Solving 1D Conservation Laws Using Pontryagin's Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax--Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions

Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations

We address finding the semi-global solutions to optimal feedback control and the Hamilton--Jacobi--Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin's maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the $6$-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with two examples of rigid body attitude control using momentum wheels

Recursive Algorithms for Distributed Forests of Octrees

The forest-of-octrees approach to parallel adaptive mesh refinement and coarsening (AMR) has recently been demonstrated in the context of a number of large-scale PDE-based applications. Although linear octrees, which store only leaf octants, have an underlying tree structure by definition, it is not often exploited in previously published mesh-related algorithms. This is because the branches are not explicitly stored, and because the topological relationships in meshes, such as the adjacency between cells, introduce dependencies that do not respect the octree hierarchy. In this work we combine hierarchical and topological relationships between octree branches to design efficient recursive algorithms. We present three important algorithms with recursive implementations. The first is a parallel search for leaves matching any of a set of multiple search criteria. The second is a ghost layer construction algorithm that handles arbitrarily refined octrees that are not covered by previous algorithms, which require a 2:1 condition between neighboring leaves. The third is a universal mesh topology iterator. This iterator visits every cell in a domain partition, as well as every interface (face, edge and corner) between these cells. The iterator calculates the local topological information for every interface that it visits, taking into account the nonconforming interfaces that increase the complexity of describing the local topology. To demonstrate the utility of the topology iterator, we use it to compute the numbering and encoding of higher-order $C^0$ nodal basis functions. We analyze the complexity of the new recursive algorithms theoretically, and assess their performance, both in terms of single-processor efficiency and in terms of parallel scalability, demonstrating good weak and strong scaling up to 458k cores of the JUQUEEN supercomputer.Comment: 35 pages, 15 figures, 3 table

Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge-Kutta time integration. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell's equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems

Stable coupling of nonconforming, high-order finite difference methods

The article of record as published may be found at http://dx.doi.org/10.1137/15M1022823A methodology for handling block-to-block coupling of nonconforming, multiblock summation-by-parts finite difference methods is proposed. The coupling is based on the construction of projection operators that move a finite difference grid solution along an interface to a space of piecewise defined functions; we specifically consider discontinuous, piecewise polynomial functions. The constructed projection operators are compatible with the underlying summation-by-parts energy norm. Using the linear wave equation in two dimensions as a model problem, energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces. To further demonstrate the power of the coupling procedure, we show how it allows for the development of a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method. The theoretical results are verified through numerical solutions on curved meshes as well as eigenvalue analysis.Approved for public release; distribution is unlimited

An energy stable approach for discretizing hyperbolic equations with nonconforming discontinuous Galerkin methods

When nonconforming, discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using linear elasticity as a model problem, we proposes a skew-symmetric formulation that has the same energy stability properties for both exact and inexact, quadrature-based integration. These stability properties are maintained even when the material properties are variable and discontinuous, and the elements non-affine (e.g., curved). The analytic stability results are confirmed through numerical experiments demonstrating the stability as well as the accuracy of the method.National Science Foundation (NSF)Office of Naval Research (ONR)EAR-1547596 (NSF)N0001416WX02190 (ONR