This volume iS a collection of lecture notes generated from the first two series of mini-courses of“Shanghai Summer Sch001 on Analysis and Numerics in Modern Sciences”held during the summers of 2004 and 2006 at Fudan University，Shanghai，China．The summer school pro-grams attracted more than 130 participants each year，including gradu-ate students，postdoctors，and junior faculty members from more than 30 universities in China and USA．
　　The purpose of the summer school i8 to promote the interaction and collaboration of researchers with expertise in scientific modeling，mathe-matical analysis and numerical simulations．The foCUS of the year 2004’S program was on the study of the multi-scale phenomena in complex flu-ids．The foCUS of the year 2006’S program was on multi-scale analysis in nonlinear partial ditierential equations and their applications．
　　The summer school hosted sever&1 mini-courses each year．During the summer of 2004，the instructors are Wleizhu Bao fNational Uni-versity of Singapore），Thomas Hou（California Institute of Technology,USA），Chun Liu（Penn State University，USA），Jianguo Liu（Univer一’sity of Maryland，USA），Tiehu Qin（Fudan University，PRC）and Qi Wang（Florida State University，USA）．During the summer of 2006，the mini-courses were taught by Zhaojun Bai and Albert Fannjiang（Univer-sity of California at Davis，USA），Thomas HOU，Wenbin Chen and Feng Qiu（Fudan University，PRC），Chun Liu，and Xiaoming Wang（Florida State University，USAl．There are also short 1ectures given by many distinguished visitors from around the world．
　　There are five chapters in this volume，covering a wide range of top-ias in both analysis and numeric&l simulation methods．a(chǎn)s well as their applications．
　　Chapter 1．by Zhaojun Bai，Wlenbin Chen，Richard Scalettar and Ichitaro Yamazaki．iS on the numerical methods for quantum Monte Carlo simulations of the Hubbard Models．
　　Chapter 2，by Albert Fannjiang，is on the wave propagation and imaging in random media．
　　Chapter 3，bv Thomas Hon，iS on multi-scale computations for flow and transport in porous media．
　　Chapter 4，by Chun Liu，iS on the energetic variational approaches of elastic complex fluids．