RESEARCH STATEMENT Xuemin Tu 1 Overview My general research interest is in developing efficient numerical algorithms and applying them to real life problems. Particularly, I have been working on particle filters, domain decomposition algorithms, nonlinear multigrid methods. I have helped develop some special finite element methods for non-convex domains. I have also worked on some biomathematical projects such as the effect of cannibalism on flour beetle population dynamics and blood flow in stenotic collapsible tubes. 2 Previous Research Experiences and Accomplishments 2.1 Domain Decomposition Algorithms and Parallel Computation Usually the first step of solving a linear elliptic partial differential equation (PDE) numerically is its discretization. Finite difference, finite element, or other discretization methods reduce the original PDE to an often huge and ill-conditioned linear system of algebraic equations Au = f. (1) Limited by the memory and speed of computers, the traditional direct solvers often cannot han- dle such large linear systems. Also, iterative methods, such as Krylov space methods, can need thousands of iterations to obtain accurate solutions due to large condition numbers of such systems. If we can find a matrix P such that P −1A has a smaller condition number than A and P −1 acting on a vector is much easier to compute than for A−1, we can then solve −1 −1 P Au = P f, (2) instead of (1). This will need much fewer iterations when we use Krylov space methods because of the much smaller condition number of P −1A. We call the matrix P −1 a preconditioner of A. Domain decomposition methods provide efficient preconditioners that can be accelerated by Krylov space methods. They have become popular in applications in computational fluid dynamics, structural engineering, electromagnetics, constrained optimization, etc. The basic idea of domain decomposition methods is to split the original huge problem into many small problems which can be handled by direct solvers, and then solve these smaller problems a number of times and accelerate the solution of the original problem with Krylov space methods. The preconditioner of a domain decomposition method can often be written as: N −1 T −1 T −1 P = Ri DiAi DiRi + R0 A0 R0. =1 kX 1 Here, we have decomposed our original domain Ω into subdomains Ωi, i = 1 · · · N. Ai is the local problem on subdomain Ωi and A0 is a coarse problem. Ri are restriction operators and Di are certain matrices of weights. There are two main classes of domain decomposition methods: overlapping Schwarz methods and iterative substructuring methods. One well-known family of the iterative substructuring do- main decomposition methods is the Balancing Domain Decomposition by Constraints (BDDC) algorithms, which were introduced and analyzed in [3, 14]. In BDDC algorithms, we first reduce the original system to a subdomain interface system (Schur complement). The local components of the BDDC preconditioners are the local Schur complements of the subdomains and the coarse com- ponent is given in term of a set of primal constraints chosen for each pair of adjacent subdomains. We can obtain condition number bounds of the form 2 −1 H κ(P A) ≤ C 1 + log . (3) BDDC h Here H is the diameter and h is the typical mesh size of subdomains and C is a constant independent of H and h. Combining this estimate and the convergence analysis of Krylov space methods, we can conclude that the rate of convergence of BDDC is independent of the number of subdomains but depends weakly on the problem size of each subdomain. These are efficient and scalable algorithms. For parallel computation, we can assign one or several subdomains to individual processors and the coarse problem to one processor or to each processor. In each iteration, the subdomain local problems of different processors are solved in parallel. The sizes of the subdomain local and coarse problems are much smaller than the original problem. Therefore, we can obtain a very good parallel efficiency with BDDC algorithms. The key issue in BDDC algorithms is how to choose the primal constraints which define the coarse problems. I have extended the BDDC algorithms to several new applications and will discuss them in some detail in Section 2.1.1. Domain decomposition methods not only provide efficient preconditioners, but can also be coupled directly to discretizations in some cases. In collaboration with Dr. Maksymilian Dryja of Warsaw University, Poland, we have developed a domain decomposition discretization for the heat equation suitable for parallel computation; see [5]. 2.1.1 Extensions of BDDC Algorithms I have extended the BDDC algorithms to scalar elliptic equation with two kinds of discretizations: namely mixed and hybrid finite element discretizations, which have many applications such as for flow in porous media. These discretizations give systems of equations of following form: T A B u F1 = . (4) " B 0 # " p # " F2 # The system matrix of (4) is symmetric indefinite with the matrix A symmetric, positive definite. The hybrid finite element discretization is equivalent to a nonconforming finite element method. We can reduce the original saddle point problem to a positive definite system for the pressure p by introducing Lagrange multipliers on the interface of the subdomains and by eliminating the velocity u in each subdomain. I then use the BDDC preconditioner to solve the interface problem for the Lagrange multipliers, which can be interpreted as an approximation to the trace of the pressure. 2 By enforcing a suitable set of constraints, I obtain the same convergence rate as for a conforming finite element case (3), see [20]. Using the mixed formulation, I obtain a saddle point problem which is closely related to that arising from the incompressible Stokes equations [13]. I first reduce the original problem to an interface problem with the solution in a benign space, which is a subspace in which the BDDC preconditioned operator is positive definite. I choose edge/face constraints to force the iterates into the benign space. The conjugate gradient methods can therefore be used to accelerate the convergence. I have proved a condition number bound as in (3) for this BDDC algorithm, see [19]. We have also considered BDDC algorithms for some symmetric indefinite and nonsymmetric positive definite problems. This is a joint work with Dr. Jing Li of Kent State University. For the symmetric indefinite case, we consider the following system of linear equations 2 Au = (K − σ M)u = f, (5) where K is the stiffness matrix, M is the mass matrix, and σ is a scalar. This system arises from a finite element discretization of the Helmholtz equation on bounded interior domains or from inverse iterations for eigenvalue problems. Our BDDC algorithm is motivated by the dual-primal finite element tearing and interconnecting algorithm (FETI-DP) for solving the time-harmonic wave propagation problems (FETI-DPH), which was first proposed by Farhat and Li in [6]. The FETI-DPH method has been shown to be parallel scalable by extensive experiments and has been applied to the simulation of elastic waves in structural dynamics problems, and to the simulation of sound waves in acoustic scattering problems. A key component in FETI-DPH, and our BDDC algorithm, is some plane waves incorporated in the coarse level problem to enhance the convergence rate. These plane waves represent exact solutions of the partial differential equation in free space. This idea was first introduced by Farhat, Macedo, and Lesoinne [7] with the FETI-H algorithm for solving the Helmholtz equations. We use the GMRES iterations for the BDDC preconditioned system. Under the condition that the diameters of the subdomains are small enough, we prove that the convergence rate of the GMRES iteration depends polylogarithmically on the dimension of the individual subdomain problems and that it improves with a decrease of the subdomain diameters. We also establish the spectral equivalence between the proposed BDDC algorithms and the FETI- DPH algorithms for solving (5). Therefore, a convergence analysis of FETI-DPH algorithms is also obtained, see [12]. The systems of linear equations arising from the finite element discretization of advection- diffusion equations are nonsymmetric, but usually positive definite. We proposed a BDDC precon- ditioner for this nonsymmetric but positive definite system. A preconditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. A key component in this BDDC algorithm, is certain constraints related the flux across the sub- domain interface incorporated in the coarse level problem to enhance the convergence rate. A convergence rate estimate for the GMRES iteration is established, under the condition that the di- ameters of subdomains are small enough. It is independent of the number of subdomains and grows only slowly with the subdomain problem size. Numerical experiments for several two-dimensional advection-diffusion problems illustrate the fast convergence of this algorithm, see [24]. 2.1.2 Three-level BDDC Algorithms One of the shortcoming of the BDDC methods is that the coarse problem needs to be generated and factored by a direct solver at the beginning of the computation. The number of primal constraints 3 selected for each subdomain must be large enough to make sure that the preconditioned system has the condition number bound as in (3). The coarse component can therefore be a bottleneck if the number of subdomains is very large. Motivated by this, I have developed two three-level BDDC algorithms to remove this difficulty. I group several subdomains together into a subregion and then use BDDC idea recursively for the coarse problem. I first reduce the original coarse problem to a subregion interface problem by eliminating the subregion interior variables independently. In the three-level BDDC algorithms, I do not solve the subregion interface problem exactly, but treat it by doing one iteration of the BDDC preconditioner.