I. Sensitivity of Numerical Methods and Adaptivity

一、数值方法的敏感性和适应性

• 批准号: RGPIN-2014-05758
• 负责人: Trummer, Manfred
• 金额: $ 0.8万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667188
• 关键词: I.; Sensitivity; Numerical; Methods; Adaptivity

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.

科学计算中的许多问题必须用复杂的方法来解决，这些方法必须适应特定问题所带来的困难。申请人的研究领域主要是由物理现象的连续数学模型引起的微分方程的数值解。当用数值方法求解这类方程时，计算的基础域必须在网格或网格上离散化。传统的方法是在选定的网格上尽可能精确地求解离散化问题。自适应方法允许网格适应特定的解决方案轮廓，或者，在时间相关的问题中，随着解决方案的演变特征而移动。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。我们开发了一种方法，通过结合坐标拉伸技术和自适应添加搭配点来解决极薄的边界层。*下一步是将这些方法扩展到内层。为了解决非常薄的层，我们需要找到一个合适的坐标变换。我们的目标是拥有不需要调整参数的鲁棒自动方法——在边界层的情况下，我们通过增加自适应来实现这一点。**高阶方法通常对舍入误差更敏感，在选择最佳数值离散化时必须小心。尽管理论结果表明，离散化越精细，精度越高，但在实际操作中，由于舍入误差的影响，得到的结果要差得多。高阶方法往往导致求解条件非常恶劣的方程组，然而，尽管条件恶劣，数值计算仍能得到非常精确的结果。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。我们也在研究微分方程离散化的重新表述，以便首先避免不良条件。**我们研究的第二个领域是医学成像中的计算和数学方法。这项研究是由核医学的应用驱动的，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模式，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，使用相同的数据来重建一系列3D图像（即3D电影）以显示动态行为。这是一个不适定的问题，面临的挑战是没有足够的数据来确定未知数。因此，必须将附加信息纳入解决算法，以排除在物理上没有意义的数学上可能的解决方案。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约束，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，空间平滑必须明确强制。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，允许对解决方案施加约束。我们同时重建所有的帧。这比逐帧重建更昂贵，但会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，本质上是重建图像的小区域（或体素）的时间演变。