Nonlinear Programming for Large-Scale Applications

9800544Nash这项拨款建立在先前对优化算法和软件的研究基础上，以开发成功的医学成像方法和其他具有挑战性的应用问题。研究的一些应用包括通过正电子发射断层扫描（PET）进行图像重建、放射治疗计划、x射线晶体学和水下含水层建模。这些问题非常大。所使用的算法是基于发展良好的通用非线性优化方法。他们使用基于截断牛顿方法的基础软件进行无约束优化（串行和并行），该方法是在先前的研究中开发的。应用的算法还使用了为屏障方法开发的复杂思想，这些技术确保了能够产生高精度解决方案的稳定数值计算。这些算法也基于原始对偶框架，这些框架被证明是成功的，适用于广泛的线性和非线性优化问题。通过利用应用问题的特殊特征，本工作扩展了这些强大的计算工具。并行性（包括底层优化方法和模型函数的计算）也是实现有效方法的必要条件。这项研究的直接目标是生产出可供科学家和临床医生使用的软件。这需要快速可靠的算法。长期的希望是，为这些应用问题开发的技术将导致通用优化的改进方法。总体目的是产生可靠的算法和软件，以尽快解决这些问题。

9800544Nash This grant builds upon prior research on optimization algorithms and software to develop successful methods for medical imaging and other challenging applied problems. Some of the applications studied are image reconstruction via Positron Emission Tomography (PET), radiation therapy planning, x-ray crystallography, and the modeling of underwater aquifers. These problems are very large. The algorithms used are based upon well developed general-purpose methods for nonlinear optimization. They use as a foundation software for unconstrained optimization (both serial and parallel) based on a truncated-Newton method developed in prior research. The applied algorithms also use sophisticated ideas developed for barrier methods, techniques that ensure stable numerical computations capable of producing high-accuracy solutions. The algorithms are also based on primal-dual frameworks that are proving to be successful for wide classes of linear and nonlinear optimization problems. The work extends these powerful computational tools, by exploiting the special features of the applied problems. Parallelism (both in the underlying optimization method and in the calculations of the model functions) is also essential to achieve effective methods. The immediate goal of this research is to produce software that can be used by scientists and clinicians. This requires algorithms that are fast and reliable. The longer-term hope is that the techniques developed for these applied problems will lead to improved methods for general-purpose optimization. The overall intent is to produce reliable algorithms and software to solve these problems as rapidly as possible.