DMLPG method for specifying a control function in two-dimensional parabolic inverse PDEs
Introduction
Inverse problems play a fundamental role in physics and applied mathematics. These problems appear in various applications and many physical phenomena are modeled with inverse partial differential equations [1], [2], [3], [4], [5], [6], [7]. Over the last few decades, specifying unknown sources in inverse source problems has attracted a lot of attention. This kind of inverse problems appears in several areas of engineering and science and covers a wide range of applications such as: electroencephalography (EEG) [1], photo/optical tomography [2], identification of pollution sources in surface water [3], [4], [5], [6], bioluminescence tomography [8] and so on [7].
An overview of various properties of the inverse problems, such as existence, uniqueness, stability and some other theoretical discussions can be found in references [9], [10], [11], [12], [13], [14], [15]. Until now, some numerical methods have been proposed for solving inverse problems such as a local meshless method for Cauchy problem of Poisson’s equation and 2D inverse heat conduction problems [16], finite difference method for identifying a control function in one, two and three-dimensional parabolic inverse problems [17], [18], [19], [20], radial basis functions collocation method for a non-local boundary value problem [21], Adomian decomposition method for identifying a control function in parabolic PDEs [22], [23], homotopy perturbation method for inverse parabolic equations [24], compact finite difference method for determining unknown control parameter in an inverse problem [25], meshless radial point interpolation method for 2D heat equation with non-classical boundary condition [26], dual-reciprocity boundary element method for determination of a control function in 2D diffusion equation [27], shifted Legendre-tau method for 1D inverse diffusion equation [28], [29], method of lines for solving a quasilinear parabolic inverse problem [30], Legendre pseudospectral method for identifying a control function in 3D parabolic equations [31] and so on.
In [32], a greedy MLPG method has been applied for identifying a control function in 2D parabolic PDEs. Unlike the global weak form methods such as Galerkin finite element and element free Galerkin techniques, MLPG does not use global background mesh to evaluate integrals. Instead, the integration process is applied on some regular, simple shape and independent sub-domains. In MLPG method, integrands are complicated moving least squares (MLS) shape functions and the numerical integrations over these shape functions lead to high computational costs in comparison with the finite elements method (FEM), in which integrations are done over simple polynomials. To overcome this drawback, based on the generalized moving least squares (GMLS) approximation [33], an improved version of MLPG has been proposed in [34]. This method is called DMLPG because it approximates functionals directly. Like MLPG method, DMLPG is a truly meshless method based on Petrov–Galerkin formulation. In DMLPG, numerical integrations are done over low-degree polynomial basis functions instead of complicated MLS shape functions. This feature overcomes the main drawback of meshless methods and significantly speeds up the procedure. Regardless of the costs of mesh production and mesh refinement, it can be claimed that DMLPG reduces the computational cost of MLPG to the level of classical FEM. Until now, many practical problems in physic and engineering have been solved via MLPG and DMLPG methods such as two and three-dimensional potential problems [35], time-dependent Maxwell equations [36], Poisson problems [37], 2D and 3D problems in elasticity [38], [39], multi-dimensional coupled damped nonlinear Schrödinger system [40], elliptic interface equations with applications in electrostatic and elastostatic [41], remediation of contaminated groundwater [42], heat conduction problems [43], some Turing-type models [44], time-fractional fourth-order reaction–diffusion problem [45], two-dimensional complex Ginzburg–Landau equation [46] and so on.
In this paper, we apply the low-cost DMLPG method for identifying the unknown control function in 2D parabolic PDEs on regular and irregular domains. The DMLPG method is compared with MLPG from the perspective of accuracy and computational efficiency and the superiority of the DMLPG over the classical MLPG is demonstrated. Then we conclude that the DMLPG technique can be a good alternative to MLPG method in solving inverse PDEs.
The organization of the rest of this paper is as follows: the inverse problem is formulated in Section 2. In Section 3, MLS and GMLS techniques are introduced. Time and spatial discretizations of the problem are provided in Section 4. In Section 5, numerical results are reported. Finally, Section 6 completes this report with a brief conclusion.
Section snippets
Problem formulation
In this article, we study a parabolic partial differential equation with unknown control function as follows [18] subject to the over specification condition over the spatial domain In particular, by choosing weight function , we get an over specification at a point as follows where , , and are known functions.
The inverse problem (2.1) models many physical processes. For
MLS And GMLS methods
This section provides a brief description about the moving least squares (MLS) and generalized moving least squares (GMLS) techniques. The MLS method is a well-known approximation technique which is used to approximate unknown functions in meshless methods. In [33], the GMLS method has been presented as a generalized version of MLS. Unlike the MLS, GMLS technique approximates functionals instead of functions. In GMLS method, functionals are directly approximated and this increases the
Time discretization
In this section, we use a finite difference formula for obtaining a time-discrete scheme. At First, we define the following notation with time step size in which is the final time. A Crank–Nicolson finite difference method is applied for discretization of time derivative as follows: where , and . By simplifying, we obtain the following
Spatial discretization
The
Numerical results
In this section, some numerical experiments are performed for two-dimensional parabolic inverse PDEs on regular and irregular computational domains. These numerical experiments are done by MLPG and DMLPG methods. We also compare these methods in terms of accuracy and computational efficiency. The simulations are performed using MATLAB 2017b software on an Intel Core i7, 2.4 GHz CPU machine with 8 GB of memory.
For numerical simulations, we use the quartic spline weight function which is
Conclusions
In this paper, the MLPG and DMLPG techniques have been applied for identifying a control function in two-dimensional parabolic inverse PDEs on regular and irregular domains. These methods have been compared with each other from the perspective of accuracy and computational efficiency and superiority of DMLPG method has been demonstrated. For this purpose, four test problems with Dirichlet and Neumann boundary conditions have been investigated. In addition, the results of these methods have been
Acknowledgments
The author thanks the reviewers for their useful comments and suggestions that improved the paper.
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