# Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

Authors: K. Atifi; Y. Balouki; El-H. Essoufi; B. Khouiti
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.


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## Body

## 1. Introduction

The inverse problem is expressed when the PDE solution is measured or specified, and we are interested in determining some properties: coefficients, forcing term, boundary, or initial condition from the partial knowledge of the system in a limited time interval (see [1, 2]).In the last recent years, an increasing interest has been devoted to degenerate parabolic equations. Indeed, many problems coming from physics (boundary layer models in [3], models of Kolmogorov type in [4], etc.), biology (Wright-Fisher models in [5] and Fleming-Viot models in [6]), and economics (Black-Merton-Scholes equations in [7]) are described by degenerate parabolic equations [8].The identification of the initial state of nondegenerate parabolic problems is well studied in the literature (see [9–11]). However, as far as we know, the degenerate case has not been analysed in the literature.In this paper, we are interested in estimating the initial condition by the variational method in data assimilation of degenerate/singular parabolic equation:(1)∂tψ-∂xax∂xψx-λxβψ=f,ax=xα,where Ω=0,1, α∈0,1, β∈0,2-α, λ⩽0, and f∈L2(Ω×0,T). With initial and boundary conditions(2)ψx,0=ψ0,ψx=0=ψx=1=0.The mathematical model leads to a nonconvex minimization problem(3)findψ0∈AadsuchthatJψ0=minu∈Aad⁡Ju,where the functional J is defined as follows:(4)Ju=12T∫0TCψt-ψobstL2Ω2,subject to ψ being the weak solution of the parabolic problem (1) with initial state u, ψobs an observation of ψ in Ω×0,T, and C the observation operator. The space Aad is the set of admissible initial states.Problem (3) is ill-posed in the sense of Hadamard. To solve this problem, we propose two approaches.The first approach is based on regularization, for the first time, applied to solve a degenerate inverse problem. The problem thus consists of minimizing a functional of the form(5)JTu=12T∫0TCψt-ψobstL2Ω2dt+ε2u-ψbL2Ω2.Here, the last term in (5) stands for the so-called Tikhonov-type regularization ([12, 13]), ε being a small regularizing coefficient that provides extra convexity to the functional JT and ψb a priori (background) knowledge of the true state ψ0exact (the state to estimate). We consider that the values of ψb are given in each point of analysis grid-points.The second approach is applied when there is a partial knowledge of values ofψb (example 20%); the regularization parameter is very difficult to determine. To overcome this problem, we propose a new approach, based on a hybrid genetic algorithm (married genetic with descent method gradient type). Finally, we make a comparison between the two mentioned approaches (with 20% of ψb).First of all, we prove that problem (3) has at least one solution. The gradient of the functional J is calculated with the adjoint method. Numerical experiments are presented to show the performance of our approaches.


## 2. Problem Statement and Main Result

Consider the following problem:(6)∂tψ+Aψ=fψ0,t=ψ1,t=0∀t∈0,Tψx,0=ψ0x∀x∈Ω,where Ω=]0,1[, f∈L2(Ω×0,T),and A is the operator defined as(7)Aψ=-∂xax∂xψx-λxβψ,ax=xα,with α∈]0,1[, β∈]0,2-α[, and λ⩽0.We want to estimateψ0 thanks to an observation ψobs(x,t) of ψ(x,t) in Ω×0,T. The minimization problem associated with this problem is(8)findψ0∈AadsuchthatJψ0=minu∈Aad⁡Ju,where the functional J is as follows:(9)Ju=12T∫0TCψt-ψobstL2Ω2,subject to ψ being the weak solution of the parabolic problem (6) with initial state u, ψb the background state, and C the observation operator. The space Aad is the set of admissible initial states (will be defined later).We now specify some notations we shall use. Let us introduce the following functional spaces (see [14–16]):(10)V=u∈L2Ω,uabsolutelycontinuouson0,1,S=u∈L2Ω,aux∈L2Ω,u0=u1=0,Ha1Ω=V∩S,Ha2Ω=u∈Ha1Ω,aux∈H1Ω,Hα,01=u∈Hα1∣u0=u1=0,Hα1=u∈L2Ω∩HLoc10,1∣xα/2ux∈L2Ω,with(11)uHa1Ω2=uL2Ω2+auxL2Ω2,uHa2Ω2=uHa1Ω2+auxxL2Ω2,u,vHα1=∫Ωuv+xαuxvxdx.We recall that (see [16]) Ha1 is an Hilbert space and it is the closure of Cc∞0,1 for the norm ·Ha1. If 1/a∈L1Ω then the injections(12)Ha1Ω↪L2Ω,Ha2Ω↪Ha1Ω,H10,T;L2Ω∩L20,T;DA↪L20,T;Ha1∩C0,T;L2Ωare compacts.Firstly, we prove that problem (6) is well-posed, the functional J is continuous, and J is G-derivable in Aad.The weak formulation of problem (6) is(13)∫Ω∂tψvdx+∫Ωax∂xψ∂xv-λxβψvdx=∫Ωfvdx,∀v∈H01Ω.Let(14)Bψ,v=∫Ωax∂xψ∂xv-λxβψvdx.We discuss the following cases.(
1) Noncoercive Case (see [14],
λ
=
0
). In this case, the bilinear form B becomes(15)Bψ,v=∫Ωax∂xψ∂xvdx.We have ax=0 at x=0, from where the bilinear form B will be noncoercive.Let(16)Aad=u∈Ha1Ω;uHa1Ω⩽r,whererisarealstrictlypositiveconstant.We recall the following theorem.Theorem 1 (see [14, 17, 18]).
For allf∈L2(Ω×0,T) and ψ0∈L2(0,1), there exists a unique weak solution which solves problem (6) such that (17)ψ∈C00,T;L2Ω∩L20,T;Ha1and there is a constant CT such that for any solution of (6)(18)supt∈0,T⁡ψtL2Ω2+∫0TaψxtL2Ω2dt≤CTψ0L2Ω2+fL2Ω×0,T2;if more ψ0∈Ha1Ω then (19)ψ∈C00;T,Ha1∩L20,T;Ha2∩H10,T;L2Ωand there is a constant CT such that(20)supt∈0,T⁡ψtHa12+∫0TψtL2Ω2+aψxxtL2Ω2dt⩽CTψ0Ha12+fL2Ω×0,T2.Theorem 2.
Letψ be the weak solution of (6) with initial state ψ0. In noncoercive case, the function(21)φ:Ha1Ω⟶C00,T;Ha1Ω∩L20,T;Ha2Ω∩H10,T;L2Ωψ0⟼ψis continuous, and the functional J has at least one minimum in Aad.Theorem 3.
Letψ be the weak solution of (6) with initial state ψ0. The function(22)φ:Ha1Ω⟶C00,T;Ha1Ω∩L20,T;Ha2Ω∩H10,T;L2Ωψ0⟼ψis G-derivable in Aad.(
2) Subcritical Potential Case (see [19, 20],
λ
≠
0
). Then the bilinear form B becomes(23)Bψ,v=∫Ωax∂xψ∂xv-λxβψvdx.Sinceax=0 at x=0 and limx→0⁡λ/xβ=+∞, the bilinear form B is noncoercive and is noncontinuous at x=0.Consider the not bounded operator(K,D(K)) where(24)Ku=xαuxx+λxβu,∀u∈DK,Dk=u∈Hα,01∩HLoc20,1∣xαuxx+λxβu∈L2Ω.Let(25)Aad=u∈L2Ω;uL2Ω⩽r,whererisarealstrictlypositiveconstant.We recall the following theorem.Theorem 4 (see [15, 19]).
Iff=0 then, for all ψ0∈L2(Ω), problem (6) has a unique weak solution(26)ψ∈C00,T;L2Ω∩C00,T;DK∩C10,T;L2Ω;if more ψ0∈D(K) then(27)ψ∈C00,T;DK∩C10,T;L2Ω;if f∈L2(]0,1[×0,T) then, for all ψ0∈L2(Ω), problem (6) has a unique solution(28)ψ∈C00,T;L2Ω.Theorem 5.
Letψ be the weak solution of (6) with initial state ψ0. In subcritical potential case, the function(29)φ:L2Ω⟶C0,T;L2Ωψ0⟼ψis continuous, and the functional J is continuous in Aad.Theorem 6.
Letψ be the weak solution of (6) with initial state ψ0. The function (30)φ:L2Ω⟶C0,T;L2Ωψ0⟼ψis G-derivable in Aad.


## 3. Proof

Proof of Theorem2.
Letδψ0∈Ha1(Ω) be a small variation such that ψ0+δψ0∈Aad.
Considerδψ=ψδ-ψ, with ψ being the weak solution of (6) with initial state ψ0 and ψδ is the weak solution of (6) with initial state ψ0δ=ψ0+δψ0.
Consequently,δψ is the solution of the variational problem:(31)∫Ω∂tδψvdx+∫Ωax∂xδψx∂xvdx=0δψ0,t=δψ1,t=0∀t∈0,Tδψx,0=δψ0x∀x∈Ω.
Hence,δψ is the weak solution of (6) with f=0. We apply the estimate in Theorem 1 with f=0. This gives the following.
There is a constantCT such that(32)supt∈0,T⁡δψtHa1Ω2+∫0T∂tδψL2Ω2+∂xa∂xδψtL2Ω2dt⩽CTδψ0Ha1Ω2;therefore(33)supt∈0,T⁡δψtHa1Ω2⩽CTδψ0Ha1Ω2(34)δψC0,T;Ha1Ω2⩽CTδψ0Ha1Ω2. And from (32) we have(35)δψtHa1Ω2+∫0T∂xa∂xδψtL2Ω2dt⩽CTδψ0Ha1Ω2,∀t∈0,T∫0TδψtHa1Ω2dt+T∫0T∂xa∂xδψtL2Ω2dt⩽TCTδψ0Ha1Ω2inf⁡1,T∫0TδψtHa1Ω2dt+∫0T∂xa∂xδψtL2Ω2dt⩽TCTδψ0Ha1Ω2∫0TδψtHa1Ω2dt+∫0T∂xa∂xδψtL2Ω2dt⩽TCTinf⁡1,Tδψ0Ha1Ω2.Hence,(36)δψL20,T,Ha2Ω2⩽TCTinf⁡1,Tδψ0Ha1Ω2.
In addition, from (32) we have(37)δψtHa1Ω2+∫0T∂tδψtL2Ω2dt⩽CTδψ0Ha1Ω2,∀t∈0,T(38)δψtL2Ω2+a∂xδψtL2Ω2+∫0T∂tδψtL2Ω2dt⩽CTδψ0Ha1Ω2,∀t∈0,T(39)δψtL2Ω2+∫0T∂tδψtL2Ω2dt⩽CTδψ0Ha1Ω2,∀t∈0,T(40)∫0TδψtL2Ω2dt+T∫0T∂tδψtL2Ω2dt⩽TCTδψ0Ha1Ω2(41)δψH10,T;L2Ω2⩽TCTinf⁡1,Tδψ0Ha1Ω2.Equations (34), (36), and (41) imply the continuity of the function(42)φ:Ha1Ω⟶C00,T;Ha1Ω∩L20,T;Ha2Ω∩H10,T;L2Ωψ0⟼ψ.Hence, the functional J is continuous in(43)Aad=u∈Ha1Ω;uHa1Ω⩽r.
We have1/ax=x-α/2∈L10,1, where α∈]0,1[, which gives Ha1Ω↪compactL2Ω. Since the set Aad is bounded in Ha1Ω, then Aad is a compact in L2(Ω). Therefore, J has at least one minimum in Aad.Proof of Theorem3.
Letψ0∈Aad and δψ0 such that ψ0+δψ0∈Aad; we define the function(44)φ′ψ0:δψ0∈Aad⟶δψ,where δψ is the solution of the variational problem(45)∫Ω∂tδψvdx+∫Ωax∂xδψ∂xvdx=0∀v∈H01Ωδψ0,t=δψ1,t=0∀t∈0,Tδψx,0=δψ0∀x∈Ωand we pose(46)ϕψ0=φψ0+δψ0-φψ0-φ′ψ0δψ0.We want to show that(47)ϕψ0=oδψ0.We easily verify that the function ϕ is solution of the following variational problem:(48)∫Ω∂tϕvdx+∫Ωax∂xϕ∂xvdx=0∀v∈H01Ωϕ0,t=ϕ1,t=0∀t∈0,Tϕx,0=δψ0-δψ02∀x∈Ω.By the same way as that used in the proof of continuity, we deduce(49)ϕC0,T,Ha1Ω2⩽CTδψ0-δψ02Ha1Ω2,ϕL20,T,Ha2Ω2⩽TCTinf⁡1,Tδψ0-δψ02Ha1Ω2,ϕH10,T;L2Ω2⩽TCTinf⁡1,Tψ0-δψ02Ha1Ω2.
Hence, the functionφ:ψ0→ψ is G-derivable in Aad and we deduce the existence of the gradient of the functional J.Proof of Theorem5.
Letδψ0∈L2(Ω) be a small variation such that ψ0+δψ0∈Aad.
Considerδψ=ψδ-ψ, with ψ being the weak solution of (6) with initial state ψ0, and ψδ is the weak solution of (6) with initial state ψ0δ=ψ0+δψ0.
Consequently,δψ is the solution of variational problem(50)∫Ω∂tδψvdx+∫Ωax∂xδψ∂xv-λxβδψvdx=0,∀v∈H01Ωδψ0,t=δψ1,t=0∀t∈0,Tδψx,0=δψ0x∀x∈Ω.
Takev=δψ; this gives(51)∫Ω∂tδψδψdx+∫Ωax∂xδψ2-λxβδψ2dx=0,since Ω is independent of t, which gives(52)12ddt∫Ωδψ2dxdt+∫Ωax∂xδψ2-λxβδψ2dx=0.By integrating between 0 and t with t∈[0,T] we obtain(53)12δψtL2Ω2+∫0t∫Ωax∂xδψ2-λxβδψ2dxdt=12δψ0L2Ω2(54)12δψtL2Ω2+∫0t∫Ωax∂xδψ2-λxβδψ2dxdt=12δψ0L2Ω2,and since a(x)⩾0 and -λ/xβ>0, ∀x∈Ω, we obtain(55)δψtL2Ω⩽δψ0L2Ω;this gives (56)Supt∈0,T⁡δψtL2Ω≤δψ0L2Ω.From where(57)δψtC0;T,L2Ω≤δψ0L2Ω.Which gives the continuity of the function(58)φ:L2Ω⟶C0,T;L2Ωψ0⟼ψ.Hence, the functional J is continuous in(59)Aad=u∈L2Ω;uL2Ω⩽r.Proof of Theorem6.
Letψ0∈Aad and δψ0 such that ψ0+δψ0∈Aad; we define the function(60)φ′ψ0:δψ0∈Aad⟶δψ,where δψ is the solution of the variational problem(61)∫Ω∂tδψvdx+∫Ωax∂xδψ∂xv-λxβδψvdx=0∀v∈H01Ωδψ0,t=δψ1,t=0∀t∈0,Tδψx,0=δψ0∀x∈Ωand we pose(62)ϕψ0=φψ0+δψ0-φψ0-φ′ψ0δψ0.We want to show that(63)ϕψ0=oδψ0.We easily verify that the function ϕ is the solution of the following variational problem:(64)∫Ω∂tϕvdx+∫Ωax∂xϕ∂xv-λxβϕvdx=0∀v∈H01Ωϕ0,t=ϕ1,t=0∀t∈0,Tϕx,0=δψ0-δψ02∀x∈Ω.By the same way as that used in the proof of continuity, we deduce(65)ϕC0,T,L2Ω⩽δψ0-δψ02L2Ω.Hence, in all cases, the function φ:ψ0→ψ is G-derivable in Aad and we deduce the existence of the gradient of the functional J.Now, we are going to compute the gradient ofJ with the adjoint state method.


## 4. Gradient ofJ

We define the Gâteaux derivative ofψ at ψ0 in the direction h∈L2(Ω), by(66)ψ^=limε→0⁡ψψ0+εh-ψψ0ε,where ψ(ψ0+εh) is the weak solution of (6) with initial state ψ0+εh, and ψ(ψ0) is the weak solution of (6) with initial state ψ0.We compute the Gâteaux (directional) derivative of (6) at ψ0 in some direction h∈L2(Ω), and we get the so-called tangent linear model:(67)∂tψ^+Aψ^=0ψ^0,t=ψ^1,t=0∀t∈0,Tψ^x,0=h∀x∈Ω.We introduce the adjoint variableP, and we integrate(68)∫01∫0T∂tψ^Pdtdx+∫01∫0TAψ^Pdx=0∫01ψ^P0T-∫0Tψ^∂tPdtdx+∫0TAψ^,PL2Ωdt=0(69)∫01ψ^TPT-ψ^0P0dx-∫0Tψ^,∂tPL2Ωdt+∫0TAψ^,PL2Ωdt=0.Let us takeP(x=0)=P(x=1)=0; then we may write ψ^,APL2(Ω)=Aψ^,PL2(Ω).And withP(T)=0 we may now rewrite (69) as(70)∫01ψ^0P0dx+∫0Tψ^,∂tP-APL2Ωdt=0;this gives(71)∫0Tψ^,∂tP-APL2Ωdt=-P0,hL2ΩPx=0=Px=1=0,PT=0.The Gâteaux derivative of the functional (72)Jψ0=12T∫0TCψt-ψobstL2Ω2dtat ψ0 in the direction h∈L2(Ω) is given by (73)J^h=limε→0⁡Jψ0+εh-Jψ0ε.After some calculations, we arrive at(74)J^h=1T∫0TC∗Cψ-ψobs,ψ^L2Ωdt.The adjoint model is(75)∂tP-AP=1TC∗Cψ-ψobsPx=0=Px=1=0∀t∈0,TPT=0.Problem (75) is retrograde; we make the change of variable t↔T-t, which gives(76)∂tP+AP=1TC∗ψ~obs-Cψ~Px=0=Px=1=0∀t∈0,TP0=0,with ψ~(t)=ψ(T-t).From (71), (74), and (75) the gradient of J is given by(77)∂J∂ψ0=-P0.With the change of variablet↔T-t, the gradient becomes(78)∂J∂ψ0=-PT.To calculate a gradient ofJ, we solve two problems: (6) and (76). The result solution of (6) is used in the second member of problem (76).


## 5. Discretization of Problem

Step 1 (full discretization).
To resolve problem (6) and (76), we use the method θ-schema in time. This method is unconditionally stable for 1>θ≥1/2.
Leth be the steps in space and Δt the steps in time.
Let(79)xi=ih,i∈0,1,2,…,N+1,cxi=axi,tj=jΔt,j∈0,1,2,…,M+1,fij=fxi,tj;we put(80)ψij=ψxi,tj.Let(81)daxi=cxi+1-cxih,bx=-λxβ.Therefore,(82)∂tψ+Aψ=fis approximated by(83)-θΔth2cxiψi-1j+1+1+2θΔth2cxi+daxiθΔth+bxiθΔtψij+1-θΔth2cxi+daxiθΔthψi+1j+1=1-θΔth2cxiψi-1j+1-1-θΔthdaxi-21-θΔth2cxi-1-θbxiΔtψij+1-θΔthdaxi+1-θΔth2cxiψi+1j+Δt·1-θfij+θfij+1.Let us define(84)g1xi=-θΔth2cxi,g2xi=1+2θΔth2cxi+daxiθΔth+bxiθΔt,g3xi=-θΔth2cxi-daxiθΔth,k1xi=1-θΔth2cxi,k2xi=1-1-θΔthdaxi-21-θΔth2cxi-1-θbxiΔt,k3xi=1-θΔthdaxi+1-θΔth2cxi.
Lettingψj=ψiji∈1,2,…,N, finally we get(85)Dψj+1=Bψj+Vjavecj∈1,2,…,Mψ0=ψ0ihi∈1,2,…,N,where(86)D=g2x1g3x100g1x2g2x2g3x200g1x3g2x3g3x300g1x4g2x4g3x400···0····00g1xN-1g2xN-1g3xN-100g1xNg2xNB=k2x1k3x100k1x2k2x2k3x200k1x3k2x3k3x300k1x4k2x4k3x400···0····00k1xN-1k2xN-1k3xN-100k1xNk2xNVj=Δt·1-θfx1,tj+θfx1,tj+ΔtΔt·1-θfx2,tj+θfx2,tj+Δt····Δt·1-θfxN-1,tj+θfxN-1,tj+ΔtΔt·1-θfxN,tj+θfxN,tj+Δt.Step 2 (discretization of the functional one has).
(87)
J
u
=
1
2
T
∫
0
T
∫
0
1
C
ψ
x
,
t
-
ψ
o
b
s
x
,
t
2
d
x
d
t
.
We recall that the method of Thomas Simpson to calculate an integral is(88)∫abfxdx≃h2fx0+2∑i=1N+1/2-1fx2i+4∑i=1N+1/2fx2i+1+fxN+1,with x0=a, xN+1=b, xi=a+ih, i∈1,…,N+1.
Let the functions(89)φx,t=Cψx,t-ψobsx,t2∀t∈0,T,x∈Ω,⌀t=∫01φx,tdx.This gives(90)⌀t≃h2φ0,t+2∑i=1N+1/2-1φx2i,t+4∑i=1N+1/2φx2i+1,t+φ1,t,where(91)∫0T∫01Cψx,t-ψobsx,t2dxdt≃∫0T⌀tdt≃dt2⌀0+2∑j=1M+1/2-1⌀t2j+4∑j=1M+1/2⌀t2j+1+⌀T,with t0=0, tM+1=T, tj=jdt, j∈1,…,M+1.
Therefore,(92)Ju≃12Tdt2⌀0+2∑j=1M+1/2-1⌀t2j+4∑j=1M+1/2⌀t2j+1+⌀T.Step 3 (discretization of∇J).
The adjoint problem (76) is discretized as (85), so,(93)∇J≃-PM+1.


## 6. Numerical Experiments and Results

In this section, we discuss two cases:In case we have a priori knowledgeψb of ψ0exact in each point of analysis grid-points, we apply the Tikhonov approach to solve the minimization problem (8). The data ψb is assumed to be corrupted by measurement errors, which we will refer to as noise. In particular, we suppose that ψb=ψ0exact+e. Here, we study the impact of err (err=e2) on the construction of the solution.In case we have a partial knowledge of values ofψb (example 20%): firstly, we apply the hybrid approach to rebuild the initial state. Secondly, we make a comparison between both hybrid and Tikhonov approaches.The tests have been performed in Matlab 2012A, on a Windows 7 platform.


### 6.1. Regularization Approach

The differentiability and continuity inAad of the functional,(94)JTψ0=12T∫0TCψt-ψobstL2Ω2dt+ε2ψ0-ψbL2Ω2,is deduced from the differentiability and continuity of the functional J, and we have(95)∂JT∂ψ0=-PT+εψ0-ψb,where P is the solution of (76).The main steps for descent method at each iteration are the following:(i)
Calculateψk solution of (6) with initial condition ψ0.(ii)
CalculatePk solution of (76).(iii)
Calculate the descent directiondk=-∇JT(ψ0).(iv)
Findtk=argmint>0⁡JT(ψ0+tdk)(v)
Update the variableψ0=ψ0+tkdk.The algorithm ends when JT(ψ0)<μ, where μ is a given small precision.t
k is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:letαi,β∈[0,1[ and α>0ifJT(ψ0k+αidk)⩽JT(ψ0k)+βαidkTdktk=αi and stopif notαi=ααi.We do all the tests on Pc with the following configurations: Intel Core i3 CPU 2.27 GHz; RAM = 4 GB (2.93 usable).In all figures, the observed function is drawn in red and built function in blue.LetN be number of points in space and M number of points in time.


#### 6.1.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 1, 2, 3, and 4.Figure 1
Initial temperature. This figure shows that we can rebuild the initial state.Figure 2
Final temperature.Figure 3
Graph ofJ.Figure 4
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. In Figures 5, 6, 7, and 8, ψ0exact is drawn in red and ψ0 (rebuilt initial condition) in blue.Figure 5
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 6
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 7
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition begins to move away from the true state.Figure 8
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


#### 6.1.2. Sub Critical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and the parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 9, 10, 11, and 12.Figure 9
Initial temperature. This figure shows that we can rebuild the initial state.Figure 10
Final temperature.Figure 11
Graph ofJ.Figure 12
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. See Figures 13, 14, 15, and 16.Figure 13
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 14
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 15
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition began to move away from the true state.Figure 16
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


### 6.2. Hybrid Algorithm

The genetic algorithms (GA) are adaptive search and optimization methods that are based on the genetic processes of biological organisms. Their principles have been first laid down by Holland. The aim of GA is to optimize a problem-defined function, called the fitness function. To do this, GA maintain a population of individuals (suitably represented candidate solutions) and evolve this population over time. At each iteration, called generation, the new population is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics, as, for instance, crossover and mutation. As the population evolves, the individuals in general tend toward the optimal solution [21–24]. The basic structure of a GA is the following:(1) Initialize a population of individuals;(2) Evaluate each individual in the population;(3) While the stop criterion is not reached do(4) Select individuals for the next population;(5) Apply genetic operators (crossover, mutation) to produce new individuals;(6) Evaluate the new individuals;(7) return the best individual.The hybrid methods combine principles from genetic algorithms and other optimization methods. In this approach, we will combine the genetic algorithm with method descent (steepest descent algorithm (FP)).We assume that we have a partial knowledge of background stateψb at certain points (xi)i∈I, I⊂1,2,…,N+1.We assume the individual is a vectorψ0; the population is a set of individuals.The initialization of individual is as follows:(96)fori=1toN+1ifi∈Iψ0xiischoseninthevicinityofψbxielseψ0xiischosenrandomlyendifend for.Starting by initial population, we apply genetic operators (crossover, mutation) to produce a new population in which each individual is an initial point for the descent method (FP). When a specific number of generations is reached without improvement of the best individual, only the fittest individuals (e.g., the first 10% fittest individuals in the population) survive. The remaining die and their place is occupied by new individuals with new genetic (45% are chosen randomly; the other 45% are chosen as (96)). At each generation we keep the best. The algorithm ends when Jψ0<μ or generation⩾Maxgen, where μ is a given precision (see Figure 17).Figure 17
Hybrid algorithm.The main steps for descent method (FP) at each iteration are the following:(i)
Calculateψk solution of (6) with initial condition ψ0.(ii)
CalculatePk solution of (76).(iii)
Calculate the descent directiondk=-∇J(ψ0).(iv)
Findtk=argmint>0⁡J(ψ0+tdk).(v)
Update the variableψ0=ψ0+tkdk.The algorithm ends when Jψ0<μ, where μ is a given small precision.t
k is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:letαi,β∈[0,1[ and α>0.ifJ(ψ0k+αidk)⩽J(ψ0k)+βαidkTdktk=αi and stopif notαi=ααi.Consider we know20% of values of background state (ψb), in this test we try to build the solution with the hybrid method.In the figures below, the observed function is drawn in red and built function in blue.LetN be number of points in space and M number of points in time.


#### 6.2.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100, number of individuals =30, and number of generations =2000.The test with simple descent gives Figure18.Figure 18
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives  Figure19.Figure 19
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure20.Figure 20
This figure shows that we can rebuild the solution.


#### 6.2.2. Subcritical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and parameters N=100, M=100, number of individuals = 30, and number of generations = 2000.The test with simple descent gives Figure21.Figure 21
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives Figure22.Figure 22
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure23.Figure 23
This figure shows that we can rebuild the solution.


### 6.3. Comparison between Hybrid Approach and Tikhonov Approach

Here, we assume that we know20% of values of background state (ψb).(i) Noncoercive Case. see Tables 1 and 2.Table 1
Results on the Tikhonov approach. Comparison between different values of regularizing coefficientε. The smallest value of J is reached when ε=10-06.
ε
Minimum value ofJ
Elapsed time (seconds)
1
0
-
08
1.106317
⋅
1
0
-
02
8.38
1
0
-
07
1.092014
⋅
1
0
-
02
24.47
1
0
-
06
6.630517
⋅
1
0
-
03
23.11
1
0
-
05
7.752620
⋅
1
0
-
03
21.50
1
0
-
04
7.857129
⋅
1
0
-
03
22.64
1
0
-
03
8.510799
⋅
1
0
-
03
18.95
1
0
-
02
8.733989
⋅
1
0
-
03
15.01
1
0
-
01
1.018406
⋅
1
0
-
02
17.33
1
1.552344
⋅
1
0
-
02
6.04Table 2
Results on the hybrid method.
Minimum value ofJ
Elapsed time
6.581908
⋅
1
0
-
03
1 min
5.850810
⋅
1
0
-
03
2 min
3.362100
⋅
1
0
-
04
7 min
1.071378
⋅
1
0
-
04
11 min
8.739839
⋅
1
0
-
05
23 min
5.958016
⋅
1
0
-
05
6 h 43 min
6.175260
⋅
1
0
-
06
11 h 20 minThe minimum value ofJ reached by the Tikhonov algorithm was 6.630517·10-03, whereas with the hybrid algorithm it was possible to reach the value 6.175260·10-06 in 11 h and 20 min with knowledge of 20% of ψb; if we take more than 20%, we got less than elapsed time.(ii) Subcritical Potential Case. see Tables 3 and 4.Table 3
Results on the Tikhonov aporoach. Comparison between different values of regularizing coefficientε. The smallest value of J is reached when ε=10-03.
ε
Minimum value ofJ
Elapsed time (seconds)
1
0
-
08
1.113538
⋅
1
0
-
02
10.91
1
0
-
07
1.188092
⋅
1
0
-
02
8.68
1
0
-
06
1.099187
⋅
1
0
-
02
8.75
1
0
-
05
1.267204
⋅
1
0
-
02
7.91
1
0
-
04
9.648060
⋅
1
0
-
03
11.17
1
0
-
03
7.320995
⋅
1
0
-
03
11.61
1
0
-
02
9.188603
⋅
1
0
-
03
10.20
1
0
-
01
9.799159
⋅
1
0
-
03
9.57
1
1.042463
⋅
1
0
-
02
9.66Table 4
Results on the hybrid method.
Minimum value ofJ
Elapsed time
7.605018
⋅
1
0
-
03
1 min
7.505279
⋅
1
0
-
03
2 min
1.762564
⋅
1
0
-
03
4 min
9.407809
⋅
1
0
-
04
43 min
2.981666
⋅
1
0
-
05
1 h 56 min
1.378356
⋅
1
0
-
05
6 h 43 min
8.203546
⋅
1
0
-
06
13 h 40 minThe minimum value ofJ reached by the Tikhonov algorithm was 7.320995·10-03, whereas with the hybrid algorithm it was possible to reach the value 8.203546·10-06 in 13 h and 40 min with knowledge of 20% of ψb; if we take more than 20%, we got less than elapsed time.


## 6.1. Regularization Approach

The differentiability and continuity inAad of the functional,(94)JTψ0=12T∫0TCψt-ψobstL2Ω2dt+ε2ψ0-ψbL2Ω2,is deduced from the differentiability and continuity of the functional J, and we have(95)∂JT∂ψ0=-PT+εψ0-ψb,where P is the solution of (76).The main steps for descent method at each iteration are the following:(i)
Calculateψk solution of (6) with initial condition ψ0.(ii)
CalculatePk solution of (76).(iii)
Calculate the descent directiondk=-∇JT(ψ0).(iv)
Findtk=argmint>0⁡JT(ψ0+tdk)(v)
Update the variableψ0=ψ0+tkdk.The algorithm ends when JT(ψ0)<μ, where μ is a given small precision.t
k is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:letαi,β∈[0,1[ and α>0ifJT(ψ0k+αidk)⩽JT(ψ0k)+βαidkTdktk=αi and stopif notαi=ααi.We do all the tests on Pc with the following configurations: Intel Core i3 CPU 2.27 GHz; RAM = 4 GB (2.93 usable).In all figures, the observed function is drawn in red and built function in blue.LetN be number of points in space and M number of points in time.


### 6.1.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 1, 2, 3, and 4.Figure 1
Initial temperature. This figure shows that we can rebuild the initial state.Figure 2
Final temperature.Figure 3
Graph ofJ.Figure 4
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. In Figures 5, 6, 7, and 8, ψ0exact is drawn in red and ψ0 (rebuilt initial condition) in blue.Figure 5
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 6
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 7
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition begins to move away from the true state.Figure 8
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


### 6.1.2. Sub Critical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and the parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 9, 10, 11, and 12.Figure 9
Initial temperature. This figure shows that we can rebuild the initial state.Figure 10
Final temperature.Figure 11
Graph ofJ.Figure 12
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. See Figures 13, 14, 15, and 16.Figure 13
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 14
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 15
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition began to move away from the true state.Figure 16
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


## 6.1.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 1, 2, 3, and 4.Figure 1
Initial temperature. This figure shows that we can rebuild the initial state.Figure 2
Final temperature.Figure 3
Graph ofJ.Figure 4
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. In Figures 5, 6, 7, and 8, ψ0exact is drawn in red and ψ0 (rebuilt initial condition) in blue.Figure 5
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 6
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 7
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition begins to move away from the true state.Figure 8
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


## 6.1.2. Sub Critical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and the parameters N=100, M=100.(i) Tests with
e
r
r
=
0. See Figures 9, 10, 11, and 12.Figure 9
Initial temperature. This figure shows that we can rebuild the initial state.Figure 10
Final temperature.Figure 11
Graph ofJ.Figure 12
Norm of gradient ofJ.(ii) Tests with
e
r
r
≠
0. See Figures 13, 14, 15, and 16.Figure 13
Initial temperature in err = 2%ψ0exact2 case. This figure shows that we can rebuild the true state.Figure 14
Initial temperature in err = 5%ψ0exact2 case. The reconstructed initial condition is not far from the true state.Figure 15
Initial temperature in err = 10%ψ0exact2 case. The reconstructed initial condition began to move away from the true state.Figure 16
Initial temperature in err = 20%ψ0exact2 case. This figure shows that we cannot rebuild the true state.


## 6.2. Hybrid Algorithm

The genetic algorithms (GA) are adaptive search and optimization methods that are based on the genetic processes of biological organisms. Their principles have been first laid down by Holland. The aim of GA is to optimize a problem-defined function, called the fitness function. To do this, GA maintain a population of individuals (suitably represented candidate solutions) and evolve this population over time. At each iteration, called generation, the new population is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics, as, for instance, crossover and mutation. As the population evolves, the individuals in general tend toward the optimal solution [21–24]. The basic structure of a GA is the following:(1) Initialize a population of individuals;(2) Evaluate each individual in the population;(3) While the stop criterion is not reached do(4) Select individuals for the next population;(5) Apply genetic operators (crossover, mutation) to produce new individuals;(6) Evaluate the new individuals;(7) return the best individual.The hybrid methods combine principles from genetic algorithms and other optimization methods. In this approach, we will combine the genetic algorithm with method descent (steepest descent algorithm (FP)).We assume that we have a partial knowledge of background stateψb at certain points (xi)i∈I, I⊂1,2,…,N+1.We assume the individual is a vectorψ0; the population is a set of individuals.The initialization of individual is as follows:(96)fori=1toN+1ifi∈Iψ0xiischoseninthevicinityofψbxielseψ0xiischosenrandomlyendifend for.Starting by initial population, we apply genetic operators (crossover, mutation) to produce a new population in which each individual is an initial point for the descent method (FP). When a specific number of generations is reached without improvement of the best individual, only the fittest individuals (e.g., the first 10% fittest individuals in the population) survive. The remaining die and their place is occupied by new individuals with new genetic (45% are chosen randomly; the other 45% are chosen as (96)). At each generation we keep the best. The algorithm ends when Jψ0<μ or generation⩾Maxgen, where μ is a given precision (see Figure 17).Figure 17
Hybrid algorithm.The main steps for descent method (FP) at each iteration are the following:(i)
Calculateψk solution of (6) with initial condition ψ0.(ii)
CalculatePk solution of (76).(iii)
Calculate the descent directiondk=-∇J(ψ0).(iv)
Findtk=argmint>0⁡J(ψ0+tdk).(v)
Update the variableψ0=ψ0+tkdk.The algorithm ends when Jψ0<μ, where μ is a given small precision.t
k is chosen by the inaccurate linear search by Rule Armijo-Goldstein as follows:letαi,β∈[0,1[ and α>0.ifJ(ψ0k+αidk)⩽J(ψ0k)+βαidkTdktk=αi and stopif notαi=ααi.Consider we know20% of values of background state (ψb), in this test we try to build the solution with the hybrid method.In the figures below, the observed function is drawn in red and built function in blue.LetN be number of points in space and M number of points in time.


### 6.2.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100, number of individuals =30, and number of generations =2000.The test with simple descent gives Figure18.Figure 18
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives  Figure19.Figure 19
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure20.Figure 20
This figure shows that we can rebuild the solution.


### 6.2.2. Subcritical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and parameters N=100, M=100, number of individuals = 30, and number of generations = 2000.The test with simple descent gives Figure21.Figure 21
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives Figure22.Figure 22
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure23.Figure 23
This figure shows that we can rebuild the solution.


## 6.2.1. The Noncoercive Case

Letα=1/2,λ=0 and parameters N=100, M=100, number of individuals =30, and number of generations =2000.The test with simple descent gives Figure18.Figure 18
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives  Figure19.Figure 19
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure20.Figure 20
This figure shows that we can rebuild the solution.


## 6.2.2. Subcritical Potential Case

Letα=1/2, λ=-1-α2/4, β=3/4 and parameters N=100, M=100, number of individuals = 30, and number of generations = 2000.The test with simple descent gives Figure21.Figure 21
This figure shows that we cannot rebuild the solution.The test with genetic algorithm gives Figure22.Figure 22
This figure shows that we cannot rebuild the solution.Now we turn the hybrid algorithm. This gives Figure23.Figure 23
This figure shows that we can rebuild the solution.


## 6.3. Comparison between Hybrid Approach and Tikhonov Approach

Here, we assume that we know20% of values of background state (ψb).(i) Noncoercive Case. see Tables 1 and 2.Table 1
Results on the Tikhonov approach. Comparison between different values of regularizing coefficientε. The smallest value of J is reached when ε=10-06.
ε
Minimum value ofJ
Elapsed time (seconds)
1
0
-
08
1.106317
⋅
1
0
-
02
8.38
1
0
-
07
1.092014
⋅
1
0
-
02
24.47
1
0
-
06
6.630517
⋅
1
0
-
03
23.11
1
0
-
05
7.752620
⋅
1
0
-
03
21.50
1
0
-
04
7.857129
⋅
1
0
-
03
22.64
1
0
-
03
8.510799
⋅
1
0
-
03
18.95
1
0
-
02
8.733989
⋅
1
0
-
03
15.01
1
0
-
01
1.018406
⋅
1
0
-
02
17.33
1
1.552344
⋅
1
0
-
02
6.04Table 2
Results on the hybrid method.
Minimum value ofJ
Elapsed time
6.581908
⋅
1
0
-
03
1 min
5.850810
⋅
1
0
-
03
2 min
3.362100
⋅
1
0
-
04
7 min
1.071378
⋅
1
0
-
04
11 min
8.739839
⋅
1
0
-
05
23 min
5.958016
⋅
1
0
-
05
6 h 43 min
6.175260
⋅
1
0
-
06
11 h 20 minThe minimum value ofJ reached by the Tikhonov algorithm was 6.630517·10-03, whereas with the hybrid algorithm it was possible to reach the value 6.175260·10-06 in 11 h and 20 min with knowledge of 20% of ψb; if we take more than 20%, we got less than elapsed time.(ii) Subcritical Potential Case. see Tables 3 and 4.Table 3
Results on the Tikhonov aporoach. Comparison between different values of regularizing coefficientε. The smallest value of J is reached when ε=10-03.
ε
Minimum value ofJ
Elapsed time (seconds)
1
0
-
08
1.113538
⋅
1
0
-
02
10.91
1
0
-
07
1.188092
⋅
1
0
-
02
8.68
1
0
-
06
1.099187
⋅
1
0
-
02
8.75
1
0
-
05
1.267204
⋅
1
0
-
02
7.91
1
0
-
04
9.648060
⋅
1
0
-
03
11.17
1
0
-
03
7.320995
⋅
1
0
-
03
11.61
1
0
-
02
9.188603
⋅
1
0
-
03
10.20
1
0
-
01
9.799159
⋅
1
0
-
03
9.57
1
1.042463
⋅
1
0
-
02
9.66Table 4
Results on the hybrid method.
Minimum value ofJ
Elapsed time
7.605018
⋅
1
0
-
03
1 min
7.505279
⋅
1
0
-
03
2 min
1.762564
⋅
1
0
-
03
4 min
9.407809
⋅
1
0
-
04
43 min
2.981666
⋅
1
0
-
05
1 h 56 min
1.378356
⋅
1
0
-
05
6 h 43 min
8.203546
⋅
1
0
-
06
13 h 40 minThe minimum value ofJ reached by the Tikhonov algorithm was 7.320995·10-03, whereas with the hybrid algorithm it was possible to reach the value 8.203546·10-06 in 13 h and 40 min with knowledge of 20% of ψb; if we take more than 20%, we got less than elapsed time.


## 7. Conclusion

In this paper, we have presented the regularization method and the hybrid method which are applied to determine an initial state from the point of measurements of parabolic degenerate/singular problem. These methods have proven efficiency to rebuild the solution. The proposed reconstruction algorithms are easily implanted.The elapsed time of the hybrid method is long enough. To reduce it, in our coming work we will use the multiprogramming to run two approaches of parallelism.



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*Source: 1467049-2017-06-14.xml*