Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

The inverse problem of using measurements to estimate unknown parameters of a system often arises in engineering practice and scientific research. This paper proposes a Collage-based parameter inversion framework for a class of partial differential equations. The Collage method is used to convert the parameter estimation inverse problem into a minimization problem of a function of several variables after the partial differential equation is approximated by a differential dynamical system. Then numerical schemes for solving this minimization problem are proposed, including grid approximation and ant colony optimization. The proposed schemes are applied to a parameter estimation problem for the Belousov-Zhabotinskii equation, and the results show that the proposed approximation method is efficient for both linear and nonlinear partial differential equations with respect to unknown parameters. At worst, the presented method provides an excellent starting point for traditional inversion methods that must first select a good starting point.


Introduction
In industrial and engineering applications there are broad classes of inverse problems that can be described as the problems that seek to go backwards from measurements to estimated parameter values 1, 2 .In this paper we concentrate on the following partial differential equation PDE with m unknown parameters: du dt f u, Du, x, t, λ 1 , . . ., λ m , u x, t 0 u 0 x 1.1 where u u 1 , . . ., u n and f f 1 , f 2 , . . ., f n are n-dimensional vector functions, and Du D 1 u, D 2 u, . . ., D K u is a K-dimensional vector consisting of spatial partial derivatives of

Formulation from Parameter Estimation to Minimization Problem
In this section, we restrict our discussion of technical details to a minimum.The reader is referred to 8, 13 for greater mathematical details.The framework presented in this paper is an extension of the Picard contraction mapping method for a class of inverse problems of ordinary differential equations 8 , the theoretical basis for which comes from the Collage theorem 14 .
Proposition 2.1.(Collage theorem) Let V, d be a complete metric space, and let P be a contractive map on V with fixed point u * and contraction factor c P ∈ 0, 1 .Then In 8 , the framework for solving the inverse problem of ODEs by Collage theorem was set up.Seek an ODE initial value problem u f u, t , u t 0 u 0 that admits u t as either a solution or an approximate solution, where f is restricted to a class of functional forms, for example, affine and quadratic.Associated with the initial value problem is the Picard integral operator P : It is well known that, subject to appropriate conditions on f, the operator P is contractive over an appropriate Banach function space V.By taking u as the target solution, the approximate vector field g u, t of f u, t associated with the operator P is found by minimizing the squared Collage distance d 2 u, P u .Now we turn to discuss the parameter estimation problem of 1.1 by the use of the Collage method.Firstly some basic assumptions on 1.1 are listed as follows.
i x, t ∈ Ω × t 0 , T , where Ω ⊂ R N is a bounded region; t 0 and T are two positive constants satisfying t 0 < T.
ii u x, t is a vector function with the form of u x, t u 1 x, t , u 2 x, t , . . ., u n x, t , u i x, • is a differentiable function, u i •, t ∈ C α Ω , and here α is the highest order number of spatial partial derivative involved in 1.1 .
iii f i u, Du, x, t, λ 1 , . . ., λ m 1 ≤ i ≤ n are, for the moment, continuous.iv The exact solution u * x, t of the system 1.In 13 , we have showed that, subject to appropriate conditions on the vector field f, the Picard operator W is contractive over a complete space V of functions supported over the domain Ω × t 0 , T .The space V is equipped with norm where Then an interesting inequality is obtained see 13 for details : 0 when u u * , so one can find the estimate values of the unknown parameters λ 1 , . . ., λ m by the use of the minimization of the squared Collage distance d 2 u, Wu .However, in many practical problems the target function u x, t will be generated by interpolating observational or experimental data points u x i , t j , collected at various locations x i at various times t j .Obviously, there needs a further discussion for the case u x, t / u * x, t for applying the minimization method of the squared Collage distance to practical problems.Proposition 2.2.Let u x, t satisfy u x, • be differentiable, and let ∂D i u/∂t x, t be continuous for i 1, 2, . . ., K. Then where Du 0 x Du x, t 0 , and C > 0 is a positive constant.
Proof.It follows from the differential mean-value theorem that for i 1, . . ., K From the continuity assumption on ∂D i u/∂t x, t , we have that where

2.15
We find from the definition of the norm

2.18
We have that

2.19
Letting C C 1 C 2 , we gain the result of Proposition 2.3.
The following theorem follows immediately from the inequality 2.10 and Proposition 2.3.Theorem 2.4.Let u * x, t and u x, t be the exact solution and the target of 1.1 , respectively.Denote u x, t 0 by u 0 x , and denot u * x, t 0 by u 0 x .Assume that ∂D i u * /∂t x, t and ∂D i u/∂t x, t are continuous for i 1, 2, . . ., K. Then where worst, the presented method can provide an excellent starting point for traditional inversion methods.
In a real problem, it is important to make the error bound of d u, u * obtained from 2.20 as small as possible.Obviously, there is no problem with the first term of the right-hand side of 2.20 , which approaches zero as T approaches t 0 .For guaranteeing the effectiveness of the proposed minimization method, it is necessary to construct the target solution u from the known measurements of 1.1 such that d Du 0 x , Du 0 x is as small as possible.If the target solution u satisfies that Du 0 x D 1 u 0 x , . . ., D K u 0 x , then the target function u and the exact solution u * have the same spatial derivatives at the initial time point t 0 , and d Du 0 x , Du 0 x 0. We have from 2.20 that In general, the Hermite interpolation method can be used to construct the target solution u x, t .When the exact solution u * x, t is given in the form of data points x i , t j , it can be expected to provide d Du 0 x , Du 0 x with a small value by taking the spatial derivative values of the exact solution u * x, t at initial time point t 0 .

Algorithm for Function of Several Variables
Differing from the ideas proposed in 8, 13 , the unknown parameters of 1.1 will be estimated by finding the minimum of the function of several variables determined by d 2 u, Wu .Let J be the vector function defined as follows: J u, Du, x, t, λ 1 , . . ., λ m t t 0 f u, Du, x, s, λ 1 , . . ., λ m ds, 3.1 then where u i and u i 0 denote the ith part of the vector function u and u 0 , respectively.Let

3.8
Denoting d 2 1 , d 2 2 by F 1 l, r and F 2 m, b , respectively, we have that

Numerical Approximation Methods
From the previous section, the function of several variables obtained from the Collage method has the form of a sum every member of which depends only upon a few variables.This leads to the conclusion that many parameter estimation problems for PDEs can be solved in the exact way known from classical analysis.However many problems can only be solved by an approximate numerical method when the function F λ 1 , . . ., λ m is especially complicated, such as when f associated with F is nonlinear.Also approximate numerical methods are suitable for the case that the number of variables is great.In this paper, we are interested in the grid approximation and ant colony optimization methods, and these methods will be applied to the unknown parameter estimation of 1.1 .Note that the ranges of unknown parameters may be assumed from the physical understanding of the problem and modified from the analysis of numerical approximation results.In this section we assume that λ 1 , . . ., λ m ∈ S, where S is a bounded domain with the form Thus, the continuous optimization problem associated with the parameter estimation of 1.1 can be phrased as  Example 4.1.We demonstrate the methods for the system 3.6 with assumptions: the domain Ω × 0, T 0, 1.0 × 0, 0.5 , the parameter domain S 0, 1 × 0, 1 × 0, 1 × 0, 1 , the initial condition u 0 x sinx, v 0 x cos x, and the target solution u x, t xt 3 sinx, v x, t x 3 t cos x.By applying the algorithm presented in Section 3, the coefficients of 3.12 and 3. 13

4.3
The estimates of the unknown parameters l, r and m, b can be obtained by solving the optimization problems of F 1 l, r and F 2 m, b , respectively.

Grid Approximation
We firstly describe a partition scheme of the parameter domain S. For i 1, 2, . . ., m, the intervals λ min i , λ max i are partitioned with step h i λ i,j 1 − λ i,j , j 0, 1, . . ., N i − 1, that is, 4.4  Let λ min λ 1,0 , . . ., λ m,0 .We define the spatial grid GR S by the formula For testing the effect of the grid approximation method, the minimization problems of 3.12 and 3.13 are solved with S 0, 1 × 0, 1 , h i 0.01 i 1, 2 , the results are shown in Figures 1 and 2. Note that the parameter estimation of 3.6 cannot be solved by the framework proposed in 13 due to f being nonlinear with respect to the unknown parameters.
In Figure 1, the red point is the global minimum position of F 1 l, r , where l * 0.18, r * 0.9 and F 1 l * , r * 2.8556e −04 .Similarly, m * , b * 0.14, 0.01 is the global minimum point of F 2 m, b with a minimum F 2 m * , b * 0.0143 see Figure 2 .Sometimes the stationarity conditions ∂F/∂λ i 0 can be used to reduce the computational complexity.For example, it follows from ∂F 1 l, r /∂l 0 that The minimum of F 1 l, r can be found by viewing F 1 l, r as a function with respect to the variable r; the result is shown in Figure 3.

Ant Colony Optimization Approximation
The ant colony optimization ACO algorithm was inspired by the observation of real ant colonies.Its inspiring source is the foraging behavior of real ants, which enables them to find the shortest paths between nest and food sources 15, 16 .Recently, ACO algorithms for continuous optimization problems have received an increasing attention in swarm computation; many researches have shown that the ACO algorithms have great potential in solving a wide range of optimization problems, including continuous optimization 17-22 .These ACO algorithms for continuous domains can be directly used for solving the minimization problem of 4.2 .In 17 , Shelokar et al. proposed a particle swarm optimization PSO hybridized with an ant colony approach PSACO for optimization of multimodal continuous functions, which applies PSO for global optimization and the idea of ant colony approach to update positions of particles to attain rapidly the feasible solution space see 17 for detail .for example, the PSACO algorithm is used for the minimization problem of 3.12 ; the results in Figures 4, 5, and 6 are obtained.

From Theorem 2 . 4 ,
the true approximation error d u, u * is controlled by T − t 0 3/2 , the spatial derivative approximation error d Du 0 x , Du 0 x ,and the Collage distance d u, Wu .For a given target solution u, the first two terms of the right-hand side of 2.20 are fixed; so the smallest upper bound of d u, u * associated with the inequality 2.20 can be obtained by the minimization of d u, Wu .Thus, Theorem 2.4 provides a theoretical basis for finding the unknown parameters of 1.1 by minimizing the squared Collage distance.At

F 1 Figure 1 :
Figure 1:The grid approximation for the minimum of F 1 l, r .

F 2 Figure 2 :
Figure 2: The grid approximation for the minimum of F 2 m, b .

F 1 Figure 6 :
Figure 6: The global minimum of F 1 l, r for every iteration process.
Let u * x, t and u x, t be the exact solution and the target solution of 1.1 , respectively.Assume that ∂D i u * /∂t x, t and ∂D i u/∂t x, t are continuous for i 1, 2, . . ., K. Then there exists a positive constant C such that Firstly, from Proposition 2.2, there are two positive constants C 1 and C 2 such that Du x, t − Du 0 here S Ω is the area or volume of the domain Ω.Thus the inequality 2.12 holds.