I. Sensitivity of Numerical Methods and Adaptivity

• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588599
• 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从物理现象的连续数学模型中产生的微分方程的数值解。当数值求解这样的方程时，计算的基本域必须在网格或网格上离散化。传统的离散化问题，然后尽可能准确地解决选定的网格。自适应方法允许网格适应特定的解决方案配置文件，或者，在时间相关的问题，移动与解决方案的不断发展的功能。我们的兴趣集中在结合网格自适应高阶方法，特别是谱方法和径向基函数方法。我们已经开发出的方法，可以解决极薄的边界层，结合坐标拉伸技术与自适应添加配置点。 下一步是将这些方法扩展到内部层。为了解决非常薄的层，我们需要找到一个适当的坐标变换。我们的目标是有强大的自动方法，不需要调整参数-在边界层的情况下，我们通过增加自适应实现这一点。 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，精度应该提高与更精细的离散化，在实践中，一个得到更差的结果占主导地位的舍入误差。高阶方法往往导致非常病态的方程组求解，然而，数值计算产生非常准确的结果，尽管病态。 我们正在努力更好地了解这种现象，并利用这些知识来改进我们的算法。我们也正在研究重新制定离散微分方程，以避免病态摆在首位。 我们的第二个调查领域是在医学成像的计算和数学方法领域。这项研究是由核医学，特别是SPECT（单光子发射计算机断层扫描）的应用驱动的，尽管我们的许多结果也适用于其他成像方式，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，根据在患者扫描期间获得的数据重建一个单个图像;在动态情况下，使用相同的数据来重建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，额外的信息必须纳入解决方案的算法，以排除数学上可能的解决方案，没有物理意义。 我们的方法之一是随机性（卡尔曼滤波器）。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。卡尔曼滤波器提供时间平滑，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。我们同时重建所有画面。这比逐帧重建更昂贵，但可获得上级图像质量，并允许我们控制时间-活动曲线的形状-即，实质上是重建图像的小区域（或体素）的时间演化。