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In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

The dense extracellular matrix (ECM) in tumor tissue severely hinders the penetration and enrichment of antitumor nanomedicines, which could significantly affect their efficiency. In this study, we used pH-sensitive nanocarriers loaded with collagenase (Col) to remold the tumor microenvironment (TME). Furthermore, we combined the collagenase delivery system with a nanomedicine to improve its penetration and enrichment in the tumor, thereby improving efficacy. We synthesized acetalated dextran (Ace-DEX) with an ideal pH-sensitivity as the carrier material of collagenase. Under mild preparation conditions, collagenase was loaded into Ace-DEX nanoparticles (NPs) with a high loading capacity (>4%) and remained highly active (>90%). Col-carrying NPs (Col-NPs) significantly reduced the tumor collagen content by 15.1%. Pretreatment with Col-NPs increased the accumulation of doxorubicin (DOX)-loaded liposome (DOX-Lipo) in the tumor by 2.8-fold. There were no safety concerns as the Col-NP showed no significant toxicity and reduced Col-induced damage to healthy tissues. Additionally, the number of circulating tumor cells remained unchanged after Col-NP treatment, suggesting no increased risk of tumor metastasis. Because the Col-NP acts essentially independent of the subsequent treatment, it has considerable potential for enhancing many existing delivery systems and drugs for cancer treatment. It may also be used for treating other collagen-related diseases.

The numerical solution of coupled nonlinear Klein-Gordon equations on unbounded domains is considered by applying the artificial boundary method. Based on the unified approach to overcome the coupled nonlinearity, local artificial boundary conditions are designed on the introduced artificial boundaries. The original problem is reduced to an initial boundary value problem on a bounded domain, which can be efficiently solved by the finite difference method. Some numerical examples are provided to verify the accuracy and effectiveness of the proposed method.

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

In this paper we investigate spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation with control integral constraint. First order optimality condition is derived and the regularity of control problem is discussed. Based on first discretize then optimize approach a spectral Galerkin approximation of the control problem is developed, where two-sided Jacobi polyfractonomials are used to approximate the state variable and variational discretization is used to discretize the control variable. A priori error analysis for state variable, adjoint state variable and control variable is presented. Numerical experiments are carried out to illustrate the theoretical findings.

In this paper we focus on the convergence and quasi-optimality of an adaptive finite element method for elliptic Robin boundary control problems. We use piecewise linear finite elements to approximate the state and the adjoint state variables, and the variational discretization to approximate the control variable. Under mild assumption on the initial mesh, we prove the contraction property, for the sum of the energy errors of the state and adjoint state and the scaled error estimator, on two consecutive adaptive loops. The resulting linear convergence yields the quasi-optimal convergence rate for the AFEM algorithm applied to our problem. Additionally, some numerical results are provided to support our theoretical analysis.

In this paper, we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space-fractional diffusion equation. Existence and uniqueness of the solution to optimal control problem is proved. The continuous first order optimality condition is derived. The eigenfunctions of two classes of fractional Strum–Liouville problems are used as basis functions to approximate state variable and adjoint state variable, respectively. The fractional spectral collocation scheme for the control problem is constructed based on ‘first optimize, then discretize’ approach. Note that the solutions of fractional differential equations are usually singular near the boundary, a generalized fractional spectral collocation scheme for the control problem is proposed based on ‘first optimize, then discretize’ approach. A projected gradient algorithm is designed based on the discrete optimality condition. Numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm.

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

In this paper a posteriori error analysis of virtual element method (VEM) for the optimal control problem governed by general elliptic equation is presented. The virtual element discrete scheme is constructed with virtual element approximation of the state equation and variational discretization of the control variable. Based on the a posteriori error estimates of virtual element method for general elliptic equation and approximated error equivalence of the solution of the optimal control problem to solutions of the state and adjoint problems we build up upper and lower a posteriori error estimates of the optimal control problem. Under the Dorfler’s marking strategy, the traditional projected gradient algorithm and adaptive VEM algorithm drived by the state and adjoint error estimators are used to solve the optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

We study in this paper the optimal convergence of finite element approximation to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter α so that the constants appeared in the derivation are independent of α. Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.

Photodynamic therapy (PDT) is highly effective in treating tumors located near body surface, offering strong tumor suppression and low damage to normal tissue nearby. PDT is also effective for treating a number of other conditions. PDT not only provide a precise and selective method for the treatment of various diseases by itself, it can also be used in combination with other traditional therapies. Because PDT uses light as the unique targeting mechanism, it has simpler and more direct targeting capability than traditional therapies. The core material of a PDT system is the photosensitizer which converts light energy to therapeutic factors/substances. Different photosensitizers have their distinct characteristics, leading to different advantages and disadvantages. These could be enhanced or compensated by using proper PDT system. Therefore, the selected type of photosensitizer would heavily influence the overall design of a PDT system. In this article, we evaluated major types of inorganic and organic PDT photosensitizers, and discussed future research directions in the field., Graphical abstract Image, graphical abstract

In this paper we investigate finite element approximation of optimal control problem governed by space fractional diffusion equation with control constraints. The control variable is approximated by piecewise constant. Regularity estimate for the control problem is proved based on the first order optimality system and a priori error estimates for the state, the adjoint state and the control variables are derived. Due to the nonlocal property of fractional derivative, which will leads to a full stiff matrix, we develop a fast primal dual active set algorithm for the control problem. Numerical examples are given to illustrate the theoretical findings and the efficiency of the fast algorithm.

This paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretization and a finite ...

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t = 0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

A spectral Galerkin approximation of a optimal control problem governed by a fractional advection–diffusion–reaction equation with integral fractional Laplacian is investigated in 1D. We first derive a first-order optimality condition and analyze the regularity of the solution based on this optimality condition. We present a spectral Galerkin scheme for the control problem using weighted Jacobi polynomials and prove optimal error estimates of the spectral method for state, adjoint state and control variables. We also propose a fast projected gradient algorithm of quasilinear complexity and present two numerical examples verifying our theoretical findings.

In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoint state, while the control is discretized by variational discretization method. A priori error estimates for the semi-discrete approximations of the state, adjoint state and control are derived. Furthermore, we also discuss the fully discrete scheme for the control problems. A finite difference method developed in Lin and Xu (2007) is used to discretize the time fractional derivative. Fully discrete first order optimality condition is developed based on 'first discretize, then optimize' approach. The stability and truncation error of the fully discrete scheme are analyzed. Numerical example is given to illustrate the theoretical findings.

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and $$L^2(0,T;L^2(\varGamma ))$$L2(0,T?L2(Γ)) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space---time $$L^2$$L2-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori $$L^2$$L2-error bounds for controls and states. We finally present numerical examples to support our theoretical findings.

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings.

We consider finite element approximations of parabolic control problems with pointwise control. The state equation exhibits low regularity due to the control imposed pointwisely; this introduces some difficulties for both theoretical and numerical analysis. To discretize the optimal control problem we use variational discretization together with piecewise linear and continuous finite elements for the space discretization of the state, and we use the backward Euler scheme for time discretization. We prove a priori error estimates for the control, state, and adjoint state. Numerical experiments are provided which support the theoretical results.

In this article, we investigate local discontinuous Galerkin approximation of stationary convection-dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first-order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection-dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L2 norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014

We investigate anH1-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

In this paper, we investigate a characteristic mixed finite element approximation of transient convection diffusion optimal control problems. The state and the adjoint state are approximated by characteristic mixed finite element method, while the control is discretized by standard finite element method. We derive the continuous and discrete first-order optimality conditions and prove a priori error estimates for the state, the adjoint state and the control. Numerical examples are presented to illustrate the theoretical findings.

In this paper, we discuss an H 1 -Galerkin mixed finite element method for a class of heat transport equations in the one-dimensional case. Optimal error estimates in L 2 -norm and H 1 -norm are obtained for both semi-discrete and fully discrete approximation scheme. Moreover, a numerical example is presented to illustrate the effectiveness of the method.