Fast inverse solver for identifying the diffusion coefficient in time-dependent problems using noisy data
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O R I G I NA L
Jinhui Jiang · M. Shadi Mohamed Hongqiu Li
· Mohammed Seaid ·
Fast inverse solver for identifying the diffusion coefficient in time-dependent problems using noisy data
Received: 10 June 2020 / Accepted: 6 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose an efficient inverse solver for identifying the diffusion coefficient based on few random measurements which can be contaminated with noise. We focus mainly on problems involving solutions with steep heat gradients common with sudden changes in the temperature. Such steep gradients can be a major challenge for numerical solutions of the forward problem as they may involve intensive computations especially in the time domain. This intensity can easily render the computations prohibitive for the inverse problems that requires many repetitions of the forward solution. Compared to the literature, we propose to make such computations feasible by developing an iterative approach that is based on the partition of unity finite element method, hence, significantly reducing the computations intensity. The proposed approach inherits the flexibility of the finite element method in dealing with complicated geometries, which otherwise cannot be achieved using analytical solvers. The algorithm is evaluated using several test cases. The results show that the approach is robust and highly efficient even when the input data is contaminated with noise. Keywords Inverse problem · Finite element method · Partition of unity method · Diffusion coefficient identification · Transient heat transfer
1 Introduction The diffusion equation in its general form is one of the fundamental equations in science and engineering. It is often used to describe the behaviour of heat and particles or organisms as they diffuse or spread into a medium [1]. Its applications can be found in a diverse range of topics such as chemically reacting flows [2], molecular peptide structures [3] and subsurface light transport [4], among many others. A key parameter in this equation is the diffusion coefficient which describes the rate at which the heat, particles or organisms spread into the J. Jiang State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China M. Shadi Mohamed (B) School of Energy, Geoscience, Infrastructure and Society, Heriot-Watt University, Edinburgh EH14 4AS, UK E-mail: [email protected]
M. Seaid Department of Engineering, University of Durham, South Road, Durham DH1 3LE, UK H. Li Mechatronic Engineering College, Jinling Institute of Technology, Nanjing 211169, China
J. Jiang et al.
medium. In a given time interval [0, T ] and in an open bounded domain ⊂ R2 with a boundary , the linear form of the diffusion equation can be written as ∂u(t, x) − D∇ 2 u(t, x) = f (t, x), ∂t
(t, x) ∈ [0, T ] × .
(1)
We consider the equation with a mixed-type boundary condition αu +
∂u = g(t, x), ∂n
(t, x) ∈ [0, T ] × ,
(2)
and a given initial conditi
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