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  • 1.
    Baravdish, G.
    et al.
    Department of Science and Technology, Linköping University, Linköping, Sweden .
    Svensson, O.
    Department of Science and Technology, Linköping University, Linköping, Sweden .
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Zhang, Ye
    Shenzhen MSU-BIT University, 518172 Shenzhen, China; School of Mathematics and Statistics, Beijing Institude of Technology, China.
    Damped second order flow applied to image denoising2019Ingår i: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 84, nr 6, s. 1082-1111Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we introduce a new image denoising model: the damped flow (DF), which is a second order nonlinear evolution equation associated with a class of energy functionals of an image. The existence, uniqueness and regularization property of DF are proven. For the numerical implementation, based on the Störmer–Verlet method, a discrete DF, SV-DDF, is developed. The convergence of SV-DDF is studied as well. Several numerical experiments, as well as a comparison with other methods, are provided to demonstrate the efficiency of SV-DDF.

  • 2.
    Cheng, Xiaoliang
    et al.
    Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China.
    Lin, Guangliang
    Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Mathematics.
    Gong, Rongfang
    Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Mathematics.
    A modified coupled complex boundary method for an inverse chromatography problem2018Ingår i: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 26, nr 1, s. 33-49Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.

  • 3.
    Dai, Xiaoxia
    et al.
    School of Computing Science, Zhejiang University City College, Hangzhou, People’s Republic of China.
    Zhang, Chengwei
    School of Computing Science, Zhejiang University City College, Hangzhou, People’s Republic of China.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Topology optimization of steady Navier-Stokes flow via a piecewise constant level set method2018Ingår i: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 57, nr 6, s. 2193-2203Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper presents a piecewise constant level set method for the topology optimization of steady Navier- Stokes flow. Combining piecewise constant level set functions and artificial friction force, the optimization problem is formulated and analyzed based on a design variable. The topology sensitivities are computed by the adjoint method based on Lagrangian multipliers. In the optimization procedure, the piecewise constant level set function is updated by a new descent method, without the needing to solve the Hamilton-Jacobi equation. To achieve optimization, the piecewise constant level set method does not track the boundaries between the different materials but instead through the regional division, which can easily create small holes without topological derivatives. Furthermore, we make some attempts to avoid updating the Lagrangian multipliers and to deal with the constraints easily. The algorithm is very simple to implement, and it is possible to obtain the optimal solution by iterating a few steps. Several numerical examples for both two- and three-dimensional problems are provided, to demonstrate the validity and efficiency of the proposed method.

  • 4.
    Flodén, L.
    et al.
    Department of Quality technology and management, mechanical engineering and mathematics, Mid Sweden University, Östersund, Sweden.
    Holmbom, A.
    Department of Quality technology and management, mechanical engineering and mathematics, Mid Sweden University, Östersund, Sweden.
    Lobkova, T.
    Department of Quality technology and management, mechanical engineering and mathematics, Mid Sweden University, Östersund, Sweden.
    Olsson Lindberg, M.
    Department of Quality technology and management, mechanical engineering and mathematics, Mid Sweden University, Östersund, Sweden.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem2017Ingår i: ICNPAA 2016 World Congress: 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, Melville, USA: American Institute of Physics (AIP), 2017, Vol. 1798, nr 1, artikel-id 020177Konferensbidrag (Refereegranskat)
    Abstract [en]

    We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ε → 0.We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ε−1∂tuε as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

  • 5.
    Gulliksson, Mårten
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Holmbom, Anders
    Department of Engineering and Sustainable Development, Mid-Sweden University, Östersund, Sweden.
    Persson, Jens
    Department of Engineering and Sustainable Development, Mid-Sweden University, Östersund, Sweden.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems2016Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 32, nr 2, artikel-id 025005Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak -limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.

  • 6.
    Gulliksson, Mårten
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Ögren, Magnus
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Oleynik, Anna
    Department of Mathematics, University of Bergen, Norway.
    Zhang, Ye
    Faculty of Mathematics, Chemnitz University of Technology, Germany.
    Damped Dynamical Systems for Solving Equations and Optimization Problems2019Ingår i: Handbook of the Mathematics of the Arts and Sciences / [ed] Bharath Sriraman, Springer , 2019Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    We present an approach for solving optimization problems with or without constrains which we call Dynamical Functional Particle Method (DFMP). The method consists of formulating the optimization problem as a second order damped dynamical system and then applying symplectic method to solve it numerically. In the first part of the chapter, we give an overview of the method and provide necessary mathematical background. We show that DFPM is a stable, efficient, and given the optimal choice of parameters, competitive method. Optimal parameters are derived for linear systems of equations, linear least squares, and linear eigenvalue problems. A framework for solving nonlinear problems is developed and numerically tested. In the second part, we adopt the method to several important applications such as image analysis, inverse problems for partial differential equations, and quantum physics.  At the end, we present open problems and share some ideas of future work on generalized (nonlinear) eigenvalue problems, handling constraints with reflection, global optimization, and nonlinear ill-posed problems.

  • 7.
    Lin, G.
    et al.
    Department of Mathematics, Zhejiang University, Hangzhou, China.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Cheng, X.
    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.
    A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography2018Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 97, nr 1, s. 13-40Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Competitive adsorption isotherms must be estimated in order to simulate and optimize modern continuous modes of chromatography in situations where experimental trial-and-error approaches are too complex and expensive. The inverse method is a numeric approach for the fast estimation of adsorption isotherms directly from overloaded elution profiles. However, this identification process is usually ill-posed. Moreover, traditional model-based inverse methods are restricted by the need to choose an appropriate adsorption isotherm model prior to estimate, which might be very hard for complicated adsorption behavior. In this study, we develop a Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem. The solvability and convergence for the proposed inverse method are studied. In particular, using a problem-adapted adjoint, we obtain a convergence rate under substantially weaker and more realistic conditions than are required by the general theory. Based on the adjoint technique, a numerical algorithm for solving the proposed optimization problem is developed. Numerical tests for both synthetic and real-world problems are given to show the efficiency of the proposed regularization method.

  • 8.
    Lin, Guangliang
    et al.
    School of Mathematics, Zhejiang University, Hangzhou, China.
    Cheng, Xiaoliang
    School of Mathematics, Zhejiang University, Hangzhou, China.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik. Faculty of Mathematics, Technische Universitat Chemnitz, Chemnitz, Germany.
    A parametric level set based collage method for an inverse problem in elliptic partial differential equations2018Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 340, s. 101-121Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this work, based on the collage theorem, we develop a new numerical approach to reconstruct the locations of discontinuity of the conduction coefficient in elliptic partial differential equations (PDEs) with inaccurate measurement data and coefficient value. For a given conductivity coefficient, one can construct a contraction mapping such that its fixed point is just the gradient of a solution to the elliptic system. Therefore, the problem of reconstructing a conductivity coefficient in PDEs can be considered as an approximation of the observation data by the fixed point of a contraction mapping. By collage theorem, we translate it to seek a contraction mapping that keeps the observation data as close as possible to itself, which avoids solving adjoint problems when applying the gradient descent method to the corresponding optimization problem. Moreover, the total variation regularizing strategy is applied to tackle the ill-posedness and the parametric level set technique is adopted to represent the discontinuity of the conductivity coefficient. Various numerical simulations are given to show the efficiency of the proposed method.

  • 9.
    Yao, Zhigang
    et al.
    Department of Statistics and Applied Probability, National University of Singapore, , Singapore.
    Zhang, Ye
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Bai, Zhidong
    KLASMOE and School of Mathematics and Statistics, Northeast Normal University, Changchun, China.
    Eddy, William F.
    Department of Statistics, Carnegie Mellon University, Pittsburgh, USA.
    Estimating the Number of Sources in Magnetoencephalography Using Spiked Population Eigenvalues2018Ingår i: Journal of the American Statistical Association, ISSN 0162-1459, E-ISSN 1537-274X, Vol. 113, nr 522, s. 505-518Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Magnetoencephalography (MEG) is an advanced imaging technique used to measure the magnetic fields outside the human head produced by the electrical activity inside the brain. Various source localization methods in MEG require the knowledge of the underlying active sources, which are identified by a priori. Common methods used to estimate the number of sources include principal component analysis or information criterion methods, both of which make use of the eigenvalue distribution of the data, thus avoiding solving the time-consuming inverse problem. Unfortunately, all these methods are very sensitive to the signal-to-noise ratio (SNR), as examining the sample extreme eigenvalues does not necessarily reflect the perturbation of the population ones. To uncover the unknown sources from the very noisy MEG data, we introduce a framework, referred to as the intrinsic dimensionality (ID) of the optimal transformation for the SNR rescaling functional. It is defined as the number of the spiked population eigenvalues of the associated transformed data matrix. It is shown that the ID yields a more reasonable estimate for the number of sources than its sample counterparts, especially when the SNR is small. By means of examples, we illustrate that the new method is able to capture the number of signal sources in MEG that can escape PCA or other information criterion based methods.

  • 10.
    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.

  • 11.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany.
    Gong, Rongfang
    Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China.
    Cheng, Xiaoliang
    Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations2018Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, nr 6, artikel-id 065001Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This study considers the inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional boundary conditions. Unlike the existing methods found in the literature, which usually employ the first-order in time gradient-like system (such as the steepest descent methods) for numerically solving the regularized optimization problem with a fixed regularization parameter, we propose a novel method with a second-order in time dissipative gradient-like system and a dynamical selected regularization parameter. A damped symplectic scheme is proposed for the numerical solution. Theoretical analysis is given for both the continuous model and the numerical algorithm. Several numerical examples are provided to show the robustness of the proposed algorithm.

  • 12.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany .
    Gong, Rongfang
    Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, P. R. China .
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Cheng, Xiaoliang
    Department of Mathematics, Zhejiang University, Hangzhou, P. R. China.
    A coupled complex boundary expanding compacts method for inverse source problems2019Ingår i: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 27, nr 1, s. 67-86Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be determined by additional boundary data. This problem is ill-posed since the dimensionality of the boundary is lower than the dimensionality of the inner domain. To overcome the ill-posed nature, using the a priori information (sourcewise representation), and based on the coupled complex boundary method, we propose a coupled complex boundary expanding compacts method (CCBECM). A finite element method is used for the discretization of CCBECM. The regularization properties of CCBECM for both the continuous and discrete versions are proved. Moreover, an a posteriori error estimate of the obtained finite element approximate solution is given and calculated by a projected gradient algorithm. Finally, numerical results show that the proposed method is stable and effective.

  • 13.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden; .
    Guang-Liang, Lin
    Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People's Republic of China.
    Forssén, Patrik
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Gulliksson, Mårten
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Fornstedt, Torgny
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Cheng, Xiao-Liang
    Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People's Republic of China.
    A regularization method for the reconstruction of adsorption isotherms in liquid chromatography2016Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 32, nr 10, artikel-id 105005Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Determining competitive adsorption isotherms is an open problem in liquid chromatography. Since traditional experimental trial-and-error approaches are too complex and expensive, a modern technique of obtaining adsorption isotherms is to solve the inverse problem so that the simulated batch separation coincides with actual experimental results. This is a typical ill-posed problem. Moreover, in almost all cases the observed concentration at the outlet is the total response of all components, which makes the problem more difficult. In this work, we tackle the ill-posedness with a new regularization method, which is based on the fact that the adsorption isotherms do not depend on the injection profile. The proposed method transfers the original problem to an optimization problem with a time-dependent convection–diffusion equation constraint. Iterative algorithms for solving constraint optimization problems for both the equilibrium-dispersive and the transport-dispersive models are developed. The mass transfer resistance is also estimated by the proposed inverse method. A regularization parameter selection method and the convergence property of the proposed algorithm are discussed. Finally, numerical tests for both synthetic problems and real-world problems are given to show the efficiency and feasibility of the proposed regularization method.

  • 14.
    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.

  • 15.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany.
    Hofmann, B.
    Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany.
    On the second-order asymptotical regularization of linear ill-posed inverse problems2018Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504XArtikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.

  • 16.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Hofmann, Bernd
    Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany.
    On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces2019Ingår i: Fractional Calculus and Applied Analysis, ISSN 1311-0454, E-ISSN 1314-2224, Vol. 22, nr 3, s. 699-721Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

  • 17.
    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.

  • 18.
    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.

  • 19.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Lukyanenko, Dmitry V.
    Department of Mathematics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russian Federation.
    Yagola, Anatoly G.
    Department of Mathematics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russian Federation.
    An optimal regularization method for convolution equations on the sourcewise represented set2015Ingår i: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 23, nr 5, s. 465-475Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article, we consider an inverse problem for the integral equation of the convolution typein a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a prioriinformation (sourcewise representation) based on optimal recovery theory we propose a new method. Theregularization and optimization properties of this method are proved. An optimal minimal a priori error ofthe problem is found. Moreover, a so-called optimal regularized approximate solution and its correspondingerror estimation are considered. Eciency and applicability of this method are demonstrated in a numericalexample of the image deblurring problem with noisy data.

  • 20.
    Zhang, Ye
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany.
    Yao, Z.
    Department of Statistics and Applied Probability, National University of Singapore, Singapore, Singapore.
    Forssén, P.
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Fornstedt, T.
    Department of Engineering and Chemical Sciences, Karlstad University, Karlstad, Sweden.
    Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach2019Ingår i: Annals of Applied Statistics, ISSN 1932-6157, E-ISSN 1941-7330, Vol. 13, nr 4, s. 2011-2042Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reac-tants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hor-mone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

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