When modeling complex physical systems with advanced dynamics, such as shocks and singularities, many classic methods for solving partial differential equations can return inaccurate or unusable results. One way to resolve these complex dynamics is through r-adaptive refinement methods, in which a fixed number of mesh points are shifted to areas of high interest. The mesh refinement map can be found through the solution of the Monge-Ampére equation, a highly nonlinear partial differential equation. Due to its nonlinearity, the numerical solution of the Monge-Ampére equation is nontrivial and has previously required computationally expensive methods. In this report, we detail our novel optimization-based, multigrid-enabled solver for a low-order finite element approximation of the Monge-Ampére equation. This fast and scalable solver makes r-adaptive meshing more readily available for problems related to large-scale optimal design. Beyond mesh adaptivity, our report discusses additional applications where our fast solver for the Monge-Ampére equation could be easily applied.
This paper continues our efforts to exploit optimization and control ideas as a common foundation for the development of property-preserving numerical methods. Here we focus on a class of scalar advection equations whose solutions have fixed mass in a given Eulerian region and constant bounds in any Lagrangian volume. Our approach separates discretization of the equations from the preservation of their solution properties by treating the latter as optimization constraints. This relieves the discretization process from having to comply with additional restrictions and makes stability and accuracy the sole considerations in its design. A property-preserving solution is then sought as a state that minimizes the distance to an optimally accurate but not property-preserving target solution computed by the scheme, subject to constraints enforcing discrete proxies of the desired properties. We consider two such formulations in which the optimization variables are given by the nodal solution values and suitably defined nodal fluxes, respectively. A key result of the paper reveals that a standard Algebraic Flux Correction (AFC) scheme is a modified version of the second formulation obtained by shrinking its feasible set to a hypercube. We conclude with numerical studies illustrating the optimization-based formulations and comparing them with AFC.
This paper considers preconditioners for the linear systems that arise from optimal control and inverse problems involving the Helmholtz equation. Specifically, we explore an all-at-once approach. The main contribution centers on the analysis of two block preconditioners. Variations of these preconditioners have been proposed and analyzed in prior works for optimal control problems where the underlying partial differential equation is a Laplace-like operator. In this paper, we extend some of the prior convergence results to Helmholtz-based optimization applications. Our analysis examines situations where control variables and observations are restricted to subregions of the computational domain. We prove that solver convergence rates do not deteriorate as the mesh is refined or as the wavenumber increases. More specifically, for one of the preconditioners we prove accelerated convergence as the wavenumber increases. Additionally, in situations where the control and observation subregions are disjoint, we observe that solver convergence rates have a weak dependence on the regularization parameter. We give a partial analysis of this behavior. We illustrate the performance of the preconditioners on control problems motivated by acoustic testing.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
As the mean time between failures on the future high-performance computing platforms is expected to decrease to just a few minutes, the development of “smart”, property-preserving checkpointing schemes becomes imperative to avoid dramatic decreases in application utilization. In this paper we formulate a generic optimization-based approach for fault-tolerant computations, which separates property preservation from the compression and recovery stages of the checkpointing processes. We then specialize the approach to obtain a fault recovery procedure for a model scalar transport equation, which preserves local solution bounds and total mass. Numerical examples showing solution recovery from a corrupted application state for three different failure modes illustrate the potential of the approach.
This LDRD project was developed around the ambitious goal of applying PDE-constrained opti- mization approaches to design Z-machine components whose performance is governed by elec- tromagnetic and plasma models. This report documents the results of this LDRD project. Our differentiating approach was to use topology optimization methods developed for structural design and extend them for application to electromagnetic systems pertinent to the Z-machine. To achieve this objective a suite of optimization algorithms were implemented in the ROL library part of the Trilinos framework. These methods were applied to standalone demonstration problems and the Drekar multi-physics research application. Out of this exploration a new augmented Lagrangian approach to structural design problems was developed. We demonstrate that this approach has favorable mesh-independent performance. Both the final design and the algorithmic performance were independent of the size of the mesh. In addition, topology optimization formulations for the design of conducting networks were developed and demonstrated. Of note, this formulation was used to develop a design for the inner magnetically insulated transmission line on the Z-machine. The resulting electromagnetic device is compared with theoretically postulated designs.
We present an abstract mathematical framework for an optimization-based additive decomposition of a large class of variational problems into a collection of concurrent subproblems. The framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions. The significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems. Consequently, assuming that stable numerical methods and efficient solvers are available for every subproblem, our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily. An application of the framework to the Oseen equations illustrates its potential.
We study a time-parallel approach to solving quadratic optimization problems with linear time-dependent partial differential equation (PDE) constraints. These problems arise in formulations of optimal control, optimal design and inverse problems that are governed by parabolic PDE models. They may also arise as subproblems in algorithms for the solution of optimization problems with nonlinear time-dependent PDE constraints, e.g., in sequential quadratic programming methods. We apply a piecewise linear finite element discretization in space to the PDE constraint, followed by the Crank-Nicolson discretization in time. The objective function is discretized using finite elements in space and the trapezoidal rule in time. At this point in the discretization, auxiliary state variables are introduced at each discrete time interval, with the goal to enable: (i) a decoupling in time; and (ii) a fixed-point iteration to recover the solution of the discrete optimality system. The fixed-point iterative schemes can be used either as preconditioners for Krylov subspace methods or as smoothers for multigrid (in time) schemes. We present promising numerical results for both use cases.
In this report we formulate eigenvalue-based methods for model calibration using a PDE-constrained optimization framework. We derive the abstract optimization operators from first principles and implement these methods using Sierra-SD and the Rapid Optimization Library (ROL). To demon- strate this approach, we use experimental measurements and an inverse solution to compute the joint and elastic foam properties of a low-fidelity unit (LFU) model.