Research/Teaching Statement Gerard Awanou 1 Trivariate Splines

Research/Teaching statement Gerard Awanou I would like to describe in this statement the research I have done and comment on future directions. My work has been on trivariate splines for scattered data interpolation and numerical solution of partial differential equations, particularly, the Navier-Stokes equations. Another line of my research is mathematical finance. I have been interested in risk minimization in regime switching and two factor stochastic volatility models. I will describe my recent success in the construction of a family of mixed elements for three dimensional elasticity, a problem which was open for over 40 years. 1 Trivariate splines Let us assume that Ω is a three dimensional domain with a triangulation 4, i.e. the union of nonoverlapping tetrahedra with the property that the intersection of any two tetrahedra is either empty, a common vertex, a common edge or a common face. A spline function of degree d and smoothness r over Ω is a r times differentiable function which is a polynomial of degree d when restricted to each tetrahedron. Finite elements are examples of spline functions. Scattered data interpolation A typical problem in surface design is the following: Given a set of scattered points in IR3, which are assumed to be at the vertices of a triangulation 4, find a spline with appropriate smoothness which interpolates the given data at the vertices. The resolution of this problem is by means of the construction of locally supported spline functions with high smoothness. I have implemented C1 spline functions in [6] on a special triangulation. In general, implementing spline functions with high smoothness is very difficult. In [7], I used an energy based method which permits one to do scattered data interpolation with splines of arbitrary degree and arbitrary smoothness. The method consists in representing splines on each tetrahedron by a vector of coefficients, encoding the smoothness conditions and interpolation conditions as linear systems of constraints, and minimizing the thin plate energy functional under these constraints. The method is parallelizable and compared to the locally supported spline method, the linear systems are easier to assemble and sparser. Navier−Stokes equations For the numerical solution of a partial differential equation defined on a domain T of the Euclidean space, one typically wants to divide the domain into small elements. This is because the error made will depend on the size of these elements. Now, many partial differential equations have more smoothness away from the boundary and for high order partial differential equations the approximate solution has to be smooth 1 across element boundaries. Using the classical finite element method, it is difficult to construct approximations which satisfy the above requirements in dimensions two and three. In my dissertation, I used spline functions of arbitrary degree and arbitrary smoothness over an arbitrary domain of IR3 to approximate the solutions of the Poisson equation, the biharmonic equation and the steady state Navier-Stokes equations in velocity pressure formulation. The numerical analysis of the Navier-Stokes equations is very important. Early work on these equations can be found in [19]. It also plays an important role in the study of the cardiovascular system. For example, blood flow in large arteries can be modelled by the Navier-Stokes equations at low Reynolds numbers, [18] and [16]. Although many computational methods exist, there is a need for more efficient methods capable of high accuracy and able to cope with irregular geometries. We start by representing the approximate solution on each tetrahedron by a vector of coefficients. Then the smoothness conditions and the boundary conditions are encoded as linear systems of constraints. A particular difficulty in treating the in- compressible Navier-Stokes equations is the divergence free condition. It is encoded as a linear constraint as well. From the weak formulation of the equations, using an energy argument, we derive the discrete problem. Lagrange multipliers are used to enforce the above constraints. Extensive details can be found in [8]. We considered two methods to linearize the nonlinear discrete equations, by a simple iterative algorithm and by Newton’s method. The linearized equations lead to consider systems of the type ALT c F = L 0 λ G with A non symmetric with symmetric part positive definite on the kernel of L. A variant of the augmented Lagrangian algorithm was used and I proved a linear convergence rate of that algorithm in [9]. The advantages of the approach to numerical PDE’s are as follows: No macro-element or locally supported spline functions are constructed; Polynomials of high degrees can be easily used locally to get better approximation properties; Smoothness can be im- posed in a flexible way across the domain at places where the solution is expected to be smooth; The mass and stiffness matrices can be assembled easily and these processes can be done in parallel; When the weak solution to the Navier-Stokes equations is strong, it satisfies the divergence-free condition exactly. For these advantages, the price to pay is that it is a computationally expensive method. Future directions: I am working on extending the spline method to the 3D time dependent Navier- Stokes equations. This requires exporting the numerical codes to a high performance 2 environment. Certainly, the method can be applied to various high order partial differential equations, e.g. the Cahn-Hilliard equations. The extension to variational inequalities for the numerical solution of the American option problem [17] is also an interesting project. Finally I am interested in possible connections with the discon- tinuous Galerkin method [15] and the FETI domain decomposition method [20]. 2 Risk minimization Contingent claims are playing an increasingly important role in modern financial markets. A contingent claim or derivative security is a financial contract whose value is derived from the value of another security such as a stock or bond. Hedging a contingent claim consists in investing in a dynamic portfolio of bonds and stocks such that the value of the portfolio is identical to the value of the claim at the expiration of the contract. Hedging contingent claims has become a prominent problem in mathematical finance. Depending on the model of stock fluctuations, it might not be possible to hedge exactly a claim. But it is always possible to find a super- replicating portfolio, a portfolio whose value dominates the value of the claim at expiration. The initial value of this portfolio is called the super-replication capital. The super-replication capital is often too high. If one lowers the initial capital, there is a possibility of shortfall and the problem becomes what trading strategies would minimize the shortfall risk for a given initial capital. There are two models of stock fluctuations I have considered: regime switching models and two-factor stochastic volatility models. In regime switching models, asset prices are generated by the realization of a Gaussian diffusion with mean and volatility which depends on the state of a finite-state Markov process. The use of a Markov chain allows to capture market trends as well as various economic factors. In two factor stochastic volatility models, the mean is constant and the volatilty is driven by two diffusion processes, one fast mean reverting and the other varying slowly. There is empiral evidence for the existence of these two factors [2]. I have solved the shortfall risk minimization problem in a discrete regime switching model [11] and characterize the super-replication capital in [12] in a continuous regime switching model using the duality method. In [13], using a singular perturbation expansion associated to the fast factor and a regular expansion associated with the slow factor, I obtained approximate trading strategies which minimize the shortfall risk in a two factor stochastic volatility model. As future direction, I plan to apply the asymptotic method as a computational tool for solving various stochastic control problems. 3 Challenging problems Stable mixed finite elements for elasticity: For over 40 years, the question of how to construct a pair of stable mixed elements for elasticity using polynomial shape 3 functions in three dimensions was open. One element of the pair is used to discretize the space of symmetric tensors in which the stress field is sought, and the other to discretize the space of vector fields in which the displacement is sought. The sym- metry of the stress tensor was the essential difficulty. The solution of this problem is expected to have a broad impact in computational relativity. Recently [5], Arnold and Winther have designed such elements for two dimensional triangular meshes. I have also very recently constructed analogues of their elements, both conforming and nonconforming, for rectangular meshes [3]. But the three dimensional problem remained open. In both cases above, the elements are related to a discrete version of the elasticity sequence, which is more complicated in three dimensions. Recently, using computational experiments, a three dimensional element was proposed in [1]. But the question remained whether there was an error in the computer program and no proof of uniqueness was immediately available. Revisiting the problem, I was able to put the puzzle together, using research notes of Arnold and Winther I improved, to design the analogues of the Arnold-Winther elements for three dimensional elasticity [4]. The Adams-Cockburn element has no analogue in two dimensions for either triangular or rectangular meshes. Moreover it has no higher order version and certainly not related to a discrete version of the elasticity sequence, a property I believe will be useful for applications. I am now completing the analysis of the nonconforming versions of the three dimensional mixed elements in the lowest order case [14].

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