Numerical Inversion of the Laplace Transform and its Applications to Evolution Equations

研究者研究向量值拉普拉斯变换数值反演的理论和应用。目标是(i)扩展研究者在无噪声情况下的拉普拉斯变换数值反演的结果，（ii）开发新的向量值拉普拉斯变换数值反演的近似方法，（iii）表明拉普拉斯变换的反演与有限元理论一起为卷积型演化方程解的数值近似提供了坚实的基础。（四）将近似方法应用于多尺度多孔材料的输运问题；（五）为本科生提供研究经验。其主要思想是将一般巴拿赫空间中具有值的连续和指数有界函数空间上的移位算子半群的逼近方法转化为这些函数的拉普拉斯变换反演的逼近方法。进化过程出现在许多科学问题中，如流体流动、图像处理、机械系统、相对论、数学金融和数学生物学。这些过程用某些积分偏微分方程的解来描述。然而，在大多数情况下，解不能被明确地计算出来，因为它们要么不能被找到，要么不能以普通代数形式得到。因此，为了获得进化过程的准确描述，人们需要对这些方程的解进行精确的近似。研究者和他的合作者最近开发的关于拉普拉斯变换的可扩展方法为卷积型积分偏微分方程的解提供了精确的近似。首席研究员进一步发展和实现了新的近似方法，并使用这些方法来准确描述不同的进化过程。通过使用不同的方法来近似解决与多尺度多孔材料运输相关的问题，如石油和天然气勘探，或控制地下污染源，如高放射性废物和地质储存的二氧化碳，为学生提供本科研究经验。

JaraDMS-1008101 The investigator studies the theory and applications of thenumerical inversion of the vector-valued Laplace transform. Theobjectives are (i) to extend the investigator's results for thenumerical inversion of the Laplace transform beyond thenoise-free case, (ii) to develop new approximation methods forthe numerical inversion of the vector-valued Laplace transform,(iii) to show that the inversion of the Laplace transformtogether with the theory of finite elements provide a solidfoundation for the numerical approximation of solutions ofevolution equations of convolution type, (iv) to apply theapproximation methods to problems arising from transport inmultiscale porous materials, and (v) to provide researchexperience for undergraduate students. The main idea is thatapproximation methods for the shift operator semigroup on thespace of continuous and exponentially bounded functions withvalues in a general Banach space translates into approximationmethods for the inversion of the Laplace transform of thesefunctions. Evolution processes arise in many scientific problems, suchas fluid flows, image processing, mechanical systems, relativity,mathematical finance, and mathematical biology. These processesare described by the solutions of certain integro-partialdifferential equations. However, in most of these cases, thesolutions cannot be calculated explicitly because either theycannot be found or they are not obtained in a plain algebraicform. Thus, in order to obtain an accurate description of theevolution process, one needs to develop accurate approximationsto the solutions of these equations. The scalable methodsrecently developed by the investigator and his collaboratorsconcerning the Laplace transform provide accurate approximationsto solutions of integro-partial differential equations ofconvolution type. The principal investigator further developsand implements new approximation methods, and uses these methodsfor the accurate description of different evolution processes. Undergraduate research experience is provided to students byusing the different methods to approximate the solutions ofproblems related to transport in multiscale porous materials,like oil and gas exploration, or controlling underground sourcesof pollution such as high-level radioactive waste andgeologically stored carbon dioxide.