Regularization Method with Two Differential Operators for Simultaneous Inversion of Source Term and Initial Value in a Time-Fractional Black-Scholes Equation

This paper is devoted to identifying source term and initial value simultaneously in a time-fractional Black–Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of unbounded operator equation system, and conditional stability is established under certain source conditions. We propose a stable approximation method to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied 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Stationary caseInverse Problems in Science and Engineering, Vol. 15, No. 8 | 18 December 2007 Cross Ref The boundary-element method for the determination of a heat source dependent on one variableJournal of Engineering Mathematics, Vol. 54, No. 4 | 3 January 2006 Cross Ref Inverse source problem for a transmission problem for a parabolic equationJournal of Inverse and Ill-posed Problems, Vol. 14, No. 1 | 1 Jan 2006 Cross Ref Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problemInverse Problems, Vol. 21, No. 3 | 9 May 2005 Cross Ref Conditional stability in determining a heat sourceJournal of Inverse and Ill-posed Problems, Vol. 12, No. 3 | 1 Jun 2004 Cross Ref Lipschitz stability in inverse parabolic problems by the Carleman estimateInverse Problems, Vol. 14, No. 5 | 1 January 1999 Cross Ref Determination of the unknown sources in the heat-conduction equationComputational Mathematics and Modeling, Vol. 8, No. 4 | 1 Oct 1997 Cross Ref An estimation method for point sources of multidimensional diffusion equationApplied Mathematical Modelling, Vol. 21, No. 2 | 1 Feb 1997 Cross Ref A reliable estimation method for locations of point sources for an n-dimensional Poisson equationApplied Mathematical Modelling, Vol. 20, No. 11 | 1 Nov 1996 Cross Ref Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary dataInverse Problems, Vol. 10, No. 6 | 1 January 1999 Cross Ref Conditional Stability in Determination of Densities of Heat Sources in a Bounded DomainControl and Estimation of Distributed Parameter Systems: Nonlinear Phenomena | 1 Jan 1994 Cross Ref Conditional stability in determination of force terms of heat equations in a rectangleMathematical and Computer Modelling, Vol. 18, No. 1 | 1 Jul 1993 Cross Ref Inverse problems and Carleman estimatesInverse Problems, Vol. 8, No. 4 | 1 January 1999 Cross Ref Determination of an unknown function in a parabolic equation with an overspecified conditionMathematical Methods in the Applied Sciences, Vol. 13, No. 5 | 1 Nov 1990 Cross Ref Some stability estimates for a heat source in terms of overspecified data in the 3-D heat equationJournal of Mathematical Analysis and Applications, Vol. 147, No. 2 | 1 Apr 1990 Cross Ref Parameter identification in hyperbolic and parabolic partial differential equations of cylindrical geometry from overspecified boundary dataInternational Journal of Engineering Science, Vol. 28, No. 10 | 1 Jan 1990 Cross Ref Identification of certain physical parameters in hyperbolic boundary value problems from overspecified boundary dataInternational Journal of Engineering Science, Vol. 26, No. 8 | 1 Jan 1988 Cross Ref An Inverse Problem for a Nonlinear Elliptic Differential EquationMichael Pilant and William RundellSIAM Journal on Mathematical Analysis, Vol. 18, No. 6 | 1 August 2006AbstractPDF (793 KB)An inverse problem for the heat equationInverse Problems, Vol. 2, No. 4 | 1 January 1999 Cross Ref A Note on an Inverse Problem Related to the 3-D Heat EquationInverse Problems | 1 Jan 1986 Cross Ref Determination of a Source Term in a Linear Parabolic Differential Equation with Mixed Boundary ConditionsInverse Problems | 1 Jan 1986 Cross Ref Nonparametric algorithm for input signals identification in static distributed-parameter systemsIEEE Transactions on Automatic Control, Vol. 29, No. 7 | 1 Jul 1984 Cross Ref Parameter determination in parabolic partial differential equations from overspecified boundary dataInternational Journal of Engineering Science, Vol. 20, No. 6 | 1 Jan 1982 Cross Ref Determination of an unknown non-homogeneous term in a linear partial differential equation .from overspecified boundary dataApplicable Analysis, Vol. 10, No. 3 | 2 May 2007 Cross Ref Distributed parameter system indentification A survey†International Journal of Control, Vol. 26, No. 4 | 16 May 2007 Cross Ref Determination of a source term in a linear parabolic partial differential equationZeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 27, No. 3 | 1 May 1976 Cross Ref Some general remarks on improperly posed problems for partial differential equationsSymposium on Non-Well-Posed Problems and Logarithmic Convexity | 22 August 2006 Cross Ref Estimation of parameters in partial differential equations from noisy experimental dataChemical Engineering Science, Vol. 26, No. 6 | 1 Jun 1971 Cross Ref Determination of an unknown forcing function in a hyperbolic equation from overspecified dataAnnali di Matematica Pura ed Applicata, Vol. 85, No. 1 | 1 Dec 1970 Cross Ref The character of non-uniqueness in the conductivity modelling problem for the earthPure and Applied Geophysics PAGEOPH, Vol. 80, No. 1 | 1 Jan 1970 Cross Ref Volume 5, Issue 2| 1968SIAM Journal on Numerical Analysis199-459 History Submitted:29 September 1967Published online:03 August 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

In this paper we prove some new converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces.

EIT image reconstruction for two-phase flow monitoring using a sub-domain based regularization method

This paper investigates the initial value inversion of a time-fractional Black–Scholes equation, which is an ill-posed inverse problem. The quasi-reversibility regularization method and the Landweber iterative regularization method are employed to address this issue. Based on specific selection criteria for both a priori and a posteriori regularization parameters, we derive error estimates for the regularized solutions. The convergence rate obtained through the quasi-reversibility method is not universally optimal, whereas the convergence rate that is derived from the Landweber iterative approach consistently achieves order optimality. In our numerical experiments section, examples demonstrate that the proposed regularization methods are effective and stable.

This paper deals with an inverse problem of determining source term and initial data simultaneously for a space-fractional diffusion equation in a strip domain, with the aid of extra measurement data at a fixed time. The uniqueness results are obtained by a simple trick based on the linear property of the proposed equation. Since this problem is ill-posed, a modified quasi-reversibility method is obtained by employing the Fourier transform. Error estimates for source term and initial value are obtained from a suitable parameter choice rule. Finally, several numerical examples show that the proposed regularization method is effective and stable.

The solution of inverse problems has many applications in mathematical physics. Regularization methods can be applied to obtain the solution of ill-conditioned inverse problems by solving a family of neighboring well-posed problems. Thus, it is significant to investigate the regularization methods to increase the accuracy and efficiency of the solution of inverse problems. In this work, a new regularization filter and the related regularization method based on the singular system theory of compact operator are proposed to solve ill-posed problems. The Cauchy problem of Laplace equation of the first kind is a kind of well-known ill-posed problem. Numerical tests show that the proposed regularization method can solve the Cauchy problems more efficiently under a proper selection of regularization parameters. Numerical results also show that the proposed method is especially effective in solving ill-posed problems with big perturbations.

This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L 1 {L_{1}} -forward difference implicit approximation for the forward problem. Numerical results using L 1 {L_{1}} -forward difference implicit approximation ( L 1 {L_{1}} -FDIA) for the inverse problem are also discussed briefly.

The main purpose of this Newsletter section is to make even more attractive to our multidisciplinary audience by providing additional information such as: informed commentaries on problems, solutions, methods and applications; news on initiatives in funding and research programmes; reports on conferences, symposia and workshops; summaries and commentaries on important papers appearing in this and other journals; forthcoming workshops and conferences; and book reviews. For further details, see the first Newsletter published in volume 8 issue 1 (February 1992).

INVERSE PROBLEMS NEWSLETTER

Computed tomography (CT) is an ill-posed problem. Reconstruction on unstructured grid reduces the computational cost and alleviates the ill-posedness by decreasing the dimension of the solution space. However, there was no systematic study on edge-preserving regularization methods for CT reconstruction on unstructured grid. In this work, we propose a novel regularization method for CT reconstruction on unstructured grid, such as triangular or tetrahedral meshes generated from the initial images reconstructed via analysis reconstruction method (e.g., filtered back-projection). The proposed regularization method is modeled as a three-term optimization problem, containing a weighted least square fidelity term motivated by the simultaneous algebraic reconstruction technique (SART). The related cost function contains two non-differentiable terms, which bring difficulty to the development of the fast solver. A fixed-point proximity algorithm with SART is developed for solving the related optimization problem, and accelerating the convergence. Finally, we compare the regularized CT reconstruction method to SART with different regularization methods. Numerical experiments demonstrated that the proposed regularization method on unstructured grid is effective to suppress noise and preserve edge features.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

In this paper, we consider an inverse problem to simultaneously reconstruct the source term and initial data associated with a parabolic equation based on the additional temperature data at a terminal time t = T and the temperature data on an accessible part of a boundary. The conditional stability and uniqueness of the inverse problem are established. We apply a variational regularization method to recover the source and initial value. The existence, uniqueness and stability of the minimizer of the corresponding variational problem are obtained. Taking the minimizer as a regularized solution for the inverse problem, under an a priori and an a posteriori parameter choice rule, the convergence rates of the regularized solution under a source condition are also given. Furthermore, the source condition is characterized by an optimal control approach. Finally, we use a conjugate gradient method and a stopping criterion given by Morozov's discrepancy principle to solve the variational problem. Numerical experiments are provided to demonstrate the feasibility of the method.

Distribution of electrical potentials over the surface of the heart, which is called the epicardial potential distribution, is a valuable tool to understand whether there is a defect in the heart. Direct measurement of these potentials requires highly invasive procedures. An alternative is to reconstruct these epicardial potentials non-invasively from the body surface potentials, which constitutes one form of the ill-posed inverse problem of electrocardiography (ECG). The goal of this study is to solve the inverse problem of ECG using several regularization methods and compare their performances. We employed Tikhonov Regularization, Truncated Singular Value Decomposition (TSVD), Least Squares QR (LSQR) methods in this study. We compared the effectiveness of these regularization methods to solve the ill-posed inverse ECG problem. Some of the regularization methods require a regularization parameter to solve the inverse problem. We used the well-known L-Curve method to obtain the regularization parameter. The performance of the regularization methods for solving the inverse ECG problem was also evaluated based on a realistic heart-torso model simulation protocol. In this paper, we also investigated the usage of genetic algorithm (GA) for regularizing the ill-posed inverse ECG problem. The results showed that GA can be applied to regularize the ill-posed problem when combined with the results of conventional regularization methods or additional information about solutions.

Inverse problems can usually be modelled as operator equations in infinite-dimensional spaces with a forward operator acting between Hilbert or Banach spaces—a formulation which quite often also serves as the basis for defining and analyzing solution methods. The additional amount of structure and geometric interpretability provided by the concept of an inner product has rendered these methods amenable to a convergence analysis, a fact which has led to a rigorous and comprehensive study of regularization methods in Hilbert spaces over the last three decades. However, for numerous problems such as x-ray diffractometry, certain inverse scattering problems and a number of parameter identification problems in PDEs, the reasons for using a Hilbert space setting seem to be based on conventions rather than an appropriate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, non-Hilbertian regularization and data fidelity terms incorporating a priori information on solution and noise, such as general Lp-norms, TV-type norms, or the Kullback–Leibler divergence, have recently become very popular. These facts have motivated intensive investigations on regularization methods in Banach spaces, a topic which has emerged as a highly active research field within the area of inverse problems. Meanwhile some of the most well-known regularization approaches, such as Tikhonov-type methods requiring the solution of extremal problems, and iterative ones like the Landweber method, the Gauss–Newton method, as well as the approximate inverse method, have been investigated for linear and nonlinear operator equations in Banach spaces. Convergence with rates has been proven and conditions on the solution smoothness and on the structure of nonlinearity have been formulated. Still, beyond the existing results a large number of challenging open questions have arisen, due to the more involved handling of general Banach spaces and the larger variety of concrete instances with special properties.