Applied Mathematics and Numerical Analysis Seminar 

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.

The complexity of the cardiovascular system makes it unfeasible to perform 3D simulations in large vascular regions. Thus, truncated 3D regions of interest must be considered, originating artificial sections. Commonly these simulations are carried out neglecting the remaining cardiovascular system. However there is a strong relation between local and global hemodynamics that must be taken into account in order to perform realistic simulations. The global circulation can be approximated through reduced (1D and 0D) models. The reduced models have lower accuracy and complexity, yet they provide the useful information to be coupled to the 3D model. On the other hand, experimental results demonstrate that blood can exhibit non-Newtonian characteristics such as shear-thinning viscosity, viscoelasticity or yield stress, which should be captured by the mathematical model. We address both issues of the geometrical multiscale modelling of the cardiovascular system and blood rheology. We describe the different models, including 3D generalized Newtonian fluids and fluid-structure interaction. We also focus on the coupling between them, detailing 1D-3D, 0D-3D and 1D-0D couplings. Several numerical results are presented to illustrate the geometrical multiscale approach.

The object of this talk is to introduce the pipeline of studying physiological flows. We start by highlighting the pipeline of going from a stack of medical images to numerical simulation. Difficulties are identified and an example demonstrating the importance of accurate virtual model boundaries is presented. Morphological variations between different patients and their correlation to disease-associated fluid mechanics properties is alsointroduced. However identifying these fluid mechanics properties is also non-trivial since the full physiological complexity and responses are too complex. The idea is to identify flow characteristics and as well as when these become abnormal. Results for both a peripheral bypass graft and nasal cavity are presented.

Efficient numerical simulation of fluid-structure interaction problems involving a viscous incompressible fluid with a moderate fluid-structure density ratio is a difficult task. Blood flow simulations in deformable arteries is a popular example. Indeed, in such situations, weak (or explicit) coupling schemes, i.e. that only involve the solution of the fluid and the structure once (or just a few times) per time step, are known to give rise to numerical instabilities. Recently, we have proposed a (stabilized) explicit coupling scheme, based on Nitsche’s method, whose stability properties are independent of the fluid and structure density ratio. Stability is obtained thanks to the dissipative structure of the Nitsche coupling and a stabilization term giving control of the time fluctuations of the interface fluid load. We will discuss some theoretical and numerical results (in 2D and 3D) illustrating the features of the method.

The talk addresses the numerical solution of singularly perturbed differential equations in one and two dimensions. Because solutions of such problems exhibit sharp boundary and interior layers (which are narrow regions where solutions change rapidly), a significant economy of computer memory and time can be attained by using special layer-adapted meshes. These meshes are fine in layer-regions and standard outside; in two dimensions they have extremely high maximum aspect ratios. Ideally, mesh layer adaptation is automated by exploiting sharp a posteriori error estimates. However, the known a posteriori error estimates are typically under the minimum angle condition, equivalent to the bounded-mesh-aspect-ratio condition, which is rather restrictive and makes a posteriori error estimates less practical for layer solutions. In contrast, we present certain new a posteriori error estimates that hold true under no mesh aspect ratio condition. These estimates are in the maximum norm, which is sufficiently strong to capture layers. Furthermore, our error estimates are uniform in the singular perturbation parameter, which is significant since in general the error constant might blow up as the perturbation parameter becomes small.

This talk is concerned with the analysis of a linear mixed type (forward-backward) functional equation, that is, a linear functional equation which has a delay and an advance term. We search for a solution of such equation which has a given form at the initial interval $[-1,0]$ and at the final interval $[k-1,k]$. This problem has been studied both analytically and numerically (see[1]). One of the most common approaches for the analysis of this problem is based on its reduction to an initial value problem for a delay differential equation (DDE). Following this approach, we search for an approximate solution in the form of a linear combination of a given set ob basis functions. The basis functions can be extended to the interval $[1,k]$ either numerically (using the finite difference method) or analytically (using recurrence formulae — “method of steps”). Finally, the coefficients of the linear combination are computed by the collocation or least squares method, so that the numerical solution fits the data at the interval $[k-1,k]$. A different approach consists on the transformation of the considered problem into a boundary value problem for a ODE. In this case, standard numerical methods for ODEs can be applied. Numerical results obtained by these methods are presented and compared with the ones, presented in previous works. The advantages and weaknesses of the introduced computational methods are discussed. This is a joint work with P. Lima, P. Lumb and N. Ford.

Reference

1. N. Ford and P. Lumb, Mixed type functional differential equations: a numerical approach (submitted).

This work is concerned with modelling the evolution of competitive chemical reactions within a small cell with a labelled and unlabelled antigen reacting with a specific antibody on the side wall. A model consisting of coupled heat conduction equations with nonlinear and nonlocal boundary conditions is considered and shown to be equivalent to a system of Volterra integral equations with weakly singular kernel. This work generalizes some previous work done on the case of the single heat equation ([1], [2]). We prove the existence and uniqueness of the nonlinear system on $[0, 1)$. The asymptotic behavior of the solution as t tends to $0$ and $t$ tends to infinity is obtained and other properties of the solution, e.g., monotonicity, are investigated. In order to obtain a numerical solution of the system of VIES we use the technique of subtracting out singularities to derive explicit and implicit Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence. Numerical results are presented. This is a joint work with T. Diogo and S. McKee.

References

1. S. Jones, B. Jumarhon, S. McKee, J. A. Scott. A mathematical model of a biosensor, Journal of Engineering Mathematics 30, Netherlands, (1996) 321-337.
2. B. Jumarhon, S. McKee. On the heat equation with nonlinear an nonlocal boundary conditions, Journal of Mathematical Analysis and Applications 190, (1995) 806-820.