On the variational adjoint method for an inverse problem for a class of elliptic equation

2010 ◽  
Author(s):  
Bingxian Wang ◽  
Kangxiu Hu
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Adjoint Method

scholarly journals The Adjoint Method for the Inverse Problem of Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shou-Lei Wang ◽  
Yu-Fei Yang ◽  
Yu-Hua Zeng
Keyword(s):  
Inverse Problem ◽  
Option Pricing ◽  
Implied Volatility ◽  
Adjoint Method ◽  
Numerical Examples ◽  
Limited Memory ◽  
Minimization Procedure ◽  
Volatility Function ◽  
The Cost ◽  
Quasi Newton

The estimation of implied volatility is a typical PDE inverse problem. In this paper, we propose theTV-L1model for identifying the implied volatility. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy. The gradient is computed via the adjoint method which provides us with an exact value of the gradient needed for the minimization procedure. We use the limited memory quasi-Newton algorithm (L-BFGS) to find the optimal and numerical examples shows the effectiveness of the presented method.

An inverse problem for the recovery of the vascularization of a tumor

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Thierry Colin ◽  
Angelo Iollo ◽  
Jean-Baptiste Lagaert ◽  
Olivier Saut
Keyword(s):  
Growth Rate ◽  
Inverse Problem ◽  
Diffusion Equation ◽  
Cancer Cells ◽  
Blood Vessels ◽  
Tumor Cells ◽  
Adjoint Method ◽  
Source Term ◽  
Simplified Model ◽  
Oxygen Distribution

AbstractIn this paper we present a simplified model of tumor growth and an associated inverse problem. Our model consists in describing the evolution of a population of cancer cells and the density of oxygen. The growth rate depends on the concentration of oxygen. The oxygen distribution is computed by means of a diffusion equation, the source term being localized on the blood vessels. We consider the inverse problem that consists in recovering the position of the blood vessels assuming the distribution of tumor cells. We use an adjoint method. Results relative to idealized clinical cases are discussed.

scholarly journals Semi-Idealized Study on Estimation of Partly and Fully Space Varying Open Boundary Conditions for Tidal Models

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Jicai Zhang ◽  
Haibo Chen
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Adjoint Method ◽  
Open Boundary ◽  
Feature Points ◽  
Open Boundary Conditions ◽  
Negative Effects ◽  
Control Variables ◽  
Estimation Problems ◽  
Minimum Number

Two strategies for estimating open boundary conditions (OBCs) with adjoint method are compared by carrying out semi-idealized numerical experiments. In the first strategy, the OBC is assumed to be partly space varying and generated by linearly interpolating the values at selected feature points. The advantage is that the values at feature points are taken as control variables so that the variations of the curves can be reproduced by the minimum number of points. In the second strategy, the OBC is assumed to be fully space varying and the values at every open boundary points are taken as control variables. A series of semi-idealized experiments are carried out to compare the effectiveness of two inversion strategies. The results demonstrate that the inversion effect is in inverse proportion to the number of feature points which characterize the spatial complexity of open boundary forcing. The effect of ill-posedness of inverse problem will be amplified if the observations contain noises. The parameter estimation problems with more control variables will be much more sensitive to data noises, and the negative effects of noises can be restricted by reducing the number of control variables. This work provides a concrete evidence that ill-posedness of inverse problem can generate wrong parameter inversion results and produce an unreal “good data fitting.”

A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Gongsheng Li ◽  
Xianzheng Jia ◽  
Chunlong Sun
Keyword(s):  
Inverse Problem ◽  
Dispersion Equation ◽  
Adjoint Method ◽  
Noisy Data ◽  
Adjoint Problem ◽  
Admissible Set ◽  
Lipschitz Stability ◽  
Optimal Perturbation ◽  
Advection Dispersion ◽  
Bilinear Functional

Abstract This paper deals with an inverse problem of determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation with final observations using the variational adjoint method. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknowns is induced. With the aid of an adjoint problem, a bilinear functional based on the variational identity is set forth with which a norm for the unknown is well-defined under suitable conditions, and then a conditional Lipschitz stability for the inverse problem is established. Furthermore, numerical inversions with random noisy data are performed using the optimal perturbation algorithm, and the inversion solutions give good approximations to the exact solution as the noise level goes to small.

scholarly journals AN INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER ELLIPTIC EQUATION IN A RECTANGLE

2014 ◽  
Vol 19 (2) ◽  
pp. 241-256 ◽  
Author(s):  
Yashar T. Mehraliyev ◽  
Fatma Kanca
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Inverse Boundary ◽  
Numerical Tests ◽  
Second Order Elliptic Equation

In this paper, the inverse problem of finding a coefficient in a second order elliptic equation is investigated. The conditions for the existence and uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite-difference scheme combined with an iteration method is presented and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated.

Using the adjoint method for solving the nonlinear GPR inverse problem

2006 ◽  
Author(s):  
Luc van Kempen ◽  
Dinh Nho Hào ◽  
Hichem Sahli
Keyword(s):  
Inverse Problem ◽  
Adjoint Method
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