ÌáÒª:
Problems that transcend a variety of strongly coupled time and length scales are ubiquitous in modern science and engineering such as physics, biology, and materials. Those multiscale problems pose major mathematical challenges in terms of analysis, modeling and simulation. In this talk, I will discuss two classes of typical multiscale methods, atomistic/continuum methods for crystalline solids with defects and numerical homogenization methods for heterogeneous media. The mathematical understanding of those methods is far from complete, and rigorous theory is absent in most physical relevant cases. The central mathematical question here, is to establish accurate and robust quantitative estimates for the consistency, stability, convergence rates and computational cost provided the dimension of original complex multiscale problems can be drastically reduced by those multiscale methods, and to construct multiscale methods which can achieve the optimal balance of the convergence rates and the computational cost based on rigorous mathematical justification.