266 research outputs found
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 -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
Large Scale Constrained Trajectory Optimization Using Indirect Methods
State-of-the-art direct and indirect methods face significant challenges when solving large scale constrained trajectory optimization problems. Two main challenges when using indirect methods to solve such problems are difficulties in handling path inequality constraints, and the exponential increase in computation time as the number of states and constraints in problem increases. The latter challenge affects both direct and indirect methods. A methodology called the Integrated Control Regularization Method (ICRM) is developed for incorporating path constraints into optimal control problems when using indirect methods. ICRM removes the need for multiple constrained and unconstrained arcs and converts constrained optimal control problems into two-point boundary value problems. Furthermore, it also addresses the issue of transcendental control law equations by re-formulating the problem so that it can be solved by existing numerical solvers for two-point boundary value problems (TPBVP). The capabilities of ICRM are demonstrated by using it to solve some representative constrained trajectory optimization problems as well as a five vehicle problem with path constraints. Regularizing path constraints using ICRM represents a first step towards obtaining high quality solutions for highly constrained trajectory optimization problems which would generally be considered practically impossible to solve using indirect or direct methods. The Quasilinear Chebyshev Picard Iteration (QCPI) method builds on prior work and uses Chebyshev Polynomial series and the Picard Iteration combined with the Modified Quasi-linearization Algorithm. The method is developed specifically to utilize parallel computational resources for solving large TPBVPs. The capabilities of the numerical method are validated by solving some representative nonlinear optimal control problems. The performance of QCPI is benchmarked against single shooting and parallel shooting methods using a multi-vehicle optimal control problem. The results demonstrate that QCPI is capable of leveraging parallel computing architectures and can greatly benefit from implementation on highly parallel architectures such as GPUs. The capabilities of ICRM and QCPI are explored further using a five-vehicle constrained optimal control problem. The scenario models a co-operative, simultaneous engagement of two targets by five vehicles. The problem involves 3DOF dynamic models, control constraints for each vehicle and a no-fly zone path constraint. Trade studies are conducted by varying different parameters in the problem to demonstrate smooth transition between constrained and unconstrained arcs. Such transitions would be highly impractical to study using existing indirect methods. The study serves as a demonstration of the capabilities of ICRM and QCPI for solving large-scale trajectory optimization methods. An open source, indirect trajectory optimization framework is developed with the goal of being a viable contender to state-of-the-art direct solvers such as GPOPS and DIDO. The framework, named beluga, leverages ICRM and QCPI along with traditional indirect optimal control theory. In its current form, as illustrated by the various examples in this dissertation, it has made significant advances in automating the use of indirect methods for trajectory optimization. Following on the path of popular and widely used scientific software projects such as SciPy [1] and Numpy [2], beluga is released under the permissive MIT license [3]. Being an open source project allows the community to contribute freely to the framework, further expanding its capabilities and allow faster integration of new advances to the state-of-the-art
Parallel software tool for decomposing and meshing of 3d structures
An algorithm for automatic parallel generation of three-dimensional unstructured computational meshes based on geometrical domain decomposition is proposed in this paper. Software package build upon proposed algorithm is described. Several practical examples of mesh generation on multiprocessor computational systems are given. It is shown that developed parallel algorithm enables us to reduce mesh generation time significantly (dozens of times). Moreover, it easily produces meshes with number of elements of order 5 · 107, construction of those on a single CPU is problematic. Questions of time consumption, efficiency of computations and quality of generated meshes are also considered
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems
Deep learning methods are emerging as popular computational tools for solving
forward and inverse problems in traffic flow. In this paper, we study a neural
operator framework for learning solutions to nonlinear hyperbolic partial
differential equations with applications in macroscopic traffic flow models. In
this framework, an operator is trained to map heterogeneous and sparse traffic
input data to the complete macroscopic traffic state in a supervised learning
setting. We chose a physics-informed Fourier neural operator (-FNO) as the
operator, where an additional physics loss based on a discrete conservation law
regularizes the problem during training to improve the shock predictions. We
also propose to use training data generated from random piecewise constant
input data to systematically capture the shock and rarefied solutions. From
experiments using the LWR traffic flow model, we found superior accuracy in
predicting the density dynamics of a ring-road network and urban signalized
road. We also found that the operator can be trained using simple traffic
density dynamics, e.g., consisting of vehicle queues and traffic
signal cycles, and it can predict density dynamics for heterogeneous vehicle
queue distributions and multiple traffic signal cycles with an
acceptable error. The extrapolation error grew sub-linearly with input
complexity for a proper choice of the model architecture and training data.
Adding a physics regularizer aided in learning long-term traffic density
dynamics, especially for problems with periodic boundary data
Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems
Subject of this thesis is the development of adaptive techniques for multiple shooting methods. The focus is on the application to optimal control problems governed by parabolic partial differential equations. In order to retain as much freedom as possible in the later choice of discretization schemes, the details of both direct and indirect multiple shooting variants are worked out on an abstract function space level. Therefore, shooting techniques do not constitute a way of discretizing a problem. A thorough examination of the connections between the approaches provides an overview of different shooting formulations and enables their comparison for both linear and nonlinear problems.
We extend current research by considering additional constraints on the control variable in the multiple shooting context. An optimization problem is developed which includes so-called box constraints in the multiple shooting context. Several modern algorithms treating control constraints are adapted to the requirements of shooting methods. The modified algorithms permit an extended comparison of the different shooting approaches.
The efficiency of numerical methods can often be increased by developing grid adaptation techniques. While adaptive discretization schemes can be readily transferred to the multiple shooting context, questions of conditioning and stability make it difficult to develop adaptive features for shooting point distribution in multiple shooting processes. We concentrate on the design and comparison of two different approaches to shooting grid adaptation in the framework of ordinary differential equations. A residual-based adaptive algorithm is transferred to parabolic optimization problems with control constraints.
The presented concepts and methods are verified by means of several examples, whereby theoretical results are numerically confirmed. We choose the test problems so that the simple shooting method becomes unstable and therefore a genuine multiple shooting technique is required
Bvps codes for solving optimal control problems
Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontrya-gin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems
Accurate and efficient hydrodynamic analysis of structures with sharp edges by the Extended Finite Element Method (XFEM): 2D studies
Achieving accurate numerical results of hydrodynamic loads based on the
potential-flow theory is very challenging for structures with sharp edges, due
to the singular behavior of the local-flow velocities. In this paper, we
introduce the Extended Finite Element Method (XFEM) to solve fluid-structure
interaction problems involving sharp edges on structures. Four different FEM
solvers, including conventional linear and quadratic FEMs as well as their
corresponding XFEM versions with local enrichment by singular basis functions
at sharp edges, are implemented and compared. To demonstrate the accuracy and
efficiency of the XFEMs, a thin flat plate in an infinite fluid domain and a
forced heaving rectangle at the free surface, both in two dimensions, will be
studied. For the flat plate, the mesh convergence studies are carried out for
both the velocity potential in the fluid domain and the added mass, and the
XFEMs show apparent advantages thanks to their local enhancement at the sharp
edges. Three different enrichment strategies are also compared, and suggestions
will be made for the practical implementation of the XFEM. For the forced
heaving rectangle, the linear and 2nd order mean wave loads are studied. Our
results confirm the previous conclusion in the literature that it is not
difficult for a conventional numerical model to obtain convergent results for
added mass and damping coefficients. However, when the 2nd order mean wave
loads requiring the computation of velocity components are calculated via
direct pressure integration, it takes a tremendously large number of elements
for the conventional FEMs to get convergent results. On the contrary, the
numerical results of XFEMs converge rapidly even with very coarse meshes,
especially for the quadratic XFEM
Structure and pressure drop of real and virtual metal wire meshes
An efficient mathematical model to virtually generate woven metal wire meshes is
presented. The accuracy of this model is verified by the comparison of virtual structures with three-dimensional
images of real meshes, which are produced via computer tomography. Virtual structures
are generated for three types of metal wire meshes using only easy to measure parameters. For these
geometries the velocity-dependent pressure drop is simulated and compared with measurements
performed by the GKD - Gebr. Kufferath AG. The simulation results lie within the tolerances of
the measurements. The generation of the structures and the numerical simulations were done at
GKD using the Fraunhofer GeoDict software
- …