Past Computational Mathematics and Applications Seminar

Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh.

The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered.

Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs.

The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves.

We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides

1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and

2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid

such that global convergence to the solution of the infinite-dimensional problem is ensured.

We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems.

Numerical results are presented that illustrate the potential of the approach.

This is joint work with Jan Carsten Ziems.

The Snaer program calculates the posterior mean and variance of variables on some of which we have data (with precisions), on some we have prior information (with precisions), and on some prior indicator ratios (with precisions) are available. The variables must satisfy a number of exact restrictions. The system is both large and sparse. Two aspects of the statistical and computational development are a practical procedure for solving a linear integer system, and a stable linearization routine for ratios. We test our numerical method for solving large sparse linear least-squares estimation problems, and find that it performs well, even when the $n \times k$ design matrix is large ( $nk = O (10^{8})$ ).

The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals.

We present algorithms for dense LU and QR factorizations that minimize the cost of communication. One of today's challenging technology trends is the increased communication cost. This trend predicts that arithmetic will continue to improve exponentially faster than bandwidth, and bandwidth exponentially faster than latency. The new algorithms for dense QR and LU factorizations greatly reduce the amount of time spent communicating, relative to conventional algorithms.

This is joint work with James Demmel, Mark Hoemmen, Julien Langou, and Hua Xiang.

A new primal-dual augmented Lagrangian merit function is proposed that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l1 linearly constrained Lagrangian (pdl1-LCL) method.

Model reduction (also called system reduction, order reduction) is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the complexity of a given model. In this introductory talk, we will survey some of the most prominent methods used for linear systems, compare their properties and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these techniques, the range of applicability of some of the methods has considerably widened.

The performance of several approaches will be demonstrated using real-world examples from a variety of engineering disciplines.

I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for parabolic partial differential equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations.

The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to ERT parabolic stability techniques can be combined with different elliptic a posteriori error analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method has the merit of unifying previously known approaches, as well as providing new ones and providing us with novel error bounds (e.g., pointwise norm error bounds for the heat equation). [These results are based on joint work with Ch. Makridakis and A. Demlow.]

Another feature, which I would like to highlight, of the ERT is its simplifying power. It allows us to derive estimates where the analysis would be very complicated otherwise. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods. [These are recent results obtained jointly with E. Georgoulis.]

We consider the computational complexity of optimizing various classes

of continuous functions over a simplex, hypercube or sphere. These

relatively simple optimization problems arise naturally from diverse

applications. We review known approximation results as well as negative

(inapproximability) results from the recent literature.

Whether the 3D incompressible Euler or Navier-Stokes equations

can develop a finite time singularity from smooth initial data has been

an outstanding open problem. Here we review some existing computational

and theoretical work on possible finite blow-up of the 3D Euler equations.

We show that the geometric regularity of vortex filaments, even in an

extremely localized region, can lead to dynamic depletion of vortex

stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform large scale computations of the 3D Euler equations

to re-examine the two slightly perturbed anti-parallel vortex tubes which

is considered as one of the most attractive candidates for a finite time

blowup of the 3D Euler equations. We found that there is tremendous dynamic

depletion of vortex stretching and the maximum vorticity does not grow faster

than double exponential in time. Finally, we present a new class of solutions

for the 3D Euler and Navier-Stokes equations, which exhibit very interesting

dynamic growth property. By exploiting the special nonlinear structure of the

equations, we prove nonlinear stability and the global regularity of this class of solutions.

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be ''close enough'' to the dynamics of the continuous system (which is typically easier to analyze) provided that small -- hence many -- time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.

In this talk we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

The development and quantitative validation of complex nonlinear differential equation models is a difficult task that requires the support by numerical methods for sensitivity analysis, parameter estimation, and the optimal design of experiments. The talk first presents particularly efficient "simultaneous" boundary value problems methods for parameter estimation in nonlinear differential algebraic equations, which are based on constrained Gauss-Newton-type methods and a time domain decomposition by multiple shooting. They include a numerical analysis of the well-posedness of the problem and an assessment of the error of the resulting parameter estimates. Based on these approaches, efficient optimal control methods for the determination of one, or several complementary, optimal experiments are developed, which maximize the information gain subject to constraints such as experimental costs and feasibility, the range of model validity, or further technical constraints.

Special emphasis is placed on issues of robustness, i.e. how to reduce the sensitivity of the problem solutions with respect to uncertainties - such as outliers in the measurements for parameter estimation, and in particular the dependence of optimum experimental designs on the largely unknown values of the model parameters. New numerical methods will be presented, and applications will be discussed that arise in satellite orbit determination, chemical reaction kinetics, enzyme kinetics and robotics. They indicate a wide scope of applicability of the methods, and an enormous potential for reducing the experimental effort and improving the statistical quality of the models.

(Based on joint work with H. G. Bock, S. Koerkel, and J. P. Schloeder.)

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.

In this talk we will mainly analyze the vibrations of a simplified 1-d model for a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. In particular we shall analyze how solutions propagate along the graph as time evolves. The problem of the observation of waves is a natural framework to analyze this issue. Roughly, the question can be formulated as follows: Can we obtain complete information on the vibrations by making measurements in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems.

Using the Fourier development of solutions and techniques of Nonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total lengths of the network in a suitable Hilbert that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree these weights can be identified.

Once this is done these results can be transferred to other models as the Schroedinger, heat or beam-type equations.

This lecture is based on results obtained in collaboration with Rene Dager.

The "secular equation" is a special way of expressing eigenvalue

problems in a variety of applications. We describe the secular

equation for several problems, viz eigenvector problems with a linear

constraint on the eigenvector and the solution of eigenvalue problems

where the given matrix has been modified by a rank one matrix. Next we

show how the secular equation can be approximated by use of the

Lanczos algorithm. Finally, we discuss numerical methods for solving

the approximate secular equation.

The numerical study of lattice quantum chromodynamics (QCD) is an attempt to extract predictions about the world around us from the standard model of physics. Worldwide, there are several large collaborations on lattice QCD methods, with terascale computing power dedicated to these problems. Central to the computation in lattice QCD is the inversion of a series of fermion matrices, representing the interaction of quarks on a four-dimensional space-time lattice. In practical computation, this inversion may be approximated based on the solution of a set of linear systems.

In this talk, I will present a basic description of the linear algebra problems in lattice QCD and why we believe that multigrid methods are well-suited to effectively solving them. While multigrid methods are known to be efficient solution techniques for many operators, those arising in lattice QCD offer new challenges, not easily handled by classical multigrid and algebraic multigrid approaches. The role of adaptive multigrid techniques in addressing the fermion matrices will be highlighted, along with preliminary results for several model problems.

In this talk we shall be looking at recent and forthcoming developments in the widely used LAPACK and ScaLAPACK numerical linear algebra libraries.

Improvements include the following: Faster algorithms, better numerical methods, memory hierarchy optimizations, parallelism, and automatic performance tuning to accommodate new architectures; more accurate algorithms, and the use of extra precision; expanded functionality, including updating and downdating and new eigenproblems; putting more of LAPACK into ScaLAPACK; and improved ease of use with friendlier interfaces in multiple languages. To accomplish these goals we are also relying on better software engineering techniques and contributions from collaborators at many institutions.

After an overview, this talk will highlight new more accurate algorithms; faster algorithms, including those for pivoted Cholesky and updating of factorizations; and hybrid data formats.

This is joint work with Jim Demmel, Jack Dongarra and the LAPACK/ScaLAPACK team.

Several formulations of structural optimization problems based on linear and nonlinear semidefinite programming will be presented. SDP allows us to formulate and solve problems with difficult constraints that could hardly be solved before. We will show that sometimes it is advantageous to prefer a nonlinear formulation to a linear one. All the presented formulations result in large-scale sparse (nonlinear) SDPs. In the second part of the talk we will show how these problems can be solved by our augmented Lagrangian code PENNON. Numerical examples will illustrate the talk.

Joint work with Michael Stingl.