Events

I³MS - Christlieb Seminar

Location: Room 115, AICES, Rogowski

Prof. Dr. Andrew Christlieb - Steps Towards a Fast O(N) Approach for Direct Inversion of Linear Operators with Applications to Nonlinear Partial Differential Equations

Department of Mathematics, Michigan State University

Abstract

Multi-scale problems in science and engineering require accurate implicit methods for solving partial differential equations. However, in the world of distributed multi-core computing, a key bottleneck is the inversion of matrices. Hence, implicit solutions to partial differential equations have difficulty scaling on these computing platforms. Our goal is to develop a fast O(N) direct approach to the inversion of linear operators in real space. The work is based on the method of lines transpose which combines Green's function methods, successive convolution, fast summation and Rothe's method. The method may also be expressed as an efficient approach for direct evaluation of pseudodifferential operators. Practically speaking, the formulation of the method based on successive convolution can be directly expressed as an O(N) method for computing the resolvent expansion of a pseudodifferential operator in real space. This method has been used to develop an A-stable arbitrary order method for solving the two way wave equation, Maxwell's equations and both linear and nonlinear parabolic problems. We are current working to extend these methods to high-order phase field models.

 

I³MS - van Bloemen Waanders Seminar

Location: Room 115, AICES, Rogowski

Dr. Bart van Bloemen Waanders - Multiscale Optimization under Uncertainty For Additive Manufacturing

Sandia National Laboratories, Albuquerque, New Mexico

Abstract

Additive manufacturing (AM) enables new and innovative designs that are not realizable using standard procedures. Unfortunately, AM-produced parts often exhibit significant variability in their material properties. For engineering systems with strict reliability requirements, these uncertainties render AM-based subassemblies of limited use. In an attempt to address reliability, improvements are continuously being made to the controlability and observability of AM processes. However these improvements need to be coupled to numerical optimization and uncertainty quantification methods to help navigate the large design spaces and manage uncertainties inherent in these problems. In particular, design, control, and inversion problems must be solved at each stage of the overall design process. This presentation outlines an approach that leverages techniques from partial differential equation (PDE) constrained optimization, stochastic optimization, and multiscale (in particular mortar) finite element methods. We endow PDE-constrained optimization formulations with risk measures to steer the solution towards satisfaction of reliability and robustness criteria while accounting for model-based uncertainties. Because the optimization problem is constrained by PDEs and can contain millions of design variables, a significant software foundation is required to obtain solutions in a flexible and efficient manner. To that end, we leverage several components from the Trilinos framework. In particular, the Rapid Optimization Library (ROL) provides Newton-based optimization, line search, trust region, and stochastic optimization algorithms. ROL enables special interfaces that carefully map the underlying linear algebra of the simulation software to function space requirements for optimization. Furthermore, to approximate disparate PDE constraints, a finite element framework has been developed that leverages other Trilinos packages and automates discretizations, adjoints, and the optimization interfaces to ROL. This Multiscale Interface for Large scale Optimization (MILO) provides a convenient prototyping capability whereby the weak form of any set of PDEs is directly mapped to the underlying finite element matrices and vectors through automatic differentiation, thus creating a fully functional 3D parallel optimization solver. Although this approach is not unique, our development enables novel optimization under uncertainty capabilities and provides a framework to tackle multiscale and multiphysics problems. Several numerical examples are demonstrated including a convection-diffusion-reaction problem motivated by the control of trace gases in atmospheric transport.

 

CANCELLED - EU Regional School - Crane Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr. Crane - Laplace-Beltrami: The Swiss Army Knife of Geometry Processing

Computer Science Department, Robotics Institute 
Carnegie Mellon University

Abstract

A remarkable variety of fundamental 3D geometry processing tools can be expressed in terms of the Laplace-Beltrami operator on a surface—understanding these tasks in terms of basic PDEs such as heat flow, Poisson equations, and eigenvalue problems leads to an efficient, unified treatment at the computational level. The central goal of this tutorial is to show students 1. how to build the Laplacian on a triangle mesh, and 2. how to use this operator to implement a diverse array of geometry processing tasks. We will also discuss alternative discretizations of the Laplacian (e.g., on point clouds and polygon meshes), recent developments in discretization (e.g., via power diagrams), and important properties of the Laplacian in the smooth setting that become essential in geometry processing (e.g., existence of solutions, boundary conditions, etc.). 

EU Regional School - Heinkenschloss Seminar - Part 2

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr. Heinkenschloss - Model Reduction in PDE-Constrained Optimization

Department of Computational and Applied Mathematics
Rice University

Abstract

The numerical solution of optimization problems governed by partial differential equations (PDEs) requires the repeated solution of coupled systems of PDEs. Model reduction can be used to substantially lower the computational cost by using reduced order models (ROMs) as surrogates for the expensive original objective and constraint functions, or to use ROMs to accelerate subproblem solves in traditional Newton-type methods. In these lectures I will present approaches for the integration of projection based ROMs into PDE-constrained optimization, discuss their computational costs and convergence properties, and demonstrate their performance on example problems. I will review the generation of projection based ROMs, as well as Newton-type optimization algorithms. The integration of projection based ROMs and optimization will first be discussed for relatively simple PDE constrained optimization problems that allow for precomputations of ROMs and computations of global error bounds, and then for nonlinear PDE constrained optimization problems where such precomputations and generations of global error bounds are typically impossible.  

Lecture Material 

EU Regional School - Heinkenschloss Seminar - Part 1

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr. Heinkenschloss - Model Reduction in PDE-Constrained Optimization

Department of Computational and Applied Mathematics
Rice University

Abstract

The numerical solution of optimization problems governed by partial differential equations (PDEs) requires the repeated solution of coupled systems of PDEs. Model reduction can be used to substantially lower the computational cost by using reduced order models (ROMs) as surrogates for the expensive original objective and constraint functions, or to use ROMs to accelerate subproblem solves in traditional Newton-type methods. In these lectures I will present approaches for the integration of projection based ROMs into PDE-constrained optimization, discuss their computational costs and convergence properties, and demonstrate their performance on example problems. I will review the generation of projection based ROMs, as well as Newton-type optimization algorithms. The integration of projection based ROMs and optimization will first be discussed for relatively simple PDE constrained optimization problems that allow for precomputations of ROMs and computations of global error bounds, and then for nonlinear PDE constrained optimization problems where such precomputations and generations of global error bounds are typically impossible.

Lecture Material