I. Sensitivity of Numerical Methods and Adaptivity

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Many problems in scientific computing must be solved with sophisticated methods that adapt themselves to the difficulties posed by the specific problem. The applicant's area of research is mostly in the numerical solution of differential equations arising from continuous mathematical models of physical phenomena. When solving such an equation numerically, the underlying domain of computation must be discretized on a mesh or a grid. Traditionally the discretized problem is then solved as accurately as possible on the chosen mesh. Adaptive methods allow the mesh to adapt to the specific solution profile, or, in time-dependent problems, to move with the evolving features of the solution. Our interest is concentrated on combining mesh adaptivity with high-order methods, in particular spectral methods and radial basis function methods. We have developed methods that can resolve extremely thin boundary layers, by combining coordinate stretching techniques with adaptively adding collocation points.*The next step is to extend these methods to interior layers. To resolve very thin layers we need to find a proper coordinate transformation. Our aim is to have robust automatic methods that do not require tuning of parameters - in the boundary layer case we achieved this by adding adaptivity.**High-order methods are often more sensitive to round-off error, and one must be careful when choosing the best numerical discretization. Even though theoretical results show that the accuracy should improve with a finer discretization, in practice one gets much worse results dominated by round-off error. High-order methods often lead to very ill-conditioned systems of equations to be solved, yet, numerical computations produce very accurate results despite the ill-conditioning. We are working on gaining a better understanding of this phenomenon, and to use this knowledge to improve our algorithms. We are also looking into reformulating discretizations of differential equations as to avoid the ill-conditioning in the first place. **Our second area of inquiry is in the field of computational and mathematical methods in medical imaging. The research is driven by applications from nuclear medicine, in particular SPECT (single photon emission computed tomography), although many of our results have applications to other image modalities such as PET and MRI. Our main mathematical interest lies in dynamic SPECT imaging. In static imaging, one single image is reconstructed from the data obtained during a patient scan; in the dynamic case, the same data are used to reconstruct a sequence of 3-D images (i.e., a 3D movie) to show the dynamic behavior. This is an ill-posed problem, with the challenge of not having enough data to determine the unknowns. Hence, additional information must be incorporated into the solution algorithms to exclude mathematically possible solutions that are not physically meaningful. *One of our approaches is of a stochastic nature (Kalman filter). Enforcing positivity of the image is an obvious constraint, but to obtain meaningful results additional regularization is required. The Kalman filter provides temporal smoothing, spatial smoothing must be forced explicitly. We are improving the computational efficiency of this approach.*Our second approach is based on iterative methods that allow for imposing constraints on the solution. We are reconstructing all frames simultaneously. This is more expensive than a frame-by-frame reconstruction, but results in superior image quality, and allows us to control the shape of time-activity curves - i.e., essentially the time evolution of small regions (or voxels) of the reconstructed image.