Fast iterative solution of reaction-diffusion control problems arising from chemical processes

PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix system, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects in our solvers are saddle point theory, mass matrix representation and effective Schur complement approximation, as well as the outer (Newton) iteration to take account of the nonlinearity of the underlying PDEs

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.Comment: 40 pages using amsart style file, and 15 figure

Preconditioning for active set and projected gradient methods as\ud semi-smooth Newton methods for PDE-constrained optimization\ud with control constraints

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semi-smooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semi-smooth Newton method that is equivalent to the primal-dual active set method. Numerical results illustrate the competitiveness of this approach

An Unstructured Mesh Convergent Reaction-Diffusion Master Equation for Reversible Reactions

The convergent reaction-diffusion master equation (CRDME) was recently developed to provide a lattice particle-based stochastic reaction-diffusion model that is a convergent approximation in the lattice spacing to an underlying spatially-continuous particle dynamics model. The CRDME was designed to be identical to the popular lattice reaction-diffusion master equation (RDME) model for systems with only linear reactions, while overcoming the RDME's loss of bimolecular reaction effects as the lattice spacing is taken to zero. In our original work we developed the CRDME to handle bimolecular association reactions on Cartesian grids. In this work we develop several extensions to the CRDME to facilitate the modeling of cellular processes within realistic biological domains. Foremost, we extend the CRDME to handle reversible bimolecular reactions on unstructured grids. Here we develop a generalized CRDME through discretization of the spatially continuous volume reactivity model, extending the CRDME to encompass a larger variety of particle-particle interactions. Finally, we conclude by examining several numerical examples to demonstrate the convergence and accuracy of the CRDME in approximating the volume reactivity model.Comment: 35 pages, 9 figures. Accepted, J. Comp. Phys. (2018

Methods for Optimal Output Prediction in Computational Fluid Dynamics.

In a Computational Fluid Dynamics (CFD) simulation, not all data is of equal importance. Instead, the goal of the user is often to compute certain critical "outputs" -- such as lift and drag -- accurately. While in recent years CFD simulations have become routine, ensuring accuracy in these outputs is still surprisingly difficult. Unacceptable levels of output error arise even in industry-standard simulations, such as the steady flow around commercial aircraft. This problem is only exacerbated when simulating more complex, unsteady flows. In this thesis, we present a mesh adaptation strategy for unsteady problems that can automatically reduce errors in outputs of interest. This strategy applies to problems in which the computational domain deforms in time -- such as flapping-flight simulations -- and relies on an unsteady adjoint to identify regions of the mesh contributing most to the output error. This error is then driven down via refinement of the critical regions in both space and time. Here, we demonstrate this strategy on a series of flapping-wing problems in two and three dimensions, using high-order discontinuous Galerkin (DG) methods for both spatial and temporal discretizations. Compared to other methods, results indicate that this strategy can deliver a desired level of output accuracy with significant reductions in computational cost. After concluding our work on mesh adaptation, we take a step back and investigate another idea for obtaining output accuracy: adapting the numerical method itself. In particular, we show how the test space of discontinuous finite element methods can be "optimized" to achieve accuracy in certain outputs or regions. In this work, we compute test functions that ensure accuracy specifically along domain boundaries. These regions -- which are vital to both scalar outputs (such as lift and drag) and distributions (such as pressure and skin friction) -- are often the most important from an engineering standpoint.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133418/1/kastsm_1.pd

Numerical analysis for the pure Neumann control problem using the gradient discretisation method

The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, nonconforming and mimetic finite difference methods confirm the theoretical rates of convergence

Optimal Control of Convective FitzHugh-Nagumo Equation

We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian

Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems

We consider a general class of nonsmooth optimal control problems with partial differential equation (PDE) constraints, which are very challenging due to its nonsmooth objective functionals and the resulting high-dimensional and ill-conditioned systems after discretization. We focus on the application of a primal-dual method, with which different types of variables can be treated individually and thus its main computation at each iteration only requires solving two PDEs. Our target is to accelerate the primal-dual method with either larger step sizes or operator learning techniques. For the accelerated primal-dual method with larger step sizes, its convergence can be still proved rigorously while it numerically accelerates the original primal-dual method in a simple and universal way. For the operator learning acceleration, we construct deep neural network surrogate models for the involved PDEs. Once a neural operator is learned, solving a PDE requires only a forward pass of the neural network, and the computational cost is thus substantially reduced. The accelerated primal-dual method with operator learning is mesh-free, numerically efficient, and scalable to different types of PDEs. The acceleration effectiveness of these two techniques is promisingly validated by some preliminary numerical results