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  • 1.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Fornstedt, T.
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Dai, X.
    School of Computing Science, Zhejiang University City College, Hangzhou, China.
    An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data2018Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, nr 10, s. 1464-1489Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present here the theoretical results and numerical analysis of a regularization method for the inverse problem of determining the rate constant distribution from biosensor data. The rate constant distribution method is a modern technique to study binding equilibrium and kinetics for chemical reactions. Finding a rate constant distribution from biosensor data can be described as a multidimensional Fredholm integral equation of the first kind, which is a typical ill-posed problem in the sense of J. Hadamard. By combining regularization theory and the goal-oriented adaptive discretization technique,we develop an Adaptive Interaction Distribution Algorithm (AIDA) for the reconstruction of rate constant distributions. The mesh refinement criteria are proposed based on the a posteriori error estimation of the finite element approximation. The stability of the obtained approximate solution with respect to data noise is proven. Finally, numerical tests for both synthetic and real data are given to show the robustness of the AIDA.

  • 2.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Hernandez Bennetts, Victor
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Schaffernicht, Erik
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Reconstructing gas distribution maps via an adaptive sparse regularization algorithm2016Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 24, nr 7, s. 1186-1204Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we present an algorithm to be used by an inspectionrobot to produce a gas distribution map and localize gas sources ina large complex environment. The robot, equipped with a remotegas sensor, measures the total absorption of a tuned laser beam andreturns integral gas concentrations. A mathematical formulation ofsuch measurement facility is a sequence of Radon transforms,which isa typical ill-posed problem. To tackle the ill-posedness, we developa new regularization method based on the sparse representationproperty of gas sources and the adaptive finite-element method. Inpractice, only a discrete model can be applied, and the quality ofthe gas distributionmap depends on a detailed 3-D world model thatallows us to accurately localize the robot and estimate the paths of thelaser beam. In this work, using the positivity ofmeasurements and theprocess of concentration, we estimate the lower and upper boundsof measurements and the exact continuous model (mapping fromgas distribution to measurements), and then create a more accuratediscrete model of the continuous tomography problem. Based onadaptive sparse regularization, we introduce a new algorithm thatgives us not only a solution map but also a mesh map. The solutionmap more accurately locates gas sources, and the mesh map providesthe real gas distribution map. Moreover, the error estimation of theproposed model is discussed. Numerical tests for both the syntheticproblem and practical problem are given to show the efficiency andfeasibility of the proposed algorithm.

  • 3.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Lin, G.
    Department of Mathematics, Zhejiang University, Hangzhou, China.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Forssén, P.
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Fornstedt, T.
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Cheng, X.
    Department of Mathematics, Zhejiang University, Hangzhou, China.
    An adjoint method in inverse problems of chromatography2017Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 25, nr 8, s. 1112-1137Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm.

  • 4.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Lukyanenko, D. V.
    Physical Faculty, Department of Mathematics, Lomonosov Moscow State University, Moscow, Russia.
    Yagola, A. G.
    Physical Faculty, Department of Mathematics, Lomonosov Moscow State University, Moscow, Russia.
    Using Lagrange principle for solving two-dimensional integral equation with a positive kernel2016Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 24, nr 5, s. 811-831Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This article is devoted to a Lagrange principle application to an inverse problem of a two-dimensional integral equation of the first kind with a positive kernel. To tackle the ill-posedness of this problem, a new numerical method is developed. The optimal and regularization properties of this method are proved. Moreover, a pseudo-optimal error of the proposed method is considered. The efficiency and applicability of this method are demonstrated in a numerical example of an image deblurring problem with noisy data.

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