Sensitivity of Numerical Methods and Adaptivity

科学计算中的许多问题必须用复杂的方法来解决，这些方法要适应具体问题所带来的困难。申请人的研究领域主要是由物理现象的连续数学模型产生的微分方程式的数值解。在数值求解这类方程时，计算的基本区域必须在网格或网格上离散。传统上，离散化的问题会在选定的网格上尽可能精确地求解。自适应方法允许网格适应特定的解轮廓，或者，在时间相关的问题中，随着解的演变特征而移动。我们的兴趣集中在将网格自适应与高阶方法相结合，特别是谱方法和径向基函数方法。通过将坐标拉伸技术与自适应添加配置点相结合，我们已经开发出了能够解析极薄边界层的方法。下一步是将这些方法扩展到内层。为了分辨非常薄的层，我们需要找到一个适当的坐标变换。我们的目标是拥有稳健的自动化方法，不需要调整参数-在边界层的情况下，我们通过增加适应性来实现这一点。高阶方法通常对舍入误差更敏感，在选择最佳的数值离散时必须小心。尽管理论结果表明，更精细的离散化可以提高精度，但在实际应用中，以舍入误差为主的结果要差得多。高阶方法经常导致要求解的方程组非常病态，然而，尽管存在病态，数值计算产生了非常精确的结果。我们正在努力更好地理解这一现象，并利用这些知识来改进我们的算法。我们还在研究重新表述微分方程的离散化，以避免首先出现的病态。我们的第二个研究领域是医学成像中的计算和数学方法领域。这项研究是由核医学的应用程序推动的，特别是SPECT(单光子发射计算机断层扫描)，尽管我们的许多结果也应用于其他图像方式，如PET和MRI。我们的主要数学兴趣在于动态SPECT成像。在静态成像中，从患者扫描期间获得的数据重建单个图像；在动态情况下，相同的数据被用于重建3D图像序列(即，3D电影)以显示动态行为。这是一个不适定的问题，挑战是没有足够的数据来确定未知数。因此，必须将额外的信息合并到解算法中，以排除在数学上没有物理意义的可能的解。我们的方法之一是随机性质的(卡尔曼滤波)。强制图像的正性是一个明显的限制，但为了获得有意义的结果，需要额外的正则化。卡尔曼滤波提供时间平滑，空间平滑必须显式强制。我们正在提高这种方法的计算效率。我们的第二种方法是基于迭代方法，允许对解施加约束。我们正在同时重建所有的帧。这比逐帧重建成本更高，但可以产生更好的图像质量，并允许我们控制时间活动曲线的形状-即基本上重建图像的小区域(或体素)的时间演变。

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.