Sobolev Gradient Approach for Huxley and Fisher Models for Gene Propagation

The application of Sobolev gradient methods for finding critical points of the Huxley and Fisher models is demonstrated. A comparison is given between the Euclidean, weighted and unweighted Sobolev gradients. Results are given for the one dimensional Huxley and Fisher models.


Introduction
The numerical solution of nonlinear problems is a topic of basic importance in numerical mathematics, as stated in [1].It has been a subject of extensive investigation in the past decades, thus having vast literature [2][3][4][5].The most widespread way of finding numerical solutions is first discretizing the given problem, then solving the arising system of algebraic equations by a solver which is generally some iterative method.For nonlinear problems most often Newton's method is used.However, when the work of compiling the Jacobians exceeds the advantage of quadratic convergence, one may prefer gradient type iterations including steepest descent or conjugate gradients.An important example in this respect is the Sobolev gradient technique, which is relying on descent methods.The Sobolev gradient technique presents a general efficient preconditioning approach where the preconditioners are derived from the representation of the Sobolev inner product.
A detailed analysis regarding the construction and the application of Sobolev gradients can be found in [6].For a quick overview of Sobolev gradients, applications and some open problems in the subject we refer to [19].
Sobolev gradients are also useful for preconditioning for linear and nonlinear problems.Sobolev preconditioning [20] has been tested on some first order and second order linear and nonlinear problems and it is found comparable in terms of efficiency and stability with other methods such as Newton's method and Jacobi method.For differential equations with nonuniform behavior on long intervals, "Sobolev gradients have proved to be effective if we divide the interval of interest into pieces and take a recursive approach (cf.[21])".Sobolev gradients have interesting applications in the field of geometric modelling [22].It has been proved therein that the Sobolev gradient is a very useful tool for minimizing functionals that pertain to the length of curves, curvatures, surface area etc.Recently, the paper [23] has shown the possible applications of Sobolev gradient technique for systems of Differential Algebraic Equations.
The idea of a weighted Sobolev gradient has been introduced by W. T. Mahavier in [7].The weighted Sobolev gradient has successfully exhibited its effectiveness in dealing with linear and nonlinear singular differential equations with regular and some typically irregular singularities.Weighted gradients have also been used for DAEs and it turns out that weighted Sobolev gradients outperform unweighted Sobolev gradients in many situa-tions.
In the field of gene technolog, Modelling of gene frequencies is of the prospective area of research.Its applications can be seen in livestocks and agricultural crops.By the modification of their genes, they can be made more resistive to infection and to produce more yield.To derive historical patterns of migration, archaeologists are expecting that study of the entire human genetic material, will facilitate to map geographical distribution of signature genes.By using gene technology, many bacteria have been developed to prescribed antibiotics.From medical point of view, it is important to study the genetic background of diseases, with implications in diagnosis, treatment and drug development.In order to make use of genetic population data, we need to understand the dynamics of gene patterns through the population.

Fisher and Huxley Models
In the 1930s, number of authors proposed reaction-diffusion equations to model changing gene frequencies in a population.One of the earliest and best known such equations was that of Fisher.In his paper in 1937 [24], he proposed a reaction-diffusion equation with quadratic source term that models the spread of a recessive advantageous gene through a population i.e.; where is the frequency of the new mutant gene, p  is the coefficient of diffusion, and is the intensity of selection in favor of the mutant gene.The equation predicts a wave front of increasing allele frequency, propagating through a population.The quadratic logistic term of Fisher's equation is more appropriate for asexual species.m Fisher's assumptions for a sexually reproducing species lead to a Huxley reaction-diffusion equation, with cubic logistic source term for the gene frequency of a mutant advantageous recessive gene.Huxley's equation is given by  

Review of Sobolev Gradient Methods
In this section we discuss the Sobolev gradient and steepest descent.A detailed analysis regarding the construction of Sobolev gradients can be seen in [6].
Let us consider is a positive integer and is a real valued function on .We can define its gradient as For as above, but with G .,. S an inner product on different from the standard inner product n R .,. n R , there is a function so that The linear functional For each When gradient is defined in a finite or infinite dimensional Sobolev space we call it Sobolev gradient.Steepest descent can be classified into two categories: the one is discrete and other continuous steepest descent.Let be a real-valued function, defined on a Hilbert space  be its gradient with respect to the inner product .,. S defined on H . Discrete steepest descent method is a process of constructing a sequence   i x so that is given and where for each , is minimal in some appropriate sense.In continuous steepest descent we construct a function so that Under suitable conditions on G ,   Continuous steepest descent is interpreted as a limiting case of discrete steepest descent.So (7) can be considered as a numerical method for approximating solutions to (9).Continuous steepest descent gives a theoretical starting point for proving convergence of discrete steepest descent.Using (7) one seeks lim and using ( 9) one seeks lve these problems by using various desc L the vecto r space M R equipped with the usual inner p uct rod

Using Second Order Operators
for and where 1, 2, , The time-step f e p ex p is th p desired at the n t time level.In terms of erators problem can be written as .We can put the solution of this problem in other of minimizing a functional via steepest descent.Define which is zer we have the desired p .The functional has a minimum of zero when is zero so we will

 
L p look for the minimum of this functional.This functional is a convex functional that guarantees global minima in  , a solution to problem (11).The aim is to find the dient of a convex functional   gra F p associated with the problem and use this gradie steepest descent minimization process to finding the zero of the functional, that is the minimum of   nt in F u and the solution of the original problem.

Gradients and Minimization
The gradient for test function .The gradient points in the directio h n of greatest increase of the functional.The direction of greatest decrease of the functional is In this partic gives the desired gradient for steepest descent in h t L .

2
The operators 2 0 2 , : are the adjoints of i.e. the initial conditions.The internodal spacing was x  .The value of  was set to 1 for all the experiments.We   so that both source because functions has the same maximum value.The function was then evolved.The updated value of for a given time step was considered to be correct when the infinity norm of p p   πL p was less than .We set Ĥ we used the same step size regardless of the nodal spacing.The total number of minimization steps for fifteen time steps, the largest value of  that can be used and CPU time were recorded in Tables 1 and 2 H and 2 2 Ĥ are found by s respectively.Here is the adjoint of .Following rical experiments for the solution of Fisher and From the tables we see that the results in 2 2 H are far better than 2 , in fact there is no 2 convergence for .The best results are in the weighted Sobolev space Ĥ .When we perform minimization in 2 2 Ĥ the convergence is three times faster for solving Huxley's model than from that Here we suggest another approach, in order to avoid second order operators.Once again consider the problem and we wish to minimize the functional G p has a mi mum when the gradient is equal to zero, and this might be considered and also in so e new roduc es the condition for finding p at the next time step.Here , : a r e t h e a d j o i n s o f 0 1 , D D respectively.We want to minimize this functional in 2 L , m inner p t spac Once again numerical experiments are conduc us ted by ing the same parameters as defined earlier.for solution of Fisher and Huxley models were conducted as follows.The updated value of p for a given time step was considered to be correct when the infinity norm of was less than H  we used the same step-size regardless of th odal spacing.The total number of minimization steps for fifteen time steps, the largest value of e n  that can be used and CPU time were recorded in Tables 3 and 4.
We note that the finer the spacing the less CPU time the Sobolev gradient technique uses in comparison to the usual steepest descent method.The step size for minimization in 2 L has to decrease as the spacing is refined.From the tables one can see that the results in H  are the best.

Using First Order Operators using
Once again consider the problem  1 .
ion which is first order in e is to solve We define functions 1 (36) The functional for the pr The problem is considered to be solved when   , F p q has been minimized, that is, when 0 S T   ty norms of S and T are less than some desired tolerance.The gradients are The Sobolev gradients in (40) 2 1 H are found by solving We want to minimize this functional in ( L H and so in the new inner product spaces 1 2 . To define these new inner products we follow the tec ique of Mahavier [7] for singular differential equations and use weighted Sobolev spaces , , , Numerical experiments are conducted by us sa (46) ing the me parameters as defined in Section 2.2 .The updated value of p for a given time step was considered to be correct when the infinity norms of both s and T were less than for the time increment.The total number of minimization steps for fifteen time steps, the largest value of  that can be used and CPU time were recorded in Tables 5 and 6.L has to decrease as the spacing is refined.
From the table one can see that the results in H are far better than 2 L and results in the space

Summary and Conclusions
In this paper, we have presented minimization schemes sed on the Sobolev for the Huxley and Fisher's models ba gradient technique [6].The Sobolev gradient technique is computationally more efficient than the usual steepest descent method as the spacing of the numerical grid is made finer.Choosing an optimal inner product can improve the performance with respect to which the Sobolev gradient works better.it is still an open question what the absolutely optimal inner product is, and it is possible that different inner products might not make large differences in computational performance in all cases.One advantage of steepest descent is that it converges even for a poor initial guess.The Sobolev gradient methods presented here converge even for rough initial guess or jumps in the initial guess.
In to increase at the origin and then spread throughout the range.As expected mutant take over is greatly retarded in the Huxley model compared to the Fisher model.So, for asexually reproducing population, a cubic source term is more appropriate than a quadratic source term and for sexually reproducing population, Fisher's equation is more appropriate.

t
must be prescribed small e ha nough ti ve 1 t m 

0 D and 2 e
Sobolev gradient a proac o the problem of minimizing functionals is to do the minimization in Sobolev spaces which correspond to the problem.In this paper only discrete Sobolev spaces D respectively.Th p-are used.We define two such spaces in which the minimization can be compared to minimization in 2 L .We are prompted to consider the space 2 Huxley's equations were conducted as follows.A system of M nodes was set up with 

1 D and 0 D
22) his takes into account the coefficients of because t in   L p and   F p .The desired Sobolev gradi for a finite dimensional version of the problem with discrete time steps is given by   ing.The asso on

15 time steps using first order operators for the Huxley's model.
ual steepest descent.For the Fisher and Huxley model the same step size  can be used for all spacings  when minimizing in the appropriate Sobolev space.The step-size for minimization in 2