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Zhang, Y., Gong, R., Gulliksson, M. & Cheng, X. (2019). A coupled complex boundary expanding compacts method for inverse source problems. Journal of Inverse and Ill-Posed Problems, 27(1), 67-86
Open this publication in new window or tab >>A coupled complex boundary expanding compacts method for inverse source problems
2019 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 27, no 1, p. 67-86Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Walter de Gruyter, 2019
Keywords
Inverse source problem, expanding compacts method, finite element method, error estimation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-68829 (URN)10.1515/jiip-2017-0002 (DOI)000457195600006 ()2-s2.0-85053166222 (Scopus ID)
Funder
Knowledge Foundation, 20170059
Note

Funding Agencies:

Alexander von Humboldt foundation  

Natural Science Foundation of China  11571311  11401304 

Fundamental Research Funds for the Central Universities  NS2018047 

Available from: 2018-09-11 Created: 2018-09-11 Last updated: 2019-02-13Bibliographically approved
Gulliksson, M., Ögren, M., Oleynik, A. & Zhang, Y. (2019). Damped Dynamical Systems for Solving Equations and Optimization Problems. In: Bharath Sriraman (Ed.), Handbook of the Mathematics of the Arts and Sciences: . Springer
Open this publication in new window or tab >>Damped Dynamical Systems for Solving Equations and Optimization Problems
2019 (English)In: Handbook of the Mathematics of the Arts and Sciences / [ed] Bharath Sriraman, Springer , 2019Chapter in book (Other academic)
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.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Optimization, damped dynamical systems, convex problems, eigenvalue problems, image analysis, inverse problems, quantum physics, Schrödinger equation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:oru:diva-71881 (URN)10.1007/978-3-319-70658-0 (DOI)978-3-319-70658-0 (ISBN)
Available from: 2019-01-29 Created: 2019-01-29 Last updated: 2019-04-01Bibliographically approved
Baravdish, G., Svensson, O., Gulliksson, M. & Zhang, Y. (2019). Damped second order flow applied to image denoising. IMA Journal of Applied Mathematics, 84(6), 1082-1111
Open this publication in new window or tab >>Damped second order flow applied to image denoising
2019 (English)In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 84, no 6, p. 1082-1111Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Oxford University Press, 2019
Keywords
Nonlinear flow, image denoising, p-parabolic, p-Laplace, inverse problems, regularization, damped Hamiltonian system, symplectic method, Störmer–Verlet.
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-79218 (URN)10.1093/imamat/hxz027 (DOI)000509388900002 ()
Note

Funding Agency:

Alexander von Humboldt Foundation

 

Available from: 2020-01-16 Created: 2020-01-16 Last updated: 2020-02-06Bibliographically approved
Zhang, Y., Yao, Z., Forssén, P. & Fornstedt, T. (2019). Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach. Annals of Applied Statistics, 13(4), 2011-2042
Open this publication in new window or tab >>Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach
2019 (English)In: Annals of Applied Statistics, ISSN 1932-6157, E-ISSN 1941-7330, Vol. 13, no 4, p. 2011-2042Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Institute of Mathematical Statistics, 2019
Keywords
Adaptive discretization algorithm, Bayesian, Biosensor, Integral equation, Rate constant, Variational method
National Category
Signal Processing
Identifiers
urn:nbn:se:oru:diva-79216 (URN)10.1214/19-AOAS1263 (DOI)000509780500001 ()2-s2.0-85076480522 (Scopus ID)
Available from: 2020-01-16 Created: 2020-01-16 Last updated: 2020-02-14Bibliographically approved
Zhang, Y. & Hofmann, B. (2019). On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces. Fractional Calculus and Applied Analysis, 22(3), 699-721
Open this publication in new window or tab >>On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces
2019 (English)In: Fractional Calculus and Applied Analysis, ISSN 1311-0454, E-ISSN 1314-2224, Vol. 22, no 3, p. 699-721Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Walter de Gruyter, 2019
Keywords
linear ill-posed operator equation, asymptotical regularization, fractional derivatives, stopping rules, source conditions, convergence rates, acceleration
National Category
Mathematics
Identifiers
urn:nbn:se:oru:diva-75799 (URN)10.1515/fca-2019-0039 (DOI)000478760500008 ()2-s2.0-85070305500 (Scopus ID)
Note

Funding Agencies:

Alexander von Humboldt Foundation  

German Research Foundation (DFG-grant)  HO 1454/12-1 

Available from: 2019-08-16 Created: 2019-08-16 Last updated: 2019-08-16Bibliographically approved
Zhang, Y., Gong, R., Cheng, X. & Gulliksson, M. (2018). A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations. Inverse Problems, 34(6), Article ID 065001.
Open this publication in new window or tab >>A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations
2018 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 6, article id 065001Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2018
Keywords
inverse source problems, dynamical system, regularization, convergence, symplectic method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-66813 (URN)10.1088/1361-6420/aaba85 (DOI)000431055900001 ()
Funder
Knowledge Foundation, 20170059
Note

Funding Agencies:

Alexander von Humboldt foundation  

Natural Science Foundation of China  11401304  11571311

Available from: 2018-04-27 Created: 2018-04-27 Last updated: 2018-05-14Bibliographically approved
Cheng, X., Lin, G., Zhang, Y., Gong, R. & Gulliksson, M. (2018). A modified coupled complex boundary method for an inverse chromatography problem. Journal of Inverse and Ill-Posed Problems, 26(1), 33-49
Open this publication in new window or tab >>A modified coupled complex boundary method for an inverse chromatography problem
Show others...
2018 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 26, no 1, p. 33-49Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Walter de Gruyter, 2018
Keywords
Chromatography; adsorption isotherm; inverse problem; coupled complex boundary method; Tikhonov regularization
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:oru:diva-58691 (URN)10.1515/jiip-2016-0057 (DOI)000423813300003 ()
Funder
Swedish Research Council, 2015-04627
Note

Funding Agencies:

NSFC  11571311  11401304 

STINT  IB2015-5989 

KK HOG  20150233 

AForsk  15/497 

Available from: 2017-07-17 Created: 2017-07-17 Last updated: 2018-02-12Bibliographically approved
Lin, G., Cheng, X. & Zhang, Y. (2018). A parametric level set based collage method for an inverse problem in elliptic partial differential equations. Journal of Computational and Applied Mathematics, 340, 101-121
Open this publication in new window or tab >>A parametric level set based collage method for an inverse problem in elliptic partial differential equations
2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 340, p. 101-121Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Inverse problem, Partial differential equations, Collage theorem, Regularization, Parametric level set, Total variation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-64605 (URN)10.1016/j.cam.2018.02.008 (DOI)000440264600007 ()2-s2.0-85043499961 (Scopus ID)
Funder
The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), IB2015-5989
Note

Funding Agencies:

Alexander von Humboldt-Stiftung

National Natural Science Foundation of China (NSFC) 11571311

Available from: 2018-01-29 Created: 2018-01-29 Last updated: 2018-08-22Bibliographically approved
Lin, G., Zhang, Y., Cheng, X., Gulliksson, M., Forssén, P. & Fornstedt, T. (2018). A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography. Applicable Analysis, 97(1), 13-40
Open this publication in new window or tab >>A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography
Show others...
2018 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 97, no 1, p. 13-40Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Taylor & Francis Group, 2018
Keywords
Chromatography; adsorption isotherm; inverse problem; Kohn–Vogelius method; convergence rate
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:oru:diva-55172 (URN)10.1080/00036811.2017.1284311 (DOI)000417831700003 ()2-s2.0-85011298596 (Scopus ID)
Funder
Knowledge Foundation, 20150233Swedish Research Council, 2015-04627
Note

Funding Agencies:

AForsk Foundation  15/497

STINT  IB2015-5989 

NSFC  11571311 

Available from: 2017-02-01 Created: 2017-02-01 Last updated: 2018-01-03Bibliographically approved
Zhang, Y., Fornstedt, T., Gulliksson, M. & Dai, X. (2018). An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data. Inverse Problems in Science and Engineering, 26(10), 1464-1489
Open this publication in new window or tab >>An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data
2018 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, no 10, p. 1464-1489Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Oxfordshire, United Kingdom: Taylor & Francis, 2018
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-64002 (URN)10.1080/17415977.2017.1411912 (DOI)000438638300005 ()2-s2.0-85037706652 (Scopus ID)
Funder
Swedish Research Council, 2015-04627
Note

Funding Agencies:

Swedish Knowledge Foundation (KKS) project HOG

AForsk Foundation

Available from: 2018-01-10 Created: 2018-01-10 Last updated: 2018-08-30Bibliographically approved
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