## Seminar:

Spring 2013, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** March 21 **

**" Multiscale $(L_2,BV)$ Decompositions and K-functional "**

**Xiaohui Wang**

__Abstract__ In this talk, the following type Tikhonov regularization problem will be systematically studied:

\[(u_t,v_t):=\argmin_{u+v=f}~\{\frac{1}{2}\|v\|^{2}_{L_2}+t|u|_{BV}\}.\]

We shall characterize the minimizing pair $(u_t,v_t)$ and give an expression of the corresponding K-functional. After that, the following multiscale decomposition scheme will be discussed:

\[u_{k+1}:=\argmin_{u\in BV(\Omega)\cap L_2(\Omega)}~\{\frac{1}{2}\|f-u\|^2_{L_2}+t_{k}|u-u_k|_{BV}\},\]

where $u_0=0$ and $t_0>t_1>...>t_k>0$.

This method was introduced by Eitan Tadmor et al. and we shall give an improved $L_2$ convergence result.