# KULBAY MAGIRA OPTIMAL CONTROL METHODS TO SOLVE INVERSE PROBLEMS FOR DIFFUSION PROCESSES ABSTRACT

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1 KULBAY MAGIRA OPTIMAL CONTROL METHODS TO SOLVE INVERSE PROBLEMS FOR DIFFUSION PROCESSES ABSTRACT to the dissertation of Kulbay M. submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (PhD) by specialty 6D Mathematical and Computer Modeling General characteristics of the work. Identification of sources intensity in the inverse problem for diffusion equation and continuation problem for steady-state diffusion are considered in this thesis. Relevance of the research. The thesis is devoted to construction of numerical optimization methods for solving inverse diffusion problems and the construction of appropriate algorithms and carrying out computational experiments. Identification of unknown sources in diffusion processes is considered in this thesis. The relevance of numerical solution of direct and inverse boundary value problems for diffusion equations with advection terms are widely used for mathematical modelling of natural and technological processes. The analytical methods for finding solutions to direct diffusion-advection problems have been studied intensivel and much of the present effort is directed towards the speed and accuracy of computational implementations of numerical techniques rather than further theoretical developments. On the other hand, the study of the corresponding inverse source problems has raised a lot of interest lately mainly due to their many practical applications including the detection of pollution sources in an environmental medium. These algorithms can be used to interpret the measured data in technological and natural processes, which simulated by inverse problems for diffusion processes with transfer and nonlinear sources depending on the final set of parameters. In many practical applications only the final measurement of diffusing values is available to explorer, thus it is necessary to determine the sources of the distribution in the environment, or the initial distribution. The problem of identifying the right hand side for the heat equation by the final temperature measurement at the given boundary conditions is considered in the first part of the thesis. Usually such problems are formulating in terms of optimal control. The necessary and sufficient optimality conditions for this kind of problem were obtained by the methods of control theory in the work of A. Egorov for the first time. Currentl the method of quasi-inversion and regularization became standard for the numerical solution of inverse problems. The researchers come from the fact that in optimal control problems gradient functionality can be expressed in terms of the relevant decision of the dual problem. It is known that a necessary condition for optimality in some cases

2 allows us to express the optimal control solutions through direct and adjoint problems. These conditions generate a coupled system of equations consisting of these problems. The solution of the system (if it exists), satisfies the necessary conditions for optimality. Provided that necessary condition of functionality convexity is also sufficient. In the literature failed to find examples of numerical implementation of the Fourier method for solving the system of equations consisting of direct and adjoint problems. Usually this related to the fact that in the process control problems, which described by partial differential equations, this system is non-linear and the Fourier method is not applicable. In the found examples the linear solution of the dual problem is only used for numerical construction of functional gradient and joint solution of the direct and conjugate problems by the Fourier method is not carried out. The numerical implementation of Fourier method for a coupled system of equations consisting of direct and adjoint problems are considered in the first and second part of the thesis. It is important to determining the spacewise dependent source term distribution F x and / or time intensity Ht in the considered area in many engineering applications. For the heat equation excluding the transfer process, this problem adequately studied. In the most of the studies the function F x is reconstructed by the final measurement. It is a traditional approach. In some formulations additional data of measurements are used within of the interest area. However, from a practical point of view, sometimes to make research within the medium is difficult or impossible, and the researcher available only measurement at the border of interest area. In the third part of the thesis we considered algorithm and numerical methods for solving inverse problems for the source of functions of the form F x H t. It is assumed that at least one of these two functions is known. A new uniform numerical method for recovery F x, and the function Htdescribing the dynamics of the intensity of the source, through the function g t u, t is developed. The diffusion equation with transfer is used to describe the processes of distribution of impurities and contaminants in the environment and in the filtration problems. Another important application is the problem of water purification from contamination and to estimate the parameters of the filter material. Thus practically important is the problem of finding sources of contaminants if the researcher only available measurements at certain times, or if the measurements are carried out only on the border of the interest area. The Cauchy problem for Laplace s equation arises from many branches of science and engineering such as medicine, geophysics and non-destructive testing. It is so important to reconstruct data on the boundar because the object of interest the boundary is not available for measurement in practice. As an additional information all possible measurements are used on the available part of the boundary. Numerical computations become very difficult if there is no a priory information on the solution. A small error in the data deduces large error in the solution. Usually the regularization technique is required to find a stable approximate solution. In order to obtain a stable numerical solution for these kinds of ill-posed problems many regularization methods have been proposed. Application of spectral methods to the

3 Cauchy problem for Laplace s equation was used for the first time by M.M. Lavrent ev. A lot of regularization methods are implemented for a Cauchy problem of Laplace s equation by many authors. 3D model for diffusion process in cylindrically layered medium is used to describe the process of heat conduction and diffusion in protective coatings of gas and oil pipelines, the shell of reactors in the chemical industry and other industrial processes. The problem of controlling temperature of oil pipeline is particularly relevant in the problems of oil transportation in areas with harsh climate and low winter temperatures. The problem is ill-posed in the sense that small errors in the Cauchy data may lead to large errors in the recovered solution, however has many practical applications and the methods and programs for its solution are relevant. Similar models can be used to control the diffusion and thermal conductivity in the protective screens and the reactor containment in the chemical and energy industries. In such cases, only the outer side of coverings is available for measurements, when it is necessary to recover the conditions inside the coverings in order to control of equipment state. As an application of this model, we can take the nuclear power plant reactors in Japan. Another example is when the observer has the opportunity on the one hand the environment (air, liquid) to measure the temperature, heat flux, but the aim is to determine the temperature on the inaccessible border of environment. This kind of problem can be described as a mathematical model of the thermal probing, used in thermal imagers. The aim of this work is a development of optimal control methods to solve inverse problems for diffusion processes in fluid dynamics, heat and mass transfer. Research problems. Task 1) It is required to recover the unknown space-dependent source F( x, y) L 2 [0,1] L2[0,1 ], in the following boundary value problem for the heat equation: u u u u u u x t u xx u yy Fx, yh t, x 0,1 0,1, t 0, T, x, 0 0, 0, t 0, x,0, t 0, y x,1, t ux,1, t 0, 1, t u1, t 0, from the final measurement temperature: u x T u x, y, 1. Task 2) Let R 1, R2 be inner and outer radii of the cylinder with height H, the cylinder material is nonuniform, and the thermal conductivity is piecewise constant function of the radius. Finding the unknown temperature on the inner surface of the cylinder r R1 is required, if the measurements of the heat flux and the temperature at the outer boundary are given.

4 Note that all of our subsequent arguments are also valid if there are data at the inner boundary and need to find a solution on the outer boundary. Task 3) It is required to identify: (1) the unknown space-dependent source F x, subsequently defined as ISP1, (2) the unknown time-dependent source Ht, subsequently defined as ISP2, in the following advection-diffusion equation with space-dependent diffusion coefficient t x x 0,, : 0, 0, ; x T,0 0, 0, 0,, 0, 0,, u au x u x F x H t x t T u x u t ux t t T from the Dirichlet boundary measured data g t u t t T,, 0,. The object of the study are inverse problems for the diffusion models with transfer with unknown source terms, as well as the inverse problem of determining the boundary values on the inaccessible border in the Cauchy problem for the Laplace equation. The method and methodology of this work. The basic methodology for the numerical solution of inverse problems are methods for solving ill-posed problems, the theory of extreme problems and the discretization methods for the numerical solution. We are based on the quasi-solution and regularization methods. Inverse problems are solved by minimizing of the residual functional method. Scientific novelty. The numerical implementation of the Fourier method for solving a system consisting of direct and adjoint problems for multidimensional inverse source problem for parabolic equation by final measurement data is considered first time in this work. System expresses a necessary condition of equality to zero first variation of functionality in a minimum point. The method allowed to implement a large number of numerical examples and set the empirical values of the parameters for calculating, in which the problem can be successfully solved. The presence of explicit formulas for the required functionality through the Fourier series allows detailed study of the obtained solutions, which is an advantage of the proposed method. An effective method for solving the continuation inverse problem for 3D steadystate diffusion model inside a cylindrical layered medium is proposed. This model is used to describe the process of heat conduction and diffusion in protective coatings of gas and oil pipelines, the shell of reactors in the chemical industry and other industrial processes. The novelty of this problem is that the inverse problem is solved numerically for the three-dimensional case, and continuation problem for elliptic equation is considered for piecewise constant diffusion coefficient medium. Numerical solutions obtained in two direct ways are compared: on the base of explicit

5 formula for regularized quasi-solution and solutions of the Cauchy problem by the Fourier method. Numerical experiments are investigated and show the capacity of proposed method. Under the suitable choice of regularization parameter we recover the distribution of temperature on the inner boundary with satisfactory quality for noisy data. We have determined more admissible values of the main parameters such as a regularization parameter and harmonic numbers for different noise level. In this study we come to conclusion that the application of the proposed approach gives an efficient way to solve the Cauchy problem for the 3D Laplace equation in a cylindrical layered medium. Numerical experiments are investigated and show the capacity of proposed method for smooth boundary condition only. Under the suitable choice of regularization parameter we recover the distribution of temperature on the inner boundary with satisfactory quality for noisy data. We have determined more admissible values of the main parameters such as a regularization parameter and harmonic numbers for different noise level. In this study we also considers the inverse problems of identifying either the unknown space-dependent heat source F x or the unknown time-dependent heat source Ht of the variable coefficient advection-diffusion equation with separable sources of the form F x H t from supplementary time-dependent temperature measurement at the right boundary of the domain. We proved that solutions of both inverse source problems can be identified uniquely under some regularity assumptions. We also present two novel non-iterative methods for the numerical identification of the unknown spacewise dependent source F x (ISP1) and of the unknown time-dependent source Ht (ISP2) in the advection-diffusion equation with variable diffusion coefficient only from boundary measured data. Another originality of the work lies in the fact that we recover the spacewise dependent source term F x from the time-dependent boundary measurement. The robustness and limitations of the two algorithms are investigated through numerical examples related to the reconstruction of continuous and discontinuous sources. Theoretical and practical significance. The theoretical significance of the research is that the direct and inverse boundary value problems for diffusion equations with advection terms are widely used for mathematical modelling of natural and technological processes. The analytical methods for finding solutions to direct diffusion-advection problems have been studied intensivel and much of the present effort is directed towards the speed and accuracy of computational implementations of numerical techniques rather than further theoretical developments. On the other hand, the study of the corresponding inverse source problems has raised a lot of interest lately mainly due to their many practical applications including the detection of pollution sources in an environmental medium. Aprobation of research work. The main results of the thesis were presented at the following international conferences:

6 - The 6-th International Conference Inverse Problems: Modeling and Simulation (May 21-26, 2012, Antalya, Turkey); - The 7th International Conference Inverse Problems: Modeling and Simulation (IPMS-2014, Fethiye, Turke May 26 31); - At the International Conference "Functional analysis and its applications", October 2-5, Astana, 2012; - The 1 st International scientific and practical conference «Intellectual information and communication technologies as a tool for realization of the third industrial revolution within the framework of the Kazakhstan-2050 strategy» (Astana, 2013). - International conference "Advanced mathematics, computations and applications 2014"; - The 6th International Youth Scientific School - Conference "Theory and Computational Methods for Inverse and ill-posed problems", Almat Kazakhstan, December 8-14,2014; Besides, the results were reported in the seminars of the mechanics and mathematics department al-farabi Kazakh National University in , Almaty. Publications. The results of the thesis are published in 13 works. 3 articles were published in journals recommended by the Committee for Control of Education and Science of Republic of Kazakhstan, two papers were published in the ranking Scopus and Tomson Reuters peer-reviewed international journals, which one of them has impact factor. 8 abstracts were published in the proceedings of international conferences, which 3 of them were published in the proceedings of abroad international conferences. The structure and scope of the thesis. The thesis is written in the form of manuscripts in Russian and consists of an introduction, three chapters, conclusion and list of references. The work is presented on 89 pages, contains 24 pictures and 15 tables. List of references 123.

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