# Application of PDE Constrained Optimization in Internal Combustion Engine Pollution Control

Authors: Hongliang Guan
Journal: Journal of Mathematics
 (2023)

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## Abstract

Aiming at the problem of secondary pollution of waters due to the difficulty of controlling the dosage of purifiers in the treatment of internal combustion engine pollution, a partial differential equation (referred to as PDE) constrained optimization algorithm based onl1-norm is proposed. The algorithm first converts the internal combustion engine control model of the scavenger dose into a constrained optimization problem with a l1-penalty term. Secondly, it introduces a dose constraint condition based on PDE and uses the inherent property of Moreau-Yosida regularization to establish a smooth minimization function. Finally, the semismooth Newton method is used to iteratively find the optimal solution. The results of the comparison experiment show that the algorithm in this paper has a great improvement in the results of Newton step number and dose area percentage.


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## Body

## 1. Introduction

In recent years, behind the rapid development of the marine economy, the state of the marine environment is not optimistic, and it is facing unprecedented challenge, and seawater pollution is the most important manifestation of marine environmental pollution [1]. Seawater pollutants have many species, a wide range of pollution, and a large impact effect, which has attracted more and more attention at home and abroad. Therefore, researching the optimal dosage of purifiers is the top priority in the treatment and control of internal combustion engine pollution, which is of great significance for the protection of the marine ecological environment and the safe management of water resources [2].Owing to realize the pollution-free management of the ocean, Lavaei and Lavaei et al. [3] proposed a comprehensive analysis method based on neural network and genetic algorithm. The algorithm can calculate the concentration of pollutants in different times and does not require the use of complex mathematical formulas. However, it only plays a role in local water pollution and cannot solve the large-scale pollution problem. In order to expand the scope of purification, Rivard et al. [4] combined the finite element algorithm and the optimal control theory into an optimal control analysis method suitable for water pollution. The method effectively controls the concentration of pollutants in water by adjusting the inflow speed of the river, which plays a role in controlling marine pollution in a vast range, but it can only control water pollution in a short period of time and cannot be used as a long-term method [5]. Combining the above two algorithms, they have some deficiencies in the governance of marine pollution, and a more efficient algorithm is needed to solve the problem of internal combustion engine pollution.PDE constrained optimization is an optimization control problem based on PDE constraint, which is usually applied in many fields, such as engineering and science [6]. In recent years, the control method of PDE constrained optimization has received extensive attention in the medical field, whose calculation process mainly has two stages. First, it uses the physical models to reconstruct medical images, whose reconstruction process can be expressed as a constrained optimization problem [7]. Secondly, the physical formula of dose deposition is introduced by precomputing the partial differential equation of fascicle discretization, but this simplified dose calculation is prone to large errors. In addition, it also requires high computational cost, which means that more efficient optimization algorithms are needed to overcome these difficulties under the constraints of state variables [8]. There are many feasible methods, among which the gradient descent method and Newton method are the two most commonly used optimization methods [9]. For the gradient descent method, its optimization thought is to use the negative gradient direction as the search direction, so that the objective function to be optimized is gradually decreased, which implies the solution does not guarantee a global optimal solution [10]. However, the Newton method is an optimization algorithm that approximates solving equations in the real and complex field, and each iteration of it requires solving the Hessian matrix of the objective function Inverse matrix, which makes the calculation more complicated [11, 12].As the difficulty of solving PDE optimization problems increases, researchers have developed a series of optimization methods to deal with the problem. An effective method is to addl0-norm or l1-norm as a penalty term in the objective equation, but the objective equation based on l0-norm is not easy to solve and the computational cost is too large, so this paper chooses the l1-norm penalty term. In 2017, Barnard et al. [14] proposed a PDE-constrained optimization algorithm based on l1-norm for the PDE optimization problem of radiotherapy. First, all inequality constraints were integrated as integral forms into the objective function of the original problem, then the smoothing techniques were used to burnish the nonsmooth penalty items, and finally the semi-smooth Newton method was used to solve [14]. Inspired by this method, this paper introduces the PDE’s l1-norm constraint optimization algorithm into the marine pollution control. Adjusting the dose limiting conditions by constraining the target area and risk area point by point, the l1-norm penalty term was used to reconstruct the optimization function of the objective function. Compared with the control variable parameterization method (CVP) [15], this algorithm has a better numerical effect, and it guarantees that the dosage of the purifier is minimized under the condition of not exceeding the desired level, effectively controlling marine pollution and optimizing the living environment of marine life.In summary, the main contributions of our work are as follows:(i)
We propose a partial differential equation optimization algorithm (PDE) based onl1 norm, which is applied to the water pollution control of internal combustion engines to achieve the combination of physical control and mathematical drive.(ii)
In our method, dose constraint conditions based on PDE are introduced and actual numerical experiments are carried out on this basis.(iii)
Compared with the traditional method CVP, we have greatly improved the convergence and the proportion of dose area, and also used in the practice waters to further verify the effectiveness of this method.We review the background theory in the second section and gave the model of internal combustion engine pollution control. In the third part, based on the physical model ofl1-norm, the optimization problem with affine operator as the core is established. Then, in the fourth part, the semismooth Newton method is used for numerical solution. In the fifth part, we use different parameter thresholds to compare different methods, which proved the effectiveness of this method in internal combustion engine pollution control. Finally, the sixth part puts forward the conclusion and future prospects.


## 2. Background Theories

First of all, assuming stable flow in the sea area, the average flow velocity is 0.125 cm\s, the density of pollutants is more than the water, and its diffusion speed is affected by the wind; the coefficient of wind drift is 0.03; reaction rate constant, longitudinal dispersion coefficient, transverse dispersion coefficient, and net surface tension coefficient of the purification agent, respectively, are 0.02 km\h, 1.35 cm2\s, 0.45 cm2\s, and 0.03 N\m [5]. Starting with the center to release the purifier in the polluted area, the purifying agent emission not only needs to consider the general flow state of the ocean current, but also must ensure that the dosage of the purifier reduces the secondary pollution of the water source while eliminating pollution; thereupon, we can establish purification agent reaction diffusion equation as follows [5]:(1)yt=mΔy+kwcu,yx,0=0,y0,t=b,y∞,t=0.Equation (1) can be abbreviated as εy,u, where y is the dose of the existing scavenger, t is the time state variable, x is the range of the risk or target area, mΔy is the diffusion term of the purifiers, mm>0 is the diffusion coefficient of the movement of water molecules, generally in between 0.12∼0.34 cm2 s−1, Δy=∑i=1N∂2y/∂2x2 represents the Laplace operator, kwcu is the reaction term of the purifiers, u represents the dose of purifiers added, kwc represents an embedded map from L20,T;L2wc to L20,t;L2Ω, and Ω⊂Rn is all waters.Secondly, according to the constraints of the dose of the purifiers, the dose of the purifiers should be higher than the dose levelU in the target area (contaminated water); contrarily, the dose of the purifiers should be lower than the dose level L in the risk area (unpolluted water), and which satisfy equation (2) as a whole [14], so the optimization problem of marine pollution is as follows:(2)minu∈Vad,y∈Y12uV2+α2y−zL2Q2,s.tεy,u=0,CωTy≥U,a.e.inωT,CωRy≤L.a.e.inωR.LetCw:L20,T;L1w⟶L1w (where w=wR or w=wT) denote integral operator, z∈L20,T;L2Ω is the expected value of the dose of the purifiers.


## 3. Principle Method

### 3.1. Optimization Problem

In equation (3), due to the dosage of the purifiers Cwy is continuous; therefore, the target area wT and the risk area wR cannot be separated, which makes it difficult to solve the marine pollution model. For this reason, in the case of Lagrange multipliers β1,β1>0, the l1-norm is introduced as a penalty term into the following objective equation [16]:(3)minu∈VAad,y∈Y12uV2+α2y−zLQ2+β1∫wTCwTy−U−L1wTdx+β2∫wRCwTRy−L+L1wRdx,s.t.εy,u=0,where let Cw:L20,T;L1w⟶L1w represents the integral operator; CwTy−U−L1wT denotes l1-norm in target area wT ; CwRy−L+L1wR denotes l1-norm in risk area wR.In order to simplify the objective equation, this paper considers introducing the operatorS:u⟶y in equation (3), and it satisfies the reaction diffusion equation yt=mΔy+kwCu of the purification agent, then S=∂t−mΔ−1kwC, where Δ=∂xx and kwCut,x=∑i=1Nxwixuit.Theorem 1.
OperatorS is a affine.Proof.
Byy=Su, where y∈Y, u∈L20,T;L2wC, and for every u∈V, there is a unique solution y in partial differential equation εy,u=0; let yt−mΔy−kwCu=0, then(4)∫0Tytdt=∫0TmΔy−kwcudt.
SincekwCu is a constant [4], and because Δy=∑i=1N∂2y/∂2x2, so(5)y+C=mY+kwcuT.
Thus,y=mY−C+kwcuT, where Y=∫0T∑i=1N∂2y/∂2xi2dt.
According to the definition of affine [16, 17], mY−C represents the translation matrix of u⟶y and kwCuT represents the transformation matrix of u⟶y, so S is an affine.
Substitutingy=Su into equation (3) translates into the following optimization problem:(6)minu∈VAad12uV2+α2Su−zLQ2+β1∫wTCwTSu−U−L1wTdx+β2∫wRCwTRSu−L+L1wRdx,where V is the feasible solution set of dose u; let V≔L20,T;L2wC, for x∈wC, t∈0,t, exists Vad=u∈L20,T;L2wC;Umin≤ut,x≤Umax⊂V, and wC⊂Ω is the observation domain.
For everyα>0,β1,andβ2>0, there is a unique minimum value u_ in equation (6), and u_∈Vad [14]. Literature [14] further discussed the convergence of the solution for β1andβ2⟶∞. Assuming Vad is bounded, there is a subsequence unn∈N that converges to the solution of equation (6).


### 3.2. Moreau-Yosida Regularization

Since the objective equation based onl1-norm is composed of the absolute value function [18], it is not differentiable at the origin. In order to solve the state variable u, Moreau-Yosida regularization is needed to burnish the nonsmooth penalty [19]. Firstly, assuming that u_∈Vad is the minimum value of equation (6), then there exists μ−+x∈L∞wR and μ−−x∈L∞wT, and the following formula holds:(7)u−=proxVad−p−,μ−+x=∂g+CwRsu−xfora.ex∈wR,μ−−x=∂g−CwTsu−xfora.ex∈wT,where(8)p−=S0∗αSu_−z+β1CwTSu_−U−L1wTdx+1CwRSu_−L+L1wRdx,g+vR=vR−L+,g−vT=vT−U−.Then, the regularization factorγ is introduced to equation (9), converting into the following regularization form:(9)uγ=proxVad−pγ,uγ+x=∂gγ+CwRSuγxfora.ex∈wR,uγ−x=∂gγ−CwTsuγxfora.ex∈wT,where(10)∂g+v=∂g+vγv≔1γv−prox∂g+vandprox∂g+v≔argminw∈R12γw−v2+g+w=Id+γ∂g+−1v,are the project mapping of g+ [14].


## 3.1. Optimization Problem

In equation (3), due to the dosage of the purifiers Cwy is continuous; therefore, the target area wT and the risk area wR cannot be separated, which makes it difficult to solve the marine pollution model. For this reason, in the case of Lagrange multipliers β1,β1>0, the l1-norm is introduced as a penalty term into the following objective equation [16]:(3)minu∈VAad,y∈Y12uV2+α2y−zLQ2+β1∫wTCwTy−U−L1wTdx+β2∫wRCwTRy−L+L1wRdx,s.t.εy,u=0,where let Cw:L20,T;L1w⟶L1w represents the integral operator; CwTy−U−L1wT denotes l1-norm in target area wT ; CwRy−L+L1wR denotes l1-norm in risk area wR.In order to simplify the objective equation, this paper considers introducing the operatorS:u⟶y in equation (3), and it satisfies the reaction diffusion equation yt=mΔy+kwCu of the purification agent, then S=∂t−mΔ−1kwC, where Δ=∂xx and kwCut,x=∑i=1Nxwixuit.Theorem 1.
OperatorS is a affine.Proof.
Byy=Su, where y∈Y, u∈L20,T;L2wC, and for every u∈V, there is a unique solution y in partial differential equation εy,u=0; let yt−mΔy−kwCu=0, then(4)∫0Tytdt=∫0TmΔy−kwcudt.
SincekwCu is a constant [4], and because Δy=∑i=1N∂2y/∂2x2, so(5)y+C=mY+kwcuT.
Thus,y=mY−C+kwcuT, where Y=∫0T∑i=1N∂2y/∂2xi2dt.
According to the definition of affine [16, 17], mY−C represents the translation matrix of u⟶y and kwCuT represents the transformation matrix of u⟶y, so S is an affine.
Substitutingy=Su into equation (3) translates into the following optimization problem:(6)minu∈VAad12uV2+α2Su−zLQ2+β1∫wTCwTSu−U−L1wTdx+β2∫wRCwTRSu−L+L1wRdx,where V is the feasible solution set of dose u; let V≔L20,T;L2wC, for x∈wC, t∈0,t, exists Vad=u∈L20,T;L2wC;Umin≤ut,x≤Umax⊂V, and wC⊂Ω is the observation domain.
For everyα>0,β1,andβ2>0, there is a unique minimum value u_ in equation (6), and u_∈Vad [14]. Literature [14] further discussed the convergence of the solution for β1andβ2⟶∞. Assuming Vad is bounded, there is a subsequence unn∈N that converges to the solution of equation (6).


## 3.2. Moreau-Yosida Regularization

Since the objective equation based onl1-norm is composed of the absolute value function [18], it is not differentiable at the origin. In order to solve the state variable u, Moreau-Yosida regularization is needed to burnish the nonsmooth penalty [19]. Firstly, assuming that u_∈Vad is the minimum value of equation (6), then there exists μ−+x∈L∞wR and μ−−x∈L∞wT, and the following formula holds:(7)u−=proxVad−p−,μ−+x=∂g+CwRsu−xfora.ex∈wR,μ−−x=∂g−CwTsu−xfora.ex∈wT,where(8)p−=S0∗αSu_−z+β1CwTSu_−U−L1wTdx+1CwRSu_−L+L1wRdx,g+vR=vR−L+,g−vT=vT−U−.Then, the regularization factorγ is introduced to equation (9), converting into the following regularization form:(9)uγ=proxVad−pγ,uγ+x=∂gγ+CwRSuγxfora.ex∈wR,uγ−x=∂gγ−CwTsuγxfora.ex∈wT,where(10)∂g+v=∂g+vγv≔1γv−prox∂g+vandprox∂g+v≔argminw∈R12γw−v2+g+w=Id+γ∂g+−1v,are the project mapping of g+ [14].


## 4. Implementation Details

Since the objective function (9) is semismooth, its directional derivative always exists, so it can be solved numerically using the semismooth Newton method. The specific steps are as follows:(1)
Lethγ+≔∂gγ+ and hγ−≔∂gγ−, differentiate hγ to get the corresponding Newton derivative.(2)
Rewrite the objective equation (9) by eliminating μγ+ and μγ−, then(11)uγ−proxVad−S0∗αSuγ−z+β1CwR∗Hγ+CwRSuγ+β2CwT∗Hγ−CwTSuγ=0,whereHγyx=hγyx.(3)
Establish the reversibility of Newton’s stepDNTukδu=−Tuk, then(12)Id+χVad−FukDNFukδu=−uk−projVad−Fuk,where(13)Fuk=S0∗αSuγ−z+β1CwR∗Hγ+CwRSuγ+β2CwT∗Hγ−CwTSuγ,DNFuk=S0∗α+β1γCwR∗χ+CwRS0ukCwR+β2γCwT∗χ−CwTS0ukCwTS0.(4)
Chooseu0∈Rn; for given uk, there is Fuk≠0, Vk∈DNFuk, then exists(14)Vkhk=Fuk,if there exists a solution, then it must satisfyhk≤bFuk.(5)
After further iteration, if the following formula is satisfied(15)Fuk+hk<γFuk,then the final state variableu can be determined.The algorithmG is given as follows:(1)
Initializeu0,β,γ,σ,β,γ∈0,1 and σ∈0,σ−, give the maximum number of iterations(2)
Letk = 0, 1, 2, ...., n, through equation (14) to calculate hk(3)
Ifhk satisfies equation (15), output uk, otherwise proceed to the next step(4)
Letdk=hk, αk=1, compute uk+1=uk+αkdk, go to step (2) until equation (15) is satisfiedIn this paper, the convergence of algorithmG is derived from the local Lipschitz continuity and B-differentiable properties [20], θu=Fu2 define the function θ, and the specific convergence proof can be found in the literature [21, 22].


## 5. Numerical Example

In order to clarify the effectiveness of the algorithm in this paper, a numerical experiment on the dosage adjustment of the purifiers is performed for thel1-norm PDE constrained optimization problem, and the experimental results are compared with the CVP method. Given the observation field Ω=−20,20, time T=20, and water molecule diffusion coefficient m=0.2, we choose objective, risk, and control area as wT:−9.5,9.5\−4,4, wR:−14,11\11,14∪−4,4, and wC:−20,20, respectively. Assuming that U = 10, L = 4, Umin=0,Umax=20, and α=0, they represent the Lebesgue measures of the target area and the risk area, respectively. In this paper, the observation area is discretized into 256 spatial nodes and 256 temporal nodes, for equation (2), the initialization γ is γ0 and γ0=1; for equation (3), the initialization γ is γ0 and γ0=maxβ1,β2, the maximum number of semi-smooth Newton iterations is 400. And in both cases, as long as the semi-smooth Newton method converges, γ and γ0 will have corresponding changes. The results of solving equation (2) and equation (9) are given in Figures 1–5, respectively. The dose-volume histogram shows the percentage of the area occupied by wR and wT, where the dose CwR in the risk area is at least 4 and the dose CwT in the target area is at least 10.For the solution of the CVP method as shown in Figure1, for the final value γ≈3.62×10−3, it is found that the numerical solution using this method obviously cannot meet the constraints of the dosage of the purifiers. For all γ, in the target area, the minimum value of the purifiers is samller than U, at the same time, the percentage of the dose above L in the risk area is 16%.The solution to the PDE’sl1-norm constrained optimization algorithm is shown in Figure 2, let β1∼=1.25×104, β2∼=1.25×106, and the final value γ/γ0=2.28×10−6, we can find that the effect of the target area is slightly improved, there is 40% wT with a dose less than U, and the volume fraction of CwRy>L becomes significantly smaller.Increaseβ1∼ to 2×104, while keeping β2∼ at the value of 1.25×106, the final value γ/γ0=1.78×10−6, as shown in Figure 3. It can be found that the target dose coverage has further improved, with a value of 80%; however, the area ratio of risk areas has increased significantly.Instead, increaseβ2∼ to 2×106, while keeping β1∼ at 1.25×104, the final value is γ/γ0=2.84×10−9, as shown in Figure 4, the coverage of the area greater than L in the risk area wR has improved to about 5%, but the area ratio of the target area wT has reduced to 40%.Finally, increaseβ1∼ to 2×104, β2∼ to 2×106, and the final value is still γ/γ0=2.84×10−9, as shown in Figure 5, the area ratio of each area has reached a good effect. Therefore, by adjusting the values of β1∼ and β2∼, the dosage of purifying agent in different areas can be balanced better, and it also shows that the PDE constrained optimization algorithm based on l1-norm can effectively control marine pollution.In order to better illustrate the advantages of PDE’sl1-norm constraint optimization algorithm, the following table represents the value of γ/γ0, the number of Newton steps required for the γ, the percentage of the target area wT below U, the risk area wR above L, and the percentage of the dose.The results of the CVP method are given in Table1. It can be seen that no matter how γ/γ0 changes, the percentage of the dose in the target area wT that is lower than U does not change, which shows that the minimum value of the purifiers dose has not reached U, the dose limit condition cannot be met, and it also shows that it is difficult to find a feasible solution between the target area wT and the risk area wR. In contrast, Table 2 shows that the optimization algorithm proposed in this paper not only requires fewer iteration steps, but also the percentage of doses higher than L in the risk area wR is almost unchanged when both β1∼ and β2∼ increase to the maximum. The percentage of the dose below U in the target area wT is also almost unchanged, which proves that the method has good convergence.Figure 1
Dose information for dose-constrained equation (2). (a) Final dose CwRy, risk level L, and target level R. (b) Dose volume histogram for risk wR and target region wT.
(a)(b)Figure 2
Dose information for dose-constrained equation (3) with β1∼=1.25×104andβ2∼=1.25×106. (a) Final dose CwRy, risk level L, and target level R. (b) Dose volume histogram for risk wR and target region wT.
(a)(b)Figure 3
Dose information for dose-constrained equation (3) with β1∼=2×104andβ2∼=2×106. (a) Final dose CwRy, risk level L, and target level R. (b) Dose volume histogram for risk wR and target region wT.
(a)(b)Figure 4
Dose information for dose-constrained equation (3) with β1∼=1.25×104andβ2∼=2×106. (a) Final dose CwRy, risk level L, and target level R. (b) Dose volume histogram for risk wR and target region wT.
(a)(b)Figure 5
Dose information for dose-constrained equation (3) with β1∼=2×104,β2∼=2×106. (a) Final dose CwRy, risk level L, target level R. (b) Dose volume histogram for risk wR and target region wT.
(a)(b)Table 1
Results for dose-constrained equation (2): number of SSN steps, volume fraction for risk, and target regions for different values of γ.
γ/γ01.25 × 10−13.45 × 10−33.12 × 10−23.91 × 10−31.95 × 10−37.81 × 10−4#SSN1131557∗%wR above L0013.6413.6414.79∗%wT below U100100100100100∗∗Failure to converge.Table 2
Results for dose-constrained equation (3): number of SSN steps, volume fraction for risk, and target regions for different values of γ.
γ/γ02.50 × 10−12.48 × 10−36.25 × 10−47.81 × 10−63.91 × 10−83.38 × 10−9#SSN11334∗%wR above L0045.4545.4527.12∗%wT below U10010093.1072.96100∗#SSN1135727%wR above L0011.6313.9513.9533.80%wT below U10010010088.7916.3816.38#SSN112359%wR above L0011.6311.6313.959.30%wT below U10010010010088.7912.07#SSN1131111∗%wR above L0011.6310.4610.46∗%wT below U10010010010.3410.34∗∗Failure to converge.


## 6. Conclusion

Despite the rapid development of technologies for controlling internal combustion engine pollution, it is still a major problem in this field to find the optimal amount of purifiers [23–27]. Aiming at the above problems, this paper proposes a l1-norm PDE constrained optimization algorithm. This algorithm introduces l1-norm penalty term in the optimal control of purifiers dosage. Using the properties of convex analysis to prove the existence and uniqueness of the optimal solution in the objective equation, at the same time, according to the definition of the adjacent operator, a regularized representation of the target equation is constructed, and finally, the semismooth Newton method is used to obtain the numerical solution. The results of the numerical experiments show that the algorithm in this paper reduces pollution while reducing the diffusion of purification agents and effectively controls internal combustion engine pollution. This proves that the PDE constrained optimization algorithm based on l1-norm is an effective method for controlling the dosage of purification agents. In the future, the algorithm can be further refined to adapt to other optimization problems.



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