Special Session 39: Adaptive and Iterative Decomposition Methods for Differential Equations: Stability, Error Analysis and Applications

We use the deformation method for moving grids to construct invertible transformation. We review the existing method by harmonic maps in 2D and discuss the difficulty it encounters in 3d domains. The proposed method is based on a div-curl-ode system, which can be used as constraints for a class of optimization problems. Preliminary numerical results demonstrated the effectiveness of the method. A theorem by Rado (1926) states that in 2d, any harmonic map from a domain to a convex domain is injective if it is one-to-one and onto between the boundaries. It turns out that its extension to 3d is false. Melas in 1993 constructed a harmonic map from a unit ball into another unit ball that is one-to-one and onto between the boundaries; but its Jacobian determinant is zero at the center. This counterexample was modified such that it maps two distinct points to the same point. The new method works for both 2d and 3d. Let f > 0 be any given function normalized to satisfy ∫ f = |D|, we construct an invertible transformation f by a div-curl-ode system such that J(f(x)) = f(f(x)). This problem is solved both theoretically and computationally. The solution method is based on Jurgen Moserś work in volume elements of Riemannian manifolds [Moser 1965]. The main technical advancement we made in [Cai 2004] is that we use div-curl in place of the Poisson equation, which enables the treatment of moving domains. It turns out that the flexibility provided by the curl equation can be explored in reconstruction of any given invertible transformation. Solution method: For a given function f(x) > 0 with ∫ f = |D|, we take f(x) − 1 as the right hand side of the div equation (Remark: the resulting divergence equation is the linearization of the equation J(f) = f). Then we take any divergence free vector field g (i.e. ∇ · g = 0) as the right hand side of the curl equation. We solve for a vector field u from the div-curl equations: ∇·u(x) = f(x)− 1,∇×u(x) = g(x) with u ·n = 0, n = the outward normal on ∂D. Then, we form a velocity vector v(x, t) by v(x, t) = u(x)/(t+(1−t)f(x)) for a parameter t ∈ [0, 1], x ∈ D. Now we solve a family of transformations T (x, t), t ∈ [0, 1], from the ordinary differential equation (ODE) for each fixed x in D : ∂T (x, t)/∂t = v(T (x, t), t) with T (x, 0) = x. Finally, we define f by setting f(x) = T (x, 1). Application of this method to image registration problem will be described. Preliminary computational results will be presented to demonstrate the effectiveness of the approach. Reference: [Cai 2004] X. Cai, D. Fleitas, B.-N. Jiang, G. Liao, Adaptive grid generation based on the least square finite element method, Computers and Mathematics 48 (2004) 1077-1085 [Moser 1965] J. Moser, Volume elements of a Riemann Manifold, Trans AMS, 120, 1965