R-3 Repository :: Browsing by Author "Heinkenschloss, Matthias"

Browsing by Author "Heinkenschloss, Matthias"

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A comparison of transcription techniques for the optimal control of the International Space Station
(2009) Reeger, Jonah; Heinkenschloss, Matthias
The numerical solution of Optimal Control Problems has received much attention over the past few decades. In particular, direct transcription methods have been studied because of their convergence properties for even relatively poor guesses at the solution. This thesis explores two of these techniques—Legendre-Gauss-Lobatto (LGL) Pseudospectral (PS) Collocation and Multiple Shooting (MS), and draws distinct comparisons between the two, allowing the reader to decide which would be better applied to a particular application. Specifically, comparisons will be made on accuracy, computation time, adjoint estimation, and storage requirements. It will be shown that the most distinct advantage for LGL PS collocation and MS methods will lie in computation time and storage requirements, respectively, using a nonlinear interior-point method described in this thesis.
A generalized trust region SQP algorithm for equality constrained optimization
(2004) Wang, Zhen; Heinkenschloss, Matthias
We introduce and analyze a class of generalized trust region sequential quadratic programming (GTRSQP) algorithms for equality constrained optimization. Unlike in standard trust region SQP (TRSQP) algorithms, the optimization subproblems arising in our GTRSQP algorithm can be generated from models of the objective and constraint functions that are not necessarily based on Taylor approximations. The need for such generalizations is motivated by optimal control problems for which model problems can be generated using, e.g., different discretizations. Several existing TRSQP algorithms are special cases of our GTRSQP algorithm. Our first order global convergence result for the GTRSQP algorithm applied to TRSQP allows one to relax the condition that the so-called tangential step lies in the null-space of the linearized constraints. The application of the GTRSQP algorithm to an optimal control problem governed by Burgers equation is discussed.
A Matrix-Free Trust-Region SQP Method for Equality Constrained Optimization
(SIAM, 2014) Heinkenschloss, Matthias; Ridzal, Denis
We develop and analyze a trust-region sequential quadratic programming (SQP) method for the solution of smooth equality constrained optimization problems, which allows the inexact and hence iterative solution of linear systems. Iterative solution of linear systems is important in large-scale applications, such as optimization problems with partial differential equation constraints, where direct solves are either too expensive or not applicable. Our trust-region SQP algorithm is based on a composite-step approach that decouples the step into a quasi-normal and a tangential step. The algorithm includes critical modifications of substep computations needed to cope with the inexact solution of linear systems. The global convergence of our algorithm is guaranteed under rather general conditions on the substeps. We propose algorithms to compute the substeps and prove that these algorithms satisfy global convergence conditions. All components of the resulting algorithm are specified in such a way that they can be directly implemented. Numerical results indicate that our algorithm converges even for very coarse linear system solves.
A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control problems
(EDP Sciences, 2024) Heinkenschloss, Matthias; Kroeger, Nathaniel J.
A new diagonalization technique for the parallel-in-time solution of linear-quadratic optimal control problems with time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization efficiently transforms the discretized optimality system into nt (=number of time steps) decoupled complex valued 2ny × 2ny systems, where ny is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.
A Nonlinear Differential Semblance Algorithm for Waveform Inversion
(2013-07-24) Sun, Dong; Symes, William W.; Heinkenschloss, Matthias; Zhang, Yin; Zelt, Colin A.
This thesis proposes a nonlinear differential semblance approach to full waveform inversion as an alternative to standard least squares inversion, which cannot guarantee a reliable solution, because of the existence of many spurious local minima of the objective function for typical data that lacks low-frequency energy. Nonlinear differential semblance optimization combines the ability of full waveform inversion to account for nonlinear physical effects, such as multiple reflections, with the tendency of differential semblance migration velocity analysis to avoid local minima. It borrows the gather-flattening concept from migration velocity analysis, and updates the velocity by flattening primaries-only gathers obtained via nonlinear inversion. I describe a general formulation of this algorithm, its main components and implementation. Numerical experiments show for simple layered models, standard least squares inversion fails, whereas nonlinear differential semblance succeeds in constructing a kinematically correct model and fitting the data rather precisely.
A numerical study of an adjoint based method for reservoir optimization
(2010) Wiegand, Klaus D.; Heinkenschloss, Matthias
A numerical reservoir simulator that uses a finite volume spatial discretization and two time discretization schemes is developed and tested. First and second order derivatives for the numerical simulator are derived, using an adjoint based approach. The adjoint and derivatives are validated in the context of the time discretization schemes, using varying time step sizes, and compared for accuracy. Two optimization algorithms are developed and used in combination to solve a numerical reservoir optimization problem. Numerical results are presented and discussed.
A Parallel-In-Time Gradient-Type Method For Optimal Control Problems
(2017-04-18) Deng, Xiaodi; Heinkenschloss, Matthias
This thesis proposes and analyzes a new parallel-in-time gradient-type method for time-dependent optimal control problems. When the classical gradient method is applied to time-dependent optimal control problems, each iteration requires the forward solution of the state equations followed by the backward solution of the adjoint equations before the gradient can be computed and the controls can be updated. The solution of the state equations and the adjoint equations is computationally expensive, is carried out sequentially in the time dimension, and consumes most of the computation time. The proposed new parallel-in-time gradient-type method introduces parallelism by splitting the time domain into N subdomains and executes the forward and backward computation in each time subdomain in parallel using state and adjoint variables at time subdomain boundaries from the last optimization iteration as initial values. Ignoring communication cost and assuming computation load balance, N parallel-in-time gradient-type iterations can be executed in the computing time required by a single classical gradient iteration. The basic proposed parallel-in-time gradient-type method is also generalized to allow for different time domain partitions for forward and backward computations and overlapping time subdomains. Due to the time domain decomposition, the state and the adjoint equations are not satisfied, since states and adjoints exhibit jump discontinuities at the time subdomain boundaries. Therefore the control update direction in the parallel gradient-type method is not a negative gradient direction and existing convergence theories for the gradient method do not apply. I provide convergence analyses for the new parallel-in-time gradient-type method applied to discrete-time optimal control problems. For linear-quadratic discrete-time optimal control problems, I show that the new parallel-in-time gradient-type method can be interpreted as a multiple-part splitting iteration scheme where control update in one iteration is determined by control variable iterates from multiple previous iterations, and I prove convergence of the new method for sufficiently small fixed step size by showing that the spectral radius of a corresponding implicitly constructed iteration matrix is less than one. For general non-linear discrete-time optimal control problems the parallel-in-time gradient-type method is combined with metric projection onto a closed convex set to handle simple control constraints. Convergence is proved for sufficiently small step sizes. Convergence theorems are given with different assumptions on the problem, such as convex objective function or compact control constraints. For linear-quadratic optimal control problems, I also interpret the parallel-in-time gradient-type method using a multiple shooting reformulation of the optimization problem. The new method can be seen as using a gradient-type iteration to solve the optimality saddle point system in the multiple shooting formulation. An alternative convergence proof is given for linear-quadratic problems by the multiple shooting point of view. The new parallel-in-time gradient-type method is applied to linear-quadratic optimal control problems governed by linear advection-diffusion partial differential equations (PDEs), and to a well-rate optimization problem governed by a system of highly non-linear PDEs that models two-phase immiscible incompressible subsurface flow. For linear-quadratic optimal control problem, each iteration of the parallel gradient-type method with N time subdomains takes roughly 1/N-th of the time required for one iteration of the classical gradient method. For moderate N (up to N=50 in one example) time subdomains, the parallel gradient-type method converges in approximately the same number of iterations as the classical gradient method and thus exhibits excellent scaling. For larger N, the parallel gradient-type method may use significantly more iterations than the classical gradient method, which negatively impacts scaling for large N. The parallelization in time is on top of parallelization already used to solve the state and adjoint equations (e.g., through parallel linear solvers/preconditioners). This is exploited for the larger and more complex well-rate optimization problem. If existing parallelism in space scales well up to K processors, the addition of time domain decomposition scales well up to K * N processors for small to moderate number N of time subdomains.
Accelerated PDE Constrained Optimization using Direct Solvers
(2018-04-17) Geldermans, Peter; Gillman, Adrianna; Heinkenschloss, Matthias
In this thesis, I propose a method to reduce the cost of computing solutions to optimization problems governed by partial differential equations (PDEs). Standard second order methods such as Newton require the solution of two PDEs per iteration of the Newton system, which can be prohibitively expensive for iterative solvers. In contrast, this work takes advantage a recently developed high order discretization method that comes with an efficient direct solver. The new technique precomputes a solution operator that can be reused for any body load, which is applied whenever a PDE solve is required. Thus the precomputation cost is amortized over many PDE solves. This approach will make second order optimization algorithms computationally affordable for practical applications such as photoacoustic tomography and optimal design problems.
Adaptive finite element methods for linear-quadratic convection dominated elliptic optimal control problems
(2010) Nederkoorn, Eelco; Heinkenschloss, Matthias
The numerical solution of linear-quadratic elliptic optimal control problems requires the solution of a coupled system of elliptic partial differential equations (PDEs), consisting of the so-called state PDE, the adjoint PDE and an algebraic equation. Adaptive finite element methods (AFEMs) attempt to locally refine a base mesh in such a way that the solution error is minimized for a given discretization size. This is particularly important for the solution of convection dominated problems where inner and boundary layers in the solutions to the PDEs need to be sufficiently resolved to ensure that the solution of the discretized optimal control problem is a good approximation of the true solution. This thesis reviews several AFEMs based on energy norm based error estimates for single convection dominated PDEs and extends them to the solution of the coupled system of convection dominated PDEs arising from the optimality conditions for optimal control problems. Keywords Adaptive finite element methods, optimal control problems, convection-diffusion equations, local refinement, error estimation.
ADMM and Diagonalization Based Parallel-in-Time Methods for Optimal Control Problems
(2023-12-12) Kroeger, Nathaniel James; Heinkenschloss, Matthias
This thesis investigates alternating direction method of multipliers (ADMM) and diagonalization - based parallel-in-time methods for linear-quadratic partial differential equation (PDE)-constrained optimization problems. The solution of such optimization problems is computing time and memory intensive, and efficient methods are essential to making such problems computationally tractable. Two parallel-in-time approaches are considered. In the first approach, ADMM is applied to a time domain decomposition (TDD) formulation. ADMM tailored to the TDD formulation requires the parallel solution of smaller subdomain problems and reduces the number of variables that need to be kept in memory globally. Thus, ADMM carries out the parallelization-in-time because the ADMM subproblems are able to be broken down by time subdomain. In the second approach, a diagonalization technique is used to parallelize-in-time. This approach is then extended to handle inequality constraints. The inequality constraints extension is handled by a combination of diagonalization and ADMM - the ADMM algorithm is the “main” algorithm, while the diagonalization method handles the computationally expensive substep in ADMM. Here, the diagonalization provides the parallelism in time, while the ADMM algorithm decouples the inequality constraints from the rest of the optimal control problem. Numerical results are provided to show the effectiveness of these methods.
ADMM Based Methods for Time-Domain Decomposition Formulations of Optimal Control Problems
(2020-08-03) Kroeger, Nathaniel James; Heinkenschloss, Matthias
This thesis investigates alternating direction method of multipliers (ADMM)-based methods for time-domain decomposition (TDD) formulations of linear-quadratic partial differential equation (PDE)-constrained optimization problems. The solution of such optimization problems is computing time and memory intensive. TDD formulations split the time-dependent PDE into coupled subdomain equations and introduce potential for parallelism and global memory reduction. This thesis tailors ADMM to the TDD structure. ADMM requires the parallel solution of smaller subdomain problems and reduces the number of variables that need to be kept in memory globally. Different TDD splittings lead to different ADMM variants. ADMM convergence analyses are derived from a matrix-splitting view and from the equivalence to the Douglas-Rachford algorithm applied to the dual problem, and are applied to these different variants. The effectiveness of ADMM as a preconditioner within GMRES is investigated. Computational results are presented for several variants of ADMM applied to an advection-diffusion problem.
An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Diﬀerential Equations with Uncertain Coeﬃcients
(2012-09-05) Kouri, Drew; Heinkenschloss, Matthias; Sorensen, Danny C.; Riviere, Beatrice M.; Cox, Dennis D.
Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.
Analysis of Inexact Trust-Region SQP Algorithms
(1999-09) Heinkenschloss, Matthias; Vicente, Luis N.
In this paper we study the global convergence behavior of a class of composite-step trust-region SQP methods that allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trust-region SQP method or from approximations of first-order derivatives. Accuracy requirements in our trust-region SQP methods are adjusted based on feasibility and optimality of the iterates. In the absence of inexactness our global convergence theory is equal to that of Dennis, El-Alem, Maciel (SIAM J. Optim. 7 (1997), 177-207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trust-region methods with inexact gradient information for unconstrained optimization.
Analysis of the Lagrange-SQP-Newton Method for the Control of a Phase Field Equation
(1998-10) Heinkenschloss, Matthias; Tröltzsch, Fredi
This paper investigates the local convergence of the Lagrange-SQP-Newton method applied to an optimal control problem governed by a phase field equation with distributed control. The phase field equation is a system of two semilinear parabolic differential equations. Stability analysis of optimization problems and regularity results for parabolic differential equations are used to prove convergence of the controls with respect to the L²(Q) norm and with respect to the L^{infinity}(Q) norm.
Balancing Neumann-Neumann Methods for Elliptic Optimal Control Problems
(2003-12) Heinkenschloss, Matthias; Nguyen, Hoang
We present Neumann-Neumann domain decomposition (DD) preconditioners for the solution of elliptic linear quadratic optimal control problems. The preconditioner is applied to the optimality system. A Schur complement formulation is derived that reformulates the original optimality system as a system in the state and adjoint variables restricted to the subdomain boundaries. The application of the Schur complement matrix requires the solution of subdomain optimal control problems with Dirichlet boundary conditions on the subdomain interfaces. The application of the inverses of the subdomain Schur complement matrices require the solution of subdomain optimal control problems with Neumann boundary conditions on the subdomain interfaces. Numerical tests show that the dependence of this preconditioner on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only.
Convergence Analysis of Discontinuous Galerkin Methods for Poroelasticity Equations
(2013-09-23) Tan, Jun; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.
This thesis analyzes a numerical method for solving the poroelasticity equations. The model incorporating the poroelasticity equations in this thesis can be applied in intestinal edema, which is a medical condition referring to the accumulation of excess fluid in the spaces between cells of intestinal wall tissue. The model has a dilatation term and can give a comprehensive prediction of pressure and displacement for intestinal edema. I approximate the pressure, displacement and dilatation by the discontinuous Galerkin method, which includes symmetric, nonsymmetric and incomplete interior penalty Galerkin cases. Moreover, in order to solve for the nonsymmetric case, I introduce an additional penalty term in the scheme. Theoretical convergence error estimates derived in a discrete-in-time setting show the a priori error can be bounded by some constant, which is related to the pressure, displacement, dilatation and the mesh size.
Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data
(2013-09-16) Liu, Kun; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Vannucci, Marina
This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.
Discrete Search Optimization for Real-Time Path Planning in Satellites
(2012-09-05) Mays, Millie; Heinkenschloss, Matthias; Symes, William W.; Riviere, Beatrice M.; Bedrossian, Nazareth
This study develops a discrete search-based optimization method for path planning in a highly nonlinear dynamical system. The method enables real-time trajectory improvement and singular configuration avoidance in satellite rotation using Control Moment Gyroscopes. By streamlining a legacy optimization method and combining it with a local singularity management scheme, this optimization method reduces the computational burden and advances the capability of satellites to make autonomous look-ahead decisions in real-time. Current optimization methods plan offline before uploading to the satellite and experience high sensitivity to disturbances. Local methods confer autonomy to the satellite but use only blind decision-making to avoid singularities. This thesis' method seeks near-optimal trajectories which balance between the optimal trajectories found using computationally intensive offline solvers and the minimal computational burden of non-optimal local solvers. The new method enables autonomous guidance capability for satellites using discretization and stage division to minimize the computational burden of real-time optimization.
Distributed Resource Allocation with Local Information
(2013-12-09) Dash, Debashis; Sabharwal, Ashutosh; Aazhang, Behnaam; Knightly, Edward W.; Heinkenschloss, Matthias
Making distributed decisions based on incomplete information is inevitable in dynamic wireless networks due to a multitude of constraints. We study the effects of incomplete information on system performance in two parts. We first analyze the effect of incomplete topology information on network capacity and then the effect of partial traffic information on the capacity of a two-flow interference network. In the first part of the thesis, we study the effect of local topology information based resource allocation on the number of conflicts (called defects) produced in the network. First we show its equivalence to sum rate maximization of the network. Then we prove the non-existence of an universal local coloring protocol that can produce defect-free coloring. Next we find the optimal protocol with no information and a local coloring protocol for path graphs that can achieve Nash equilibrium. We develop a general framework to analyze any local coloring protocol based on a randomized starting point that can be applied to any graph. Finally we develop a graph decomposition method to apply it to any graph with non-overlapping cliques and cycles. In the second part of the thesis, we study a two-user cognitive channel, where the primary flow is sporadic, cannot be re-designed and operating below its link capacity. To study the impact of primary traffic uncertainty, we propose a block activity model that captures the random on-off periods of primary's transmissions. Each block in the model can be split into parallel Gaussian-mixture channels, such that each channel resembles a multiple user channel from the point of view of the secondary user. The secondary senses the current state of the primary at the start of each block. We show that the optimal power transmitted depends on the sensed state and the optimal power profile is either growing or decaying in power as a function of time. We show that such a scheme achieves capacity when there is no noise in the sensing. The optimal transmission for the secondary performs rate splitting and follows a layered water-filling power allocation for each parallel channel to achieve capacity.
Domain decomposition methods for linear-quadratic elliptic optimal control problems
(2005) Nguyen, Hoang Q.; Heinkenschloss, Matthias
This thesis is concerned with the development of domain decomposition (DD) based preconditioners for linear-quadratic elliptic optimal control problems (LQ-EOCPs), their analysis, and numerical studies of their performance on model problems. The solution of LQ-EOCPs arises in many applications, either directly or as subproblems in Newton or Sequential Quadratic Programming methods for the solution of nonlinear elliptic optimal control problems. After a finite element discretization, convex LQ-EOCPs lead to large scale symmetric indefinite linear systems. The solution of these large systems is a very time consuming step and must be done iteratively, typically with a preconditioned Krylov subspace method. Developing good preconditioners for these linear systems is an important part of improving the overall performance of the solution method. The DD preconditioners for LQ-EOCPs studied in this thesis are extensions of overlapping and nonoverlapping Neumann-Neumann DD preconditioners applied to single elliptic partial differential equations (PDEs). In our case, DD is applied on the optimization level. In particular, the proposed preconditioners require the parallel solution of subdomain optimal control problems that are related to restrictions of the original LQ-EOCP to a subdomain. Numerical results on several test problems have shown that the new preconditioners are effective. Their performance relative to decreases in finite element mesh size or increase in number of subdomains seem to be numerically comparable to that of overlapping and Neumann-Neumann preconditioners for single PDEs. Remarkably, the proposed preconditioners seem to be rather insensitive to control regularization parameters. For overlapping methods, theoretical results are provided to support the numerical observations.

Browsing by Author "Heinkenschloss, Matthias"

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