The Survey Of Numerical Methods For Inverse Problem And Their Application To The Field Of Oil Exploitation

In the process of development of human society, people are accustomed to, "by the result and the fruit" of this causal relationship and to try to attempt to establish such a causal relationship between qualitative and quantitative description of the (forward problem), while rarely mentioned "by the fruit and the result of "(anti-issue). And because the inverse problem (Inverse Problem) is to detect unknown, generally more difficult. In this paper, the concept of inverse problems, forms, classification, numerical method and the application of several aspects related to inverse problems, introduced some knowledge.The unified form of the inverse problem: even though in many fields of science and engineering technology, there are a large number of inverse problems in the form of variety, vary, but in the framework of operator theory can be unified in the form of: Az = u (2.1)The following from a mathematical point of view we have the inverse problem of differential equations for a rough classification:1.To be determined differential equations in the unknown parameters of the inverse problem--Operator identification problem。2.The inverse problem of initial conditions to be determined——Time course inverse problem。3.Unknown boundary conditions to be determined—Border Control。4.The inverse problem of the boundary shape to be determined—Geometric inverse problems。The inverse problem of the theoretical approaches and regularization method: For the mathematical physics equations in inverse problems, one major concern is the well-posedness and numerical method. The so-called well-posedness problem is that in mathematical physics, the existence and uniqueness and continuous dependence on initial data solution of the problem. If it is at the same time meet the following three conditions are:①(Existence) ?u∈U,Exists z∈F Satisfy the equation(1.2.1)②(Uniqueness)If u1 , u2∈U,z1 and z2 are the solutions of equation(1.2.1) u1≠u2,then z1≠z2③(Stability)Solution is stability for( F, U),Claimed that equation (2.1) well posed. If the above three conditions is not satisfied with any one, then known as the ill-posed. The inverse problem of a particularly important attribute is that it is often "ill-posed" mathematical problem, which makes it both in theoretical analysis or numerical calculations are carrying out a particular difficulty.Of ill-posed problems can be divided into three types:Non-existence: If A is not surjective, then the equation Az = u Not for all u∈U Broken; Is not unique: if A is not injective, then the equation Az = u May have more than one solution;Instability: If the inverse operator A ?1 :U→F Exists, but not continuous, that is, equation Az = u The solution can not be continuously dependent on the data; that the data have small errors, the obtained solution with the true solution of a great error.In the ill-posed problems, people generally concerned about the stability condition does not meet a class of ill-posed problem. Thus, roughly speaking whatever solution is not continuously dependent on the data, all the mathematical problems are called ill-posed problem.The numerical solution of inverse problems: inverse problems for different have different adaptive numerical method, for different numerical method.One of the most universal, in theory, the most comprehensive and effective approach is well-known scholar from the Tikhonov first-class operator (in particular, integral operator) equation as the basic framework, in the 20th century, 60 years creative and subsequently be in-depth development of the regularization method (or strategy). In this paper, the adaptive regularization, iterative regularization, trust region and the truncated conjugate gradient method is discussed.Adaptive regularization method: Given a linear system d = Wr+n (2.2.1)Where W is a linear operator, r as input, d as its output. n for the random vector, its covariance matrix isΣ=EnnT。May be ill-posed problems of its discrete Euler equation [1] (Φ+αH )r =~r (2.2.2) WhereΦ=W TΣ?1W For the Fisher matrix or Fisher operator,, H is a positive definite matrix, and H> 0.In this adaptive regularization mainly discussed the convergence issue. The so-called adaptive regularization refers to the Euler equation (3.1.2), the selection operator H is, you can getThe definition of filter operator ( ) ( )Rαadaptλ=λλ2 +α?1 And Kee (2.2.3) is interpreted as being, then the following formula can be expressed asIn the ideal case, that there is no noise or under the influence of errors n,(2.2.1)Can write dture = Wr+,Which states that (2.2.1) of the true solution. At this point, (2.2.3) into the solution Where ~rt u re =WTdture。Adapted regularization method has some good properties [5]:Property 2.2.1 rnα? rα2≤31 63nα2 Property 2.21.2 ( )rα:= RαadaptΦ~rture→r+,At that time, where r + that the problem of the true solution.Iterative regularization method: This section focuses on iterative Tikhonov regularization, Landweber iterative regularization method and its improvement.Iterative Tikhonov regularization method: Using the following iterative Tikhonov regularization method, we can obtain higher order convergence speed. Iterative Tikhonov regularization method defined as follows: Commonly used in practical calculations the following ways: If the smoothness condition: Satisfied, and the selection strategy according to a priori: Selection parameters, according to iterative regularization data points generated by the convergence rate for:And the optimal convergence rate in time to reach ( )Oδ2 n2n+1。LandweberIterative regularization method: the general iterative scheme for:The first category is used to solve the approximate solution of operator equation, in which 0 <ω<1A2 For the relaxation factor, x 0 = x*, x * The initial guess value. Without loss of generality, it is desirable x * =0。Landweber Iterative regularization method has the advantage of relatively stable, when the right-hand side, when relatively large disturbances can still get better results. But the Landweber iteration convergence slower convergence speed on the improvement of the problem, there are many improvements [11-16]. (2.3.12)Type can be rewritten as Iterative solving equation (2.3.13), to be operator Rk : Y→X For the operator Rk : Y→X,Be redefined as:v-method: v-iterative Landweber method is a kind of acceleration. This method is introduced by Brakhage, which is a broader approach than the Chebyshev polynomial acceleration method [5-7]. v-method refers to the following iterative process: at each iteration step, the solution whereμ1 = 0,ω1=(4 v+2) (4 v+1),for k≥1,Has the following iteration Accurate for a given item in the right-hand side of the case, that is, yδ=y,Obtain the following convergence theor。Theorem 2.3.1 v is located is given, then for the v-method iterative point out that if the Then (( ))x∈RangeA*Avand x = A+y。What's more, if ( )Trust region method: using trust region method for unconstrained optimization problems in generalWe first set to take a trust region radius, and then solving (2.4.1) is a quadratic approximation model for TRSAlgorithm derived therefrom, which we call trust region algorithm, in which the trust region radius, usually a generalized ball, which characterizes the extent to which we believe that the trust domain model [5,17]. Trust region algorithm implementation process is designed to make objective functional minimization, to prove the value of the objective functional is indeed down. Trust region algorithm for the regularity, availability of the following theorem [17,19].Theorem 2.4.2 for the full small, set up, then, when.Theorem 2.4.3 is located in the operator F satisfy the aboveassumptions, the orders, as previously defined. Iterative algorithm is generated by the above-mentioned point Orobanchaceae and time converge to the solution of the problem. Truncated conjugate gradient method: cut into linear and nonlinear conjugate gradient method truncated conjugate gradient method Best photo momentum method is based on the regularization of the inverse problem solving method, and can be classified as operator theory optimization method. This paper describes the theory and practical solution process.Best photo of momentum law, the basic iteration process is as follows:(1) First, given the unknown amount of c (x) the initial guess;(2) The application of numerical method for solving initial boundary value problem (3.5.1) as well as the numerical solution u ( c0 (x) ;xm ,tm),(m=1,2,…,M)(3) with the numerical differential calculation Respectively for c 0 (x)+τ?i(x), (i = 1,2,,n) Numerical method, to calculate equation (3.5.1) and numerical solutions, and seek u ( c0 (x)+τ?i (x);xm,tm)(4) Solving equations (3.5.9), and using equation (3.5.7) obtained the amount of disturbance, take a new initial guess c1 ( x)=c0(x)+δc0(x) Repeat this process until you meet the accuracy requirements so far.Anti-problem-solving in related fields of applications: This first focuses on research more regularization method and Conjugate Gradient (CG) method, the numerical model and numerical solution is as follows.(1) The regularization method: The equation (4.2.1) written in an abstract operator equation h = Kf+e (4.2.2) And using a quadrature formula into its discrete form of the following: h =κf +e (4.2.3)Because the characteristic values concentrated in the vicinity of the origin and thus has great condition number. For the stability of the numerical solution of (4.2.3) the use of discrete regularization method, that is, consider the following unconstrained optimization problem minκf ? h2+αLf2 (4.2.4)In which the so-called regularization parameter, L for the penalty matrix, ie solution treated condition to impose a certain degree of smoothness, typically L is a (semi) definite matrices. For example, we preferred L is defined in the L ? 2 on the identity matrix I, or defined in the Sobolev space W1, 2 on the discrete non-negative Laplace matrix. Take fixed penalty matrix L, the crux of the matter is to determine the appropriate regularization parameter, making regularization functionals both a good approximation of the original problem will be able to overcome high-frequency interference (not stability). Usually signal processing problems, such parameter is the rule of thumb selected.(2) conjugate gradient method: You can use the previously mentioned conjugate gradient method to solve the image restoration problem, this time the image model can be described as a form of the following Where herr ( x, y) = h( x,y) +n( x,y)。The numerical solution of the problem, we get the discretization of the problem The purpose is to enable That is, minimizing the objective function Clearly the objective function for the following quadratic( h→herr) Its gradient and Hessian matrix can be displayed to be counted asThis requires many iterations to complete, in the process, machine errors and problems inherent in the cumulative errors are unreservedly were tried. Improved method is to make the appropriate cut-off iterationThe second, numerical differentiation problem: The following focuses on the following three kinds of numerical differentiation method of seeking。( 1 ) Spline Interpolation : Considerations are defined in [0 , 1] function y ( x),δ>0 Given data is the error level [5,19]. Numerical differentiation of the task is to error data from the discrete(2) smoothing method: This section describes the numerical solution of differential smooth (Mollification) method, which essentially is a regularization method. The basic idea is to inaccurate measurement data smoothing, into an approximation of the smooth function of the derivative problems [5-6]. This smoothing is through a smooth convolution kernel (Polished function) to be achieved, but note that the selection of this kernel function is not the only, the method can be used for all kinds of ill-posed problems, such as high-dimensional inverse When heat conduction problems, numerical differentiation and so on.(3) The integral operator method: This section first describes a derivative of the integral operator method, and then import demand higher derivative integral operator method [26-27]. Under certain conditions, can be regarded as approximate to solve the high-order derivative of a function, define integral operator Where h is the parameter, r = 1,2,,k,(D hr f)( x) Can be used as f ( r )( x) Approximate。And Such that∫11 ( ) =1? J xdx,and J ( i )(1 ) = J( i)(? 1) =0,i=0,1,,k?1。3, the inverse problem of mathematical physics in the oil development of the applications: reservoir numerical simulation of the process is admitted into the reservoir physical parameters substituted into line with the law of reservoir flow model to seek a reservoir output, pressure, water, gas oil ratio and other dynamic parameters. This process is a positive process of solving it is a direct problem. The history matching have to turn, according to the actual observed dynamic parameters to counter the sum of the amendment to these reservoir physical parameters, therefore, it is a reverse process of inversion. That is, the inverse problem discussed in this article.This inversion process can be expressed in two ways: one is to use more rigorous mathematical method to directly solve the inverse process. This method is currently only in some relatively simple issues of theoretical exploration stage, there is no practical application. The other is to repeatedly modify the physical parameters of repeated calculations and trial and error approach, which is currently widely used method. This inversion problems are often multiple solutions, meaning that there are many possible combinations of physical parameters can get similar results. Testing pressure method, using the pressure derivative Analytic Type in the instantaneous permeability; h is the reservoir thickness; q is the bottom flow; is viscosity; K is a kernel function; radial radius of penetration. Reverse only the average reservoir permeability, pressure, skin factor, and several other parameters, can not get the whole parameters of the distribution of reservoir properties, not found solution uniqueness and convergence of the solution.The pressure in the well test or production data, reservoir fluid parameters are known, the model parameters to be determined is the grid block permeability and porosity. In order to obtain the number of observation data is consistent with the model parameters, namely, the grid block permeability and porosity values, the introduction of a priori assumptions geostatistics, using a priori probability density function characterized by a priori model.Assumptions:①3 respectively, the direction of reservoir permeability, and consistent with log-normal distribution, ie Gaussian distribution;②the porosity, also in line with normal distribution, mean and variance are also known;○3 covariance function associated directly with the variogram;④known correlation between porosity and permeability coefficient, porosity and permeability of the natural logarithm of the variogram. Variance and mean from the static data (cores, well logs, seismic) surveying and geological interpretation and statistical analysis of geological obtained.If the output fitting condition that the output of the calculation, the value corresponds to the first i-real output, so that M i , Niare the upper and lower limits of m. By the method of Gauss-Newton, we can get Used by the Gauss-Newton algorithm, can greatly speed up the iterative solution convergence and speed.Multiple calculation results show that good convergence of the method to determine the market penetration of the market more accurately than the porosity of example and practical application results show that this method is feasible.The final article describes the study of inverse problems and development problems facing the prospect of:Mathematics is an unprecedented breadth and depth of penetration to other subject areas, mathematics is a strong impact on the development of economic production and social life in progress. As an emerging discipline of mathematics, inverse problems and human production and everyday life. The emergence of the inverse problem for the traditional mathematical equation has opened up new areas of research, but also promoted the active involvement of production workers in mathematics and life practical problems. Inverse problem has a very broad and practical development prospects, the inverse problem of the rich fruits of research will continue to benefit mankind.