学术报告

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6月23日 学术报告（三个）

发布时间:2015-06-23

1.题目：Several Aspects of Statistical Inverse Problems

报告人：林逵博士

时  间：6月23日（周二）15:00-16:00

地  点：新数学楼415室

Abstract:
In this talk, we focus on several aspects of statistical inverse problems. Firstly, we discuss Bayesian inverse problems in Hilbert spaces. The focus is on a fast concentration of the posterior probability around the unknown exact solution as expressed in the concept of posterior contraction rates. This concentration is dominated by a parameter which controls the variance of the prior distribution. Previous results determine posterior contraction rates based on known the smoothness of the truth. Here we show that an oracle-type parameter choice is possible. In addition, we show that the tail probability, which usually is bounded by using Chebyshev's inequality, actually has exponential decay.

Secondly, by introducing an artificial dynamic, we connect the filter approach with iterative regularization methods. We study the convergence behavior of Kalman Filter and 3DVAR as the noise level goes to zero. For different data models and different filter schemes, we obtain corresponding error estimates, which allow for proposing parameter selection and stopping criterions.

Finally, as an example of application, we establish a rigorous function-space based Bayesian formulation for a subsurface flow geometric inverse problem, which aims at determining the permeability of the subsurface from hydraulic head measurements. We study geometrically defined prior permeability fields, which admit layered, fault and channel structures, in order to mimic realistic subsurface features; within each layer we adopt either constant or continuous function representation of the permeability. We adopt a Bayesian framework showing the existence and well-posedness of the posterior distribution. We also introduce novel Markov Chain-Monte Carlo (MCMC) methods, which exploit the different character of the geometric and permeability parameters, and build on recent advances in function space MCMC. These algorithms provide rigorous estimates of the permeability, as well as the uncertainty associated with it, and only require forward model evaluations. No adjoint solvers are required and hence the methodology is applicable to black-box forward models. We then use these methods to explore the posterior and to illustrate the methodology with numerical experiments.


2.题目：打印的交叉课题研究

报告人：杨周旺副教授
        (中国科技大学)

时  间：6月23日(周二)16:00-17:00

地  点：新数学楼415室

摘要:
    快速发展的3D打印技术使得人们对几何建模与优化处理提出了更高的要求和挑战。我们从结构强度、稳定性及打印效率等方面开展3D打印的交叉课题研究，提出3D打印的几何与物理优化的一些关键理论问题与算法，开发基于隐式体表示的3D打印引擎，并通过实际打印试验来验证所提出方法的有效性。



3.题目：Wavelet Frame Transforms and Differential Operators: Bridging Discrete and Continuum for Image Restoration and Data Analysis

报告人：董彬教授
        (北京大学)

时  间：6月23日(周二)17:00-18:00

地  点：新数学楼415室

Abstract:
My talk is mainly based on a series of three papers ([1-3] below). In [1], we established connections between wavelet frame transforms and differential operators in variational framework. In [2], we established their connections for nonlinear evolution PDEs. Based on [1,2], we proposed a new piecewise smooth image restoration model based on wavelet frames in [3], and linked it with a brand new variational model, a special case of which resembles, but is superior to, the well-known Mumford-Shah model. The connections established in [1-3] provide us with new insights and inspiring interpretations of both wavelet frame and differential operator based approaches, which enable us to create new models and algorithms for image restoration that combine the merits of both approaches.  The significance of our findings is beyond what it may appear. In fact, our analysis and discussions in [1-3] already indicate that wavelet frame based approach is a new and useful tool in numerical analysis to discretize and solve variational and PDE models in general, which enriches the existing theory and applications of numerical PDEs, variational techniques, wavelet frames, etc.

Although the main application considered is image restoration, I will also discuss possible extensions to high-dimensional unstructured data analysis [4]. I will present a unified theory of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process and analyze graph data.

[1].  J. Cai, B. Dong, S. Osher and Z. Shen, Image restoration: total variation; wavelet frames; and beyond, Journal of AMS, 25(4), 1033-1089, 2012.

[2].  B. Dong, Q. Jiang and Z. Shen, Image restoration: wavelet frame shrinkage, nonlinear evolution PDEs, and beyond, preprint, December 2013.

[3].  Jian-Feng Cai, Bin Dong and Zuowei Shen, Image restorations: a wavelet frame based model for piecewise smooth functions and beyond, preprint, April 2014.

[4].  Bin Dong, Sparse Representation on Graphs by Tight Wavelet Frames and Applications, preprint, June, 2015.

欢迎师生们参加！