王宏，美国南卡罗来纳大学数学系终身教授，山东大学长江学者讲座教授。主要从事油气田勘探开发、环境污染的预测与治理和二氧化碳埋存等领域的数学模型、数值模拟与大规模科学计算的理论及应用方面的研究。迄今为止已在美国工业与应用数学会的多种期刊（SIAM J Numer. Anal.、 SIAM Sci. Comput.）、计算物理杂志（J Comput Phys）、Numer. Methods PDEs和英国IMA J. Numer. Anal.等国际权威学术杂志发表论文百余篇。王宏教授还是Numer. Methods PDEs、 Computing and Visualization in Sciences、Intl J. Numer. Anal. Modeling等国际知名杂志的编委。王宏教授的研究得到了美国国家自然科学基金会、挪威自然科学基金会、南卡州以及世界排名前列的石油公司等的多项基金资助。
In recent year, fractional partial differential equations and related nonlocal models, which provide an adequate and accurate description of phenomena that exhibits long-range spatial interactions, attract increasingly more attention.
Computationally, because of the nonlocal property of fractional differential operators, the numerical methods for these problems often generate dense/full stiffness matrices. Traditionally, direct methods were used to solve these problems, which require O(N3) computations (per time step) and O(N2) mememy, where N is the number of unknowns.
We go over the development of accurate and efficient numerical methods for fractional PDEs and related nonlocal models, which has an optimal order storage and almost linear computational complexity. These methods were developed by utilizing the structure of the stiffness matrices. No lossy compression or approximation was used. Hence, these methods retaining the same accuracy and approximation/conservation property of the underlying numerical methods. We will also discuss open problems that occur in related research.