Elvander, F.
    et al.
    Haasler, Isabel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Jakobsson, A.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Tracking and sensor fusion in direction of arrival estimation using optimal mass transport2018Ingår i: 2018 26th European Signal Processing Conference (EUSIPCO), European Signal Processing Conference, EUSIPCO , 2018, s. 1617-1621Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this work, we propose new methods for information fusion and tracking in direction of arrival (DOA) estimation by utilizing an optimal mass transport framework. Sensor array measurements in DOA estimation may not be consistent due to misalignments and calibration errors. By using optimal mass transport as a notion of distance for combining the information obtained from all the sensor arrays, we obtain an approach that can prevent aliasing and is robust to array misalignments. For the case of sequential tracking, the proposed method updates the DOA estimate using the new measurements and an optimal mass transport prior. In the case of sensor fusion, information from several, individual, sensor arrays is combined using a barycenter formulation of optimal mass transport.

  • 7.
    Elvander, Filip
    et al.
    Lund Univ, Div Math Stat, Lund, Sweden..
    Haasler, Isabel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Jakobsson, Andreas
    Lund Univ, Div Math Stat, Lund, Sweden..
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    NON-COHERENT SENSOR FUSION VIA ENTROPY REGULARIZED OPTIMAL MASS TRANSPORT2019Ingår i: 2019 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING (ICASSP), IEEE , 2019, s. 4415-4419Konferensbidrag (Refereegranskat)
    Abstract [en]

    This work presents a method for information fusion in source localization applications. The method utilizes the concept of optimal mass transport in order to construct estimates of the spatial spectrum using a convex barycenter formulation. We introduce an entropy regularization term to the convex objective, which allows for low-complexity iterations of the solution algorithm and thus makes the proposed method applicable also to higher-dimensional problems. We illustrate the proposed method's inherent robustness to misalignment and miscalibration of the sensor arrays using numerical examples of localization in two dimensions.

  • 8.
    Elvander, Filip
    et al.
    Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden..
    Jakobsson, Andreas
    Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden..
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport2018Ingår i: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 66, nr 20, s. 5285-5298Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semidefiniteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes.

  • 9.
    Elvander, Filip
    et al.
    Lund Univ, Div Math Stat, Lund, Sweden..
    Jakobsson, Andreas
    Lund Univ, Div Math Stat, Lund, Sweden..
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    USING OPTIMAL MASS TRANSPORT FOR TRACKING AND INTERPOLATION OF TOEPLITZ COVARIANCE MATRICES2018Ingår i: 2018 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING (ICASSP), IEEE , 2018, s. 4469-4473Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this work, we propose a novel method for interpolation and extrapolation of Toeplitz structured covariance matrices. By considering a spectral representation of Toeplitz matrices, we use an optimal mass transport problem in the spectral domain in order to define a notion of distance between such matrices. The obtained optimal transport plan naturally induces a way of interpolating, as well as extrapolating, Toeplitz matrices. The constructed covariance matrix interpolants and extrapolants preserve the Toeplitz structure, as well as the positive semi-definiteness and the zeroth covariance of the original matrices. We demonstrate the proposed method's ability to model locally linear shifts of spectral power for slowly varying stochastic processes, illustrating the achievable performance using a simple tracking problem.

  • 10.
    Enqvist, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Minimal Itakura-Saito distance and Covariance interpolation2008Ingår i: 47TH IEEE CONFERENCE ON DECISION AND CONTROL, 2008 (CDC 2008), 2008, s. 137-142Konferensbidrag (Refereegranskat)
    Abstract [en]

    Identification of power spectral densities rely on measured second order statistics such as, e.g. covariance estimates. In the family of power spectra consistent with such an estimate a representative spectra is singled out; examples of such choices are the Maximum entropy spectrum and the Correlogram. Here, we choose a prior spectral density to represent a priori information, and the spectrum closest to the prior in the Itakura-Saito distance is selected. It is known that this can be seen as the limit case when the cross-entropy principle is applied to a gaussian process. This work provides a quantitative measure of how close a finite covariance sequence is to a spectral density in the Itakura-Saito distance. It is given by a convex optimization problem and by considering its dual the structure of the optimal spectrum is obtained. Furthermore, it is shown that strong duality holds and that a covariance matching coercive spectral density always exists. The methods presented here provides tools for discrimination between power spectrum, identification of power spectrum, and for incorporating given data in this process.

  • 11.
    Fanizza, Giovanna
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Nagamune, Ryozo
    A global analysis approach to passivity preserving model reduction2006Ingår i: Proceedings Of The 45th IEEE Conference On Decision And Control, Vols 1-14, 2006, s. 3399-3404Konferensbidrag (Refereegranskat)
    Abstract [en]

    Passivity-preserving model reduction for linear time-invariant systems amounts to approximating a positive-real rational transfer function with one of lower degree. Recently Antoulas and Sorensen have proposed such a model-reduction method based on Krylov projections. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models.

  • 12.
    Fanizza, Giovanna
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Nagamune, Ryozo
    Univ British Columbia, Dept Mech Engn, Vancouver.
    Passivity-preserving model reduction by analytic interpolation2007Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 425, nr 2-3, s. 608-633Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Antoulas and Sorensen have recently proposed a passivity-preserving model-reduction method of linear systems based on Krylov projections. The idea is to approximate a positive-real rational transfer function with one of lower degree. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models. We also show that, in the multi-input/multi-output case, Sorensen's algorithm actually amounts to tangential Nevanlinna-Pick interpolation.

  • 13. Gattami, Ather
    et al.
    Ringh, Emil
    Ericsson Research, Sweden.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Time localization and capacity of faster-than-nyquist signaling2015Ingår i: 2015 IEEE Global Communications Conference, GLOBECOM 2015, Institute of Electrical and Electronics Engineers (IEEE), 2015, artikel-id 7417358Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this paper, we consider communication over the bandwidth limited analog white Gaussian noise channel using non-orthogonal pulses. In particular, we consider non-orthogonal transmission by signaling samples at a rate higher than the Nyquist rate. Using the faster-than- Nyquist (FTN) framework, Mazo showed that one may transmit symbols carried by sinc pulses at a higher rate than that dictated by Nyquist without loosing bit error rate. However, as we will show in this paper, such pulses are not necessarily well localized in time. In fact, assuming that signals in the FTN framework are well localized in time, one can construct a signaling scheme that violates the Shannon capacity bound. We also show directly that FTN signals are in general not well localized in time. We also consider FTN signaling in the case of pulses that are different from the sinc pulses. We show that one may use a precoding scheme of low complexity, in order to remove the intersymbol interference. This leads to the possibility of increasing the number of transmitted samples per time unit and compensate for spectral inefficiency due to signaling at the Nyquist rate of the non sinc pulses. We demonstrate the power of the precoding scheme by simulations.

  • 14.
    Georgiou, Tryphon T.
    et al.
    Department of Electrical Engineering, University of Minnesota.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Takyar, Mir Shahrouz
    Department of Electrical Engineering, University of Minnesota.
    Metrics for Power Spectra: An Axiomatic Approach2009Ingår i: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 57, nr 3, s. 859-867Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present an axiomatic framework for seeking distances between power spectral density functions. The axioms require that the sought metric respects the effects of additive and multiplicative noise in reducing our ability to discriminate spectra, as well as they require continuity of statistical quantities with respect to perturbations measured in the metric. We then present a particular metric which abides by these requirements. The metric is based on the Monge-Kantorovich transportation problem and is contrasted with an earlier Riemannian metric based on the minimum-variance prediction geometry of the underlying time-series. It is also being compared with the more traditional Itakura-Saito distance measure, as well as the aforementioned prediction metric, on two representative examples.

  • 15. Glentis, G. -O
    et al.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Jakobsson, A.
    Li, J.
    Efficient spectral analysis in the missing data case using sparse ML methods2014Konferensbidrag (Refereegranskat)
    Abstract [en]

    Given their wide applicability, several sparse high-resolution spectral estimation techniques and their implementation have been examined in the recent literature. In this work, we further the topic by examining a computationally efficient implementation of the recent SMLA algorithms in the missing data case. The work is an extension of our implementation for the uniformly sampled case, and offers a notable computational gain as compared to the alternative implementations in the missing data case.

  • 16.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Inverse Problems in Analytic Interpolation for Robust Control and Spectral Estimation2008Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis is divided into two parts. The first part deals with theNevanlinna-Pick interpolation problem, a problem which occursnaturally in several applications such as robust control, signalprocessing and circuit theory. We consider the problem of shaping andapproximating solutions to the Nevanlinna-Pick problem in a systematicway. In the second part, we study distance measures between powerspectra for spectral estimation. We postulate a situation where wewant to quantify robustness based on a finite set of covariances, andthis leads naturally to considering the weak*-topology. Severalweak*-continuous metrics are proposed and studied in this context.In the first paper we consider the correspondence between weighted entropyfunctionals and minimizing interpolants in order to find appropriateinterpolants for, e.g., control synthesis. There are two basic issues that weaddress: we first characterize admissible shapes of minimizers bystudying the corresponding inverse problem, and then we developeffective ways of shaping minimizers via suitable choices of weights.These results are used in order to systematize feedback controlsynthesis to obtain frequency dependent robustness bounds with aconstraint on the controller degree.The second paper studies contractive interpolants obtained as minimizersof a weighted entropy functional and analyzes the role of weights andinterpolation conditions as design parameters for shaping theinterpolants. We first show that, if, for a sequence of interpolants,the values of the corresponding entropy gains converge to theoptimum, then the interpolants converge in H_2, but not necessarily inH-infinity. This result is then used to describe the asymptoticbehaviour of the interpolant as an interpolation point approaches theboundary of the domain of analyticity.A quite comprehensive theory of analytic interpolation with degreeconstraint, dealing with rational analytic interpolants with an apriori bound, has been developed in recent years. In the third paper,we consider the limit case when this bound is removed, and only stableinterpolants with a prescribed maximum degree are sought. This leadsto weighted H_2 minimization, where the interpolants areparameterized by the weights. The inverse problem of determining theweight given a desired interpolant profile is considered, and arational approximation procedure based on the theory is proposed. Thisprovides a tool for tuning the solution for attaining designspecifications. The purpose of the fourth paper is to study the topology and develop metricsthat allow for localization of power spectra, based on second-orderstatistics. We show that the appropriate topology is theweak*-topology and give several examples on how to construct suchmetrics. This allows us to quantify uncertainty of spectra in anatural way and to calculate a priori bounds on spectral uncertainty,based on second-order statistics. Finally, we study identification ofspectral densities and relate this to the trade-off between resolutionand variance of spectral estimates.In the fifth paper, we present an axiomatic framework for seekingdistances between power spectra. The axioms requirethat the sought metric respects the effects of additive andmultiplicative noise in reducing our ability to discriminate spectra.They also require continuity of statistical quantities withrespect to perturbations measured in the metric. We then present aparticular metric which abides by these requirements. The metric isbased on the Monge-Kantorovich transportation problem and iscontrasted to an earlier Riemannian metric based on theminimum-variance prediction geometry of the underlying time-series. Itis also being compared with the more traditional Itakura-Saitodistance measure, as well as the aforementioned prediction metric, ontwo representative examples.

  • 17.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Enqvist, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Gattami, A.
    Confidence assessment for spectral estimation based on estimated covariances2016Ingår i: ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, Institute of Electrical and Electronics Engineers (IEEE), 2016, s. 4343-4347Konferensbidrag (Refereegranskat)
    Abstract [en]

    In probability theory, time series analysis, and signal processing, many identification and estimation methods rely on covariance estimates as an intermediate statistics. Errors in estimated covariances propagate and degrade the quality of the estimation result. In particular, in large network systems where each system node of the network gather and pass on results, it is important to know the reliability of the information so that informed decisions can be made. In this work, we design confidence regions based on covariance estimates and study how these can be used for spectral estimation. In particular, we consider three different confidence regions based on sets of unitarily invariant matrices and bound the eigenvalue distribution based on three principles: uniform bounds; arithmetic and harmonic means; and the Marcenko-Pastur Law eigenvalue distribution for random matrices. Using these methodologies we robustly bound the energy in a selected frequency band, and compare the resulting spectral bound from the respective confidence regions.

  • 18.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM.
    Georgiou, Tryphon
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    The inverse problem of analytic interpolation with degree constraint2006Ingår i: Proceedings Of The 45th IEEE Conference On Decision And Control, Vols 1-14, 2006, s. 559-564Konferensbidrag (Refereegranskat)
    Abstract [en]

    In [7], (6] a theory for degree-constrained analytic interpolation was developed in terms of the minimizers of certain convex entropy functionals. In the present paper, we introduce and study relevant inverse problems. More specifically, we answer the following two questions. First, given a function f which satisfies specified interpolation conditions, when is it that f can be obtained as the minimizer of a suitably chosen entropy functional? Second, given a function g, when does there exist a suitably entropy functional so that the unique minitnizer f which is subject to interpolation constraints also satisfies vertical bar f vertical bar = vertical bar g vertical bar on the unit circle. The theory and answers to these questions suggest an approach to identifying interpolants of a given degree and of a given approximate shape.

  • 19.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Georgiou, Tryphon T.
    Localization of power spectraArtikel i tidskrift (Övrigt vetenskapligt)
  • 20.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Georgiou, Tryphon T.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Signal analysis, moment problems & uncertainty measures2005Ingår i: IEEE Proceedings: Conference on Decision and Control (CDC), ISSN 0191-2216, s. 5710-5715Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Modern spectral estimation techniques often rely on second order statistics of a time-series to determine a power spectrum consistent with data. Such statistics provide moment constraints on the power spectrum. In this paper we study possible distance functions between spectra which permit a reasonable quantitative description of the uncertainty in moment problems. Typically, there is an infinite family or spectra consistent with given moments. A distance function between power spectra should permit estimating the diameter of the uncertainty family, a diameter which shrinks as new data accumulates. Abstract properties of such distance functions are discussed and certain specific options are put forth. These distance functions permit alternative descriptions of uncertainty in moment problems. While the paper focuses on the role of such measures in signal analysis, moment problems are ubiquitous in science and engineering, and the conclusions drawn herein are relevant over a wider spectrum of problems.

  • 21.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Georgiou, Tryphon T.
    Department of Electrical Engineering, University of Minnesota.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis2010Ingår i: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 55, nr 2, s. 405-418Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. This leads to a new procedure for feedback control synthesis.

  • 22.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Li, Jian
    Stoica, Petre
    Filter design with hard spectral constraints2014Ingår i: European Signal Processing Conference, 2014, s. 391-395Konferensbidrag (Refereegranskat)
    Abstract [en]

    Filter design is a fundamental problem in signal processing and important in many applications. In this paper we consider a communication application with spectral constraints, using filter designs that can be solved globally via convex optimization. Tradeoffs are discussed in order to determine which design is the most appropriate, and for these applications, finite impulse response filters appear to be more suitable than infinite impulse response filters since they allow for more flexible objective functions, shorter transients, and faster filter implementations.

  • 23.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    On Degree-Constrained Analytic Interpolation With Interpolation Points Close to the Boundary2009Ingår i: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 54, nr 6, s. 1412-1418Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the recent article [4], a theory for complexity-constrained interpolation of contractive functions is developed. In particular, it is shown that any such interpolant may be obtained as the unique minimizer of a (convex) weighted entropy gain. In this technical note we study this optimization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions. We first show that, if, for a sequence of interpolants, the values of the entropy gain of the interpolants converge to the optimum, then the interpolants converge in H-2, but not in H-infinity This result is then used to describe the asymptotic behavior of the interpolant as an interpolation point approaches the boundary of the domain of analyticity. For loop shaping to specifications in control design, it might at first seem natural to place strategically additional interpolation points close to the boundary. However, our results indicate that such a strategy will have little effect on the shape. Another consequence of our results relates to model reduction based on minimum-entropy principles, where one should avoid placing interpolation points too close to the boundary.

  • 24.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Stability-preserving rational approximation subject to interpolation constraints2008Ingår i: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 53, nr 7, s. 1724-1730Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H-2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic interpolants.

  • 25.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Stable rational approximation in the context of interpolation and convex optimization2007Ingår i: Proceedings Of The 46th IEEE Conference On Decision And Control, Vols 1-14, 2007, s. 2214-2221Konferensbidrag (Refereegranskat)
    Abstract [en]

    A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational interpolants with an a priori bound, has been developed in recent years. In this paper we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H-2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight and the interpolation points given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded interpolants.

  • 26.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong University, China.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    The Multidimensional Moment Problem with Complexity Constraint2016Ingår i: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 84, nr 3, s. 395-418Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper.

  • 27.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Ning, L.
    On robustness of ℓ1-regularization methods for spectral estimation2014Ingår i: Proceedings of the IEEE Conference on Decision and Control, IEEE conference proceedings, 2014, nr February, s. 1767-1773Konferensbidrag (Refereegranskat)
    Abstract [en]

    The use of ℓ<inf>1</inf>-regularization in sparse estimation methods has received huge attention during the last decade, and applications in virtually all fields of applied mathematics have benefited greatly. This interest was sparked by the recovery results of Candès, Donoho, Tao, Tropp, et al. and has resulted in a framework for solving a set of combinatorial problems in polynomial time by using convex relaxation techniques. In this work we study the use of ℓ<inf>1</inf>-regularization methods for high-resolution spectral estimation. In this problem, the dictionary is typically coherent and existing theory for robust/exact recovery does not apply. In fact, the robustness cannot be guaranteed in the usual strong sense. Instead, we consider metrics inspired by the Monge-Kantorovich transportation problem and show that the magnitude can be robustly recovered if the original signal is sufficiently sparse and separated. We derive both worst case error bounds as well as error bounds based on assumptions on the noise distribution.

  • 28.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Ning, Lipeng
    Harvard Univ, Brigham & Womens Hosp, Sch Med, Boston, MA 02115 USA..
    On Robustness of l(1)-Regularization Methods for Spectral Estimation2014Ingår i: 2014 IEEE 53RD ANNUAL CONFERENCE ON DECISION AND CONTROL (CDC), IEEE , 2014, s. 1767-1773Konferensbidrag (Refereegranskat)
    Abstract [en]

    The use of l(1)-regularization in sparse estimation methods has received huge attention during the last decade, and applications in virtually all fields of applied mathematics have benefited greatly. This interest was sparked by the recovery results of Cands, Donoho, Tao, Tropp, et al. and has resulted in a framework for solving a set of combinatorial problems in polynomial time by using convex relaxation techniques. In this work we study the use of l(1)-regularization methods for high-resolution spectral estimation. In this problem, the dictionary is typically coherent and existing theory for robust/exact recovery does not apply. In fact, the robustness cannot be guaranteed in the usual strong sense. Instead, we consider metrics inspired by the Monge-Kantorovich transportation problem and show that the magnitude can be robustly recovered if the original signal is sufficiently sparse and separated. We derive both worst case error bounds as well as error bounds based on assumptions on the noise distribution.

  • 29.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Generalized Sinkhorn Iterations for Regularizing Inverse Problems Using Optimal Mass Transport2017Ingår i: SIAM Journal on Imaging Sciences, ISSN 1936-4954, E-ISSN 1936-4954, Vol. 10, nr 4, s. 1935-1962Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The optimal mass transport problem gives a geometric framework for optimal allocation and has recently attracted significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be formulated as a linear programming problem, it is in many cases intractable for large problems due to the vast number of variables. A recent development addressing this builds on an approximation with an entropic barrier term and solves the resulting optimization problem using Sinkhorn iterations. In this work we extend this methodology to a class of inverse problems. In particular we show that Sinkhorn-type iterations can be used to compute the proximal operator of the transport problem for large problems. A splitting framework is then used to solve inverse problems where the optimal mass transport cost is used for incorporating a priori information. We illustrate this method on problems in computerized tomography. In particular we consider a limited-angle computerized tomography problem, where a priori information is used to compensate for missing measurements.

  • 30.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Rowe, William
    Xu, Luzhou
    Glentis, George-Othon
    Li, Jian
    Fast Missing-Data IAA With Application to Notched Spectrum SAR2014Ingår i: IEEE Transactions on Aerospace and Electronic Systems, ISSN 0018-9251, E-ISSN 1557-9603, Vol. 50, nr 2, s. 959-971Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Recently, the spectral estimation method known as the iterative adaptive approach (IAA) has been shown to provide higher resolution and lower sidelobes than comparable spectral estimation methods. The computational complexity is higher than methods such as the periodogram (matched filter method). Fast algorithms have been developed that considerably reduce the computational complexity of IAA by using Toeplitz and Vandermonde structures. For the missing-data case, several of these structures are lost, and existing fast algorithms are only efficient when the number of available samples is small. In this work, we consider the case in which the number of missing samples is small. This allows us to use low-rank completion to transform the problem to the structured problem. We compare the computational speed of the algorithm with the state of the art and demonstrate the utility in a frequency-notched synthetic aperture radar imaging problem.

  • 31.
    Karlsson, Johan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Takyar, Mir Shahrouz
    Georgiou, Tryphon T.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Transport metrics for power spectra2008Ingår i: 47TH IEEE CONFERENCE ON DECISION AND CONTROL, 2008 (CDC 2008), 2008, s. 143-148Konferensbidrag (Refereegranskat)
    Abstract [en]

    We present a family of metrics for power spectra based on the Monge-Kantorivic transportation distances. These metrics are constructed so that distances reduce with additive and multiplicative noise, reflecting the intuition that noise typically reduces our ability to discriminate spectra. In addition, perturbations measured in these metrics are continuous with respect to the statistics of the underlying time series. A general framework for constructing such metrics is put forth and these are contrasted with an earlier Riemannian metric which is based on prediction theory and the relevant geometry of the underlying time-series.

  • 32.
    Lindquist, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Georgiou, Tryphon
    Weight selection for gap robustness with degree-constrained controllers2008Ingår i: Proc. 47th IEEE Conference on Decision and Control, 2008, s. 4127-4134Konferensbidrag (Refereegranskat)
    Abstract [en]

    In modern robust control, control synthesis may be cast as an interpolation problem where the interpolant relates to robustness and performance criteria. In particular,robustness in the gap fits into this framework and the magnitude of the corresponding interpolant dictate the robustness to perturbations of the plant as a function of frequency. In this paper we consider the correspondence between weighted entropy functionals and minimizing interpolants in order to find appropriate interpolants for e.g. control synthesis. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. These results are used in order to systematize feedback control synthesis to obtain frequency dependentrobustness bounds with a constraint on the controller degree. 

  • 33.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China; Shanghai Jiao Tong Univ, Sch Math, Shanghai, Peoples R China.
    Further results on multidimensional rational covariance extension with application to texture generation2017Ingår i: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE , 2017Konferensbidrag (Refereegranskat)
    Abstract [en]

    The rational covariance extension problem is a moment problem with several important applications in systems and control as, for example, in identification, estimation, and signal analysis. Here we consider the multidimensional counterpart and present new results for the well-posedness of the problem. We apply the theory to texture generation by modeling the texture as the output of a Wiener system. The static nonlinearity in the Wiener system is assumed to be a thresholding function and we identify both the linear dynamical system and the thresholding parameter.

  • 34.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China; Shanghai Jiao Tong Univ, Sch Math, Shanghai, Peoples R China.
    Lower bounds on the maximum delay margin by analytic interpolation2018Ingår i: 2018 IEEE 57th Annual Conference on Decision and Control (CDC), Institute of Electrical and Electronics Engineers (IEEE), 2018, s. 5463-5469, artikel-id 8618930Konferensbidrag (Refereegranskat)
    Abstract [en]

    We study the delay margin problem in the context of recent works by T. Qi, J. Zhu, and J. Chen, where a sufficient condition for the maximal delay margin is formulated in terms of an interpolation problem obtained after introducing a rational approximation. Instead we omit the approximation step and solve the same problem directly using techniques from function theory and analytic interpolation. Furthermore, we introduce a constant shift in the domain of the interpolation problem. In this way we are able to improve on their lower bound for the maximum delay margin.

  • 35.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Tidigare Institutioner (före 2005), Matematik. Department of Automation, Shanghai Jiao Tong University, Shanghai, China.
    Multidimensional rational covariance extensionManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The rational covariance extension problem (RCEP) is an important problem in systems and control occurring in such diverse fields as control, estimation, system identification, and signal and image processing, leading to many fundamental theoretical questions. In fact, this inverse problem is a key component in many identification and signal processing techniques and plays a fundamental role in prediction, analysis, and modeling of systems and signals. It is well-known that the RCEP can be reformulated as a (truncated) trigonometric moment problem subject to a rationality condition. In this paper we consider the more general multidimensional trigonometric moment problem with a similar rationality constraint. This generalization creates many interesting new mathematical questions and also provides new insights into the original one-dimensional problem. A key concept in this approach is the complete smooth parametrization of all solutions, allowing solutions to be tuned to satisfy additional design specifications without violating the complexity constraints. As an illustration of the potential of this approach we apply our results to multidimensional spectral estimation, Wiener system identification, and image compression.

  • 36.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong Univ, Dept Automat & Math, Shanghai 200240, Peoples R China..
    MULTIDIMENSIONAL RATIONAL COVARIANCE EXTENSION WITH APPLICATIONS TO SPECTRAL ESTIMATION AND IMAGE COMPRESSION2016Ingår i: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 54, nr 4, s. 1950-1982Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The rational covariance extension problem (RCEP) is an important problem in systems and control occurring in such diverse fields as control, estimation, system identification, and signal and image processing, leading to many fundamental theoretical questions. In fact, this inverse problem is a key component in many identification and signal processing techniques and plays a fundamental role in prediction, analysis, and modeling of systems and signals. It is well known that the RCEP can be reformulated as a (truncated) trigonometric moment problem subject to a rationality condition. In this paper we consider the more general multidimensional trigonometric moment problem with a similar rationality constraint. This generalization creates many interesting new mathematical questions and also provides new insights into the original one-dimensional problem. A key concept in this approach is the complete smooth parameterization of all solutions, allowing solutions to be tuned to satisfy additional design specifications without violating the complexity constraints. As an illustration of the potential of this approach we apply our results to multidimensional spectral estimation and image compression. This is just a first step in this direction, and we expect that more elaborate tuning strategies will enhance our procedures in the future.

  • 37.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong University, China.
    Multidimensional rational covariance extension with approximate covariance matching2018Ingår i: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 56, nr 2, s. 913-944Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In our companion paper [A. Ringh, J. Karlsson, and A. Lindquist, SIAM T. Control Opton., 54 (2016), pp. 1950-1982] we discussed the multidimensional rational covariance extension problem (RCEP), which has important applications in image processing and spectral estimation in radar, sonar, and medical imaging. This is an inverse problem where a power spectrum with a rational absolutely continuous part is reconstructed from a finite set of moments. However, in most applications these moments are determined from observed data and are therefore only approximate, and the RCEP may not have a solution. In this paper we extend the results of our companion paper to handle approximate covariance matching. We consider two problems, one with a soft constraint and the other one with a hard constraint, and show that they are connected via a homeomorphism. We also demonstrate that the problems are well-posed and illustrate the theory by examples in spectral estimation and texture generation.

  • 38.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Lindquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. Shanghai Jiao Tong University, China.
    The Multidimensional Circulant Rational Covariance Extension Problem: Solutions and Applications in Image Compression2016Ingår i: 2015 54th IEEE Conference on Decision and Control (CDC), 2015, Institute of Electrical and Electronics Engineers (IEEE), 2016, s. 5320-5327Konferensbidrag (Refereegranskat)
    Abstract [en]

    Rational functions play a fundamental role in systems engineering for modelling, identification, and control applications. In this paper we extend the framework by Lindquist and Picci for obtaining such models from the circulant trigonometric moment problems, from the one-dimensional to the multidimensional setting in the sense that the spectrum domain is multidimensional. We consider solutions to weighted entropy functionals, and show that all rational solutions of certain bounded degree can be characterized by these. We also consider identification of spectra based on simultaneous covariance and cepstral matching, and apply this theory for image compression. This provides an approximation procedure for moment problems where the moment integral is over a multidimensional domain, and is also a step towards a realization theory for random fields.

  • 39.
    Ringh, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan Mikael
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    A fast solver for the circulant rational covariance extension problem2015Ingår i: 2015 European Control Conference, ECC 2015, Institute of Electrical and Electronics Engineers (IEEE), 2015, s. 727-733Konferensbidrag (Refereegranskat)
    Abstract [en]

    The rational covariance extension problem is to parametrize the family of rational spectra of bounded degree that matches a given set of covariances. This article treats a circulant version of this problem, where the underlying process is periodic and we seek a spectrum that also matches a set of given cepstral coefficients. The interest in the circulant problem stems partly from the fact that this problem is a natural approximation of the non-periodic problem, but is also a tool in itself for analysing periodic processes. We develop a fast Newton algorithm for computing the solution utilizing the structure of the Hessian. This is done by extending a current algorithm for Toeplitz-plus-Hankel systems to the block-Toeplitz-plus-block-Hankel case. We use this algorithm to reduce the computational complexity of the Newton search from O(n3) to O(n2), where n corresponds to the number of covariances and cepstral coefficients.

  • 40.
    Ringh, Emil
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Mele, Giampaolo
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Jarlebring, Elias
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Sylvester-based preconditioning for the waveguide eigenvalue problem2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, nr 1, s. 441-463Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large-scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation AX+XB+A1XB1+A2XB2+K(ring operator)X=C, where (ring operator) denotes the Hadamard product. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed by using the matrix equation version of the Sherman-Morrison-Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.

    Publikationen är tillgänglig i fulltext från 2020-05-03 09:14
  • 41. Rowe, W.
    et al.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Xu, L.
    Glentis, G. -O
    Li, J.
    SAR imaging in the presence of spectrum notches via fast missing data IAA2013Ingår i: Automatic Target Recognition XXIII, SPIE - International Society for Optical Engineering, 2013, s. UNSP 87440X-Konferensbidrag (Refereegranskat)
    Abstract [en]

    A synthetic aperture radar system operating in congested frequency bands suffers from radio frequency inter- ference (RFI) from narrowband sources. When RFI interference is suppressed by frequency notching, gaps are introduced into the fast time phase history. This results in a missing data spectral estimation problem, where the missing data increases sidelobe energy and degrades image quality. The adaptive spectral estimation method Iterative Adaptive Approach (IAA) has been shown to provide higher resolution and lower sidelobes than comparable methods, but at the cost of higher computationally complexity. Current fast IAA algorithms reduce the computational complexity using Toeplitz/Vandermonde structures, but are not applicable for missing data cases because these structures are lost. When the number of missing data samples is small, which often is the case in SAR with RFI, we use a low rank completion to restore the Toeplitz/Vandermonde structures. We show that the computational complexity of the proposed algorithm is considerably lower than the state-of-the-art and demonstrate the utility on a simulated frequency notched SAR imaging problem.

  • 42. Sadeghian, A.
    et al.
    Lim, D.
    Karlsson, Johan Mikael
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Li, J.
    Automatic target recognition using discrimination based on optimal transport2015Ingår i: ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, IEEE conference proceedings, 2015, s. 2604-2608Konferensbidrag (Refereegranskat)
    Abstract [en]

    The use of distances based on optimal transportation has recently shown promise for discrimination of power spectra. In particular, spectral estimation methods based on ℓ1 regularization as well as covariance based methods can be shown to be robust with respect to such distances. These transportation distances provide a geometric framework where geodesics corresponds to smooth transition of spectral mass, and have been useful for tracking. In this paper we investigate the use of these distances for automatic target recognition. We study the use of the Monge-Kantorovich distance compared to the standard ℓ2 distance for classifying civilian vehicles based on SAR images. We use a version of the Monge-Kantorovich distance that applies also for the case where the spectra may have different total mass, and we formulate the optimization problem as a minimum flow problem that can be computed using efficient algorithms.

  • 43.
    Shariati, Nafiseh
    et al.
    KTH, Skolan för elektroteknik och datavetenskap (EECS), Teknisk informationsvetenskap.
    Zachariah, Dave
    Uppsala University.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Bengtsson, Mats
    KTH, Skolan för elektroteknik och datavetenskap (EECS), Teknisk informationsvetenskap.
    Robust Optimal Power Distribution for Hyperthermia Cancer Treatment2019Ingår i: Medical Internet of Things (m-IoT) / [ed] Hamed Farhadi, IntechOpen , 2019, s. 55-70Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    We consider an optimization problem for spatial power distribution generated by an array of transmitting elements. Using ultrasound hyperthermia cancer treatment as a motivating example, the signal design problem consists of optimizing the power distribution across the tumor and healthy tissue regions, respectively. The models used in the optimization problem are, however, invariably subject to errors. To combat such unknown model errors, we formulate a robust signal design framework that can take the uncertainty into account using a worst-case approach. This leads to a semi-infinite programming (SIP) robust design problem, which we reformulate as a tractable convex problem that potentially has a wider range of applications.

  • 44.
    Zhang, Silun
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Hu, Xiaoming
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    A moment-based approach to modeling collective behaviors2018Ingår i: 2018 IEEE Conference on Decision and Control (CDC), Institute of Electrical and Electronics Engineers (IEEE), 2018, s. 1681-1687, artikel-id 8619389Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this work we introduce an approach for modeling and analyzing collective behavior of a group of agents using moments. We represent the occupation measure of the group of agents by their moments and show how the dynamics of the moments can be modeled. Then approximate trajectories of the moments can be computed and an inverse problem is solved to recover macro-scale properties of the group of agents. To illustrate the theory, a numerical example with interactions between the agents is given.

  • 45.
    Zhang, Silun
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Hu, Xiaoming
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM.
    Modeling collective behaviors: A moment-based approachManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    Abstract—In this work we introduce an approach for modeling and analyzing collective behavior of a group of agents using moments. We represent the group of agents via their distribution and derive a method to estimate the dynamics of the moments. We use this to predict the evolution of the distribution of agents by first computing the moment trajectories and then use this to reconstruct the distribution of the agents. In the latter an inverse problem is solved in order to reconstruct a nominal distribution and to recover the macro-scale properties of the group of agents. The proposed method is applicable for several types of multi-agent systems, e.g., leader-follower systems. We derive error bounds for the moment trajectories and describe how to take these error bounds into account for computing the moment dynamics. The convergence of the moment dynamics is also analyzed for cases with monomial moments. To illustrate the theory, two numerical examples are given. In the first we consider a multi-agent system with interactions and compare the proposed methods for several types of moments. In the second example we apply the framework to a leader-follower problem for modeling pedestrian crowd dynamics.

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