Applied Mathematics and Numerical Analysis Seminar 

In this presentation we address the parallel implementation of the Multi-Power Defect-Correction method for the computation of eigenpairs of compact integral operators. The integral operator is discretized by a projection method on a subspace of moderate dimension and the eigenelements of its representing matrix are computed. These are then iteratively refined, by a defect correction type procedure accelerated by a few power iterations, to yield a better approximation to the spectral elements of the operator. The algorithm is rich in matrix-vector multiplications involving matrices whose construction is distributed among the available processors. The defect correction phase requires the solution of a distributed linear system of moderate dimension. Computational approaches of parallelization using state-of-the-art packages will be discussed, and numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, will be presented. Co-authors: Filomena Dias d'Almeida (Faculdade de Engenharia da Universidade do Porto) e José E. Roman (Instituto ITACA, Universidad Politécnica de Valencia, Spain)

This talk is motivated by the modeling and simulation of fluid-structure interaction phenomena in the vicinity of heart valves. On the one hand, the interaction of the vessel wall is dealt with an Arbitrary Lagrangian Eulerian (ALE) formulation. On the other hand the interaction of the valves is treated with the help of Lagrange multipliers in a Fictitious Domains-like (FD) formulation. After a synthetic presentation of the several methods available for the fluid-structure interaction in blood flows, a method that permits capture the dynamics of a valve immersed in an incompressible fluid is described. The coupling algorithm is partitioned which allows the fluid and structure solvers to remain independent. In order to follow the vessel walls, the fluid mesh is mobile, but it remains none the less independent of the valve mesh. In this way we allow large displacements without the need to perform remeshing. We propose a strategy to manage contact between several immersed structures. The algorithm is completely independent of the structure solver and is well adapted to the partitioned fluid-structure coupling. The methods considered are followed with several numerical tests in 2D and 3D.

In this talk, we consider Navier-Stokes equations with Navier's slip boundary conditions. We show their weak solvability. A remarkable observation says that Navier's boundary conditions enable us to accept a class of possible collisions, in contrast with the no-slip Dirichlet boundary conditions.

Balloon angioplasty followed by stent implantation has been of great benefit in coronary applications, whereas in peripheral applications, success rates remain low. Analysis of healing patterns in successful deployments shows that six months after implantation the artery has reorganized itself and there is no purpose for the stent to remain, potentially provoking inflammation and foreign body reaction. Thus, a fully absorbable stent that fulfills the mission and steps away is of great benefit. Biodegradable polymers have a widespread usage in the biomedical field, such as sutures, scaffolds and implants. Aliphatic polyesters (the main class of biodegradable polymers used in biomedical applications) degrade by random scission due to passive hydrolysis which results in molecular weight reduction, loss of strength, and ultimately, loss of mass. A constitutive model involving degradation and its impact on mechanical properties was developed through an extension of a material which response depends on the history of the motion and on a scalar parameter reflecting the local extent of degradation. The material properties decrease with degradation and a rate equation describing the chain scission process confers characteristics of stress relaxation, creep and hysteresis. These phenomena arise due to the entropy-producing nature of degradation and are markedly different from their viscoelastic counterparts.

In this paper we discuss Nordsieck formulas applied to ordinary differential equations. We focus on local and global error evaluation techniques in the mentioned numerical schemes. The error estimators are derived for both consistent Nordsieck methods and quasi-consistent ones. It is also shown how quasi-consistent Nordsieck formulas, which suffer, on variable grids, from the order reduction phenomenon, can be modified in an optimal way to avoid the order reduction. A special task here is to study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one attractive property for practical use. The global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion. We apply this idea to Nordsieck Adams-Moulton methods, which are known to be quasi-consistent. Moreover, some Nordsieck Adams-Moulton methods are even super-quasi-consistent. The latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods. In turn, this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control. We also introduce Implicitly Extended Nordsieck methods such that in some sense they form pairs of embedded formulas with their source Nordsieck counterparts. This facilitates the local error control in quasi-consistent Nordsieck schemes. As the most promising result, we study double quasi-consistency of numerical schemes. This property means that the principal terms of the global and local errors of a doubly quasi-consistent numerical technique coincide. This benefits the global error control, significantly. We show that doubly quasi-consistent schemes do exist and belong to the class of general linear methods. The notion of double quasi-consistency creates potentially a new area of research in numerical methods for differential equations. Numerical examples presented in this paper confirm clearly the power of the above-mentioned global error estimators in practice.

In this talk, two types of precondtioned iterative methods and SOR-like methods are proposed to solving saddle point problems. The upper bound of the condition number of the preconditioned systems are estimated. Semi-conjugate direction methods (SCD) are introduced as well. The properties of the SCD method are studied. The equivalence of the SCD methods and the LU decomposition is established. Some remedies for breakdown problem are presented. Some applications and open problems are mentioned for future research.

Convex hulls and shortest paths are fundamental problems in Computational Geometry. Firstly, based on the idea of the Method of Orienting Curves (introduced by Phu in Optimization, 18 (1987) pp. 65-81 for solving optimal control problems with state constraints), we present an efficient algorithm to determine the convex hull of a finite planar set. The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Graham’s convex hull algorithm are presented. Secondly, we discuss the use of the Method of Orienting Curves in finding the shortest path between two points in a polygon. We deal with the convex rope problem, posed by Peshkin and Sanderson in IEEE J. Robotics Automat, 2 (1986) pp. 53-58, for finding the counterclockwise and clockwise convex ropes starting at the vertex a and ending at the vertex b of a simple polygon, where a is on the boundary of the convex hull of the polygon and b is visible from infinity. Based on the idea of the Method of Orienting Curves, an algorithm to determine the convex ropes, without resorting to a linear-time triangulation algorithm and without resorting to a convex hull algorithm for the polygon, is proposed. Next, we present an efficient algorithm for finding the Euclidean shortest path in the polygon between two vertices a and b without resorting to a linear-time triangulation algorithm, that provides a contribution to the solution of the open question raised by J. S. B. Mitchell in J. R. Sack and J. Urrutia, eds, Handbook of Computational Geometry, Elsevier Science B. V., 2000, p. 642. The use of the idea of the direct multiple shooting method (introduced by Bock in 1984 for direct solution of optimal control problems) in finding the shortest path problem above is also discussed.

In the present seminar the BEM and meshless LBIE methods for solving problems in linear elasticity, elastoplasticity, viscoelasticity and Navier - Stokes equations are presented in brief. The constitutive differential equations and the corresponding integral representations are addressed. Moreover we explain the treatment of terms involving possible non-linearities, time derivatives and the numerical implementation. Finally, we solve representative examples that demonstrate the accuracy and efficiency of the aforementioned methods and discuss the numerical results.

The landmark papers by Hutchinson and Barnsley and Demko showed how systems of contractive maps with associated probabilities (called iterated function systems by the latter), acting in a parallel manner, either deterministically or probabilistically, could be used to construct fractal sets and measures.

There is an ongoing research programme (see http://links.uwaterloo.ca) on the construction of appropriate IFS-type operators, or generalized fractal transforms (GFT), over various spaces, i.e., function spaces and distributions, vector-valued measures, integral transforms and wavelet transforms.

The action of a GFT on an element $u$ of the complete metric space $(X,d)$ under consideration can be summarized as follows:

1. it produces a set of $N$ spatially-contracted copies of $u$,
2. it then modifies the values of these copies by means of a suitable range-mapping,
3. it recombines these copies using an appropriate operator to produce the element $v$ in $X$, $v=Tu$.

In each of the above-mentioned cases, the fractal transform $T$ is guaranteed to be contractive when the parameters defining it satisfy appropriate conditions specific to the metric space of concern. In this situation, Banach's fixed point theorem guarantees the existence of a unique fixed point $u=Tu$.

The inverse problem of fractal-based approximation is as follows: given an element $y$, can we find a fractal transform $T$ with fixed point $u$ so that $d(y,u)$ is sufficiently small?

However, the search for such transforms is enormously complicated. Thanks to a simple consequence of Banach's fixed point theorem known as the Collage Theorem, most practical methods of solving the inverse problem seek to find an operator $T$ for which the collage distance $d(u,Tu)$ is as small as possible. The aim of this talk is to present some recent developments and extensions of fractal transforms and show interesting applications in image processing and economics.

Recent references

• La Torre, D., Vrscay, E.R., Ebrahimi A., Barnsley M., A method of fractal coding for measure-valued images, SIAM Journal on Imaging Sciences (SIIMS), revised submission.
• Kunze H., La Torre D., Vrscay E.R., Inverse problems for random differential equations using the collage method for random contraction mappings (2008) - available on line at the Journal of Computational and Applied Mathematics.
• Capasso V., Kunze H., La Torre D., Vrscay E.R., Parametric estimation for deterministic and stochastic differential equations with applications (2008) - Cambridge University Press - Advances in nonlinear analysis theory methods and applications (S. Sivasundaram ed.).
• La Torre, D., Mendivil, F., Iterated function systems on multifunctions and inverse problems (2008) - 340, 2, 1469-1479 - Journal of Mathematical Analysis and Applications.
• H. Kunze, D. La Torre, E. R. Vrscay, Contractive multifunctions, fixed point inclusions and iterated multifunction systems (2007) - 330, 159-173 - Journal of Mathematical Analysis and Applications.
• H. Kunze, D. La Torre, E. R. Vrscay, Random fixed point equations and inverse problems by collage theorem (2007) - 334, 1116-1129 - Journal of Mathematical Analysis and Applications.

We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of diffeomorphisms of a flat torus. We construct a representation of the strong solution to the Navier-Stokes equations in terms of diffusion processes (joint work with A. B. Cruzeiro).

A system of differential equations modeling the evolution of (concentrations of) several chemical species whose interaction leads to blood coagulation is considered. We show how this system can be obtained using a multiscaling approach on a different system, of enzymatic reactions. This approach can also be helpful in finding stable solutions.

The plant growth and particularly the appearance of new branches is a consequence of the concentration of two hormones. One of them is produced in the roots and the other one is produced in the growing parts of the plant. Many dynamical models of this phenomena have already been studied. In this work we study a model of multiple branching. We deal with the transport equation in domains of different sizes by using a variational reduction type method based on the asymptotic partial decomposition for an elliptic problem. In general this is applied to the case of a geometrical heterogeneous domain when the right hand side does not depend on the striking variable. In our case we consider the transport equation in an heterogeneous domain with a general right hand side.

This talk is divided into two parts. In the first part, we introduce the theory of inverse problems and present some methods: regularization technique (operators, variational methods and artificial neural networks). In the second part, we present and discuss some inverse problems arising in different fields of space research: space science (maps of cosmic background radiation, magneto-teluric inversion), space engineering (damage identification in aerospace structures), oceanography (ocean optical properties), and meteorology (atmospheric temperature and humidity from satellite data).

The existence of (super-exponentially decaying)} solutions to an equation is a potential problem to the Mathematical modeller. In the light of recent work on Mixed Type Functional Equations (MFDEs), also known as Forward-Backward Equations, we discuss research relating to Delay Differential Equations (DDES) that might enhance our insight into MFDEs. We will:

1. discuss the concept of a small solution;
2. outline some of the potential problems associated with the existence of small solutions;
3. give details of the methodology underpinning our numerical approach to the detection of small solutions (to Delay Differential Equations).

We include a summary of known analytical results and of our numerical approach. We consider the characteristic shapes of the resulting eigenspectra and hence justify our decision to pursue a statistical analysis of the eigenvalue data. We consider the following statistical parameters as possible ways of determining whether or not an equation admits small solutions.

1. Standard deviation.
2. Skewness (Is the distribution symmetrical?)
3. Kurtosis (Is the distribution ‘peaked’?)
4. Spearman’s rank correlation (Can we identify a monotonic relationship?)

We justify our conclusion that, based on the statistical analysis carried out to date, this method of ‘decision making’ was not reliable and indicate how this part of our project informed the next stage of our work.

For a class of free boundary problems with applications in plasma physics, an analytical-numerical approach is proposed, based on the asymptotic expansion of the solution in the neighborhood of the singular points. This approach was already used to approximate the solution of certain classes of singular boundary value problems on bounded and unbounded domains. Here, one-parameter families of solutions of suitable singular Cauchy problems, describing the behavior of the solution at the singularities, are derived and based on these families numerical methods are constructed.

In this talk we present and analyse a numerical method for the solution of a distributed order differential equation of the general form $\int_0^m \mathcal{A}(r, D^r_*u(t)) \, dr = f(t)$, where the derivative $D^r_*$ is taken to be a fractional derivative of Caputo type of order $r$. We give a convergence theory for our method and conclude with some numerical examples.

The Boundary Element Method (BEM) method for solving two dimensional incompressible Navier-Stokes equations is presented. The weak integral formulation of BEM methodology which involves the velocity-vorticity scheme is presented in details. The treatment of terms involving non-linearities and time derivatives is explained and the numerical implementation is addressed. Some representative fluid flow examples are solved and the numerical results are discussed.