Mirror descent method and its stochastic version for solving inverse problems


Due to its simplicity of implementation and low complexity per iteration, Landweber method has been extensively studied for solving ill-posed inverse problems in Hilbert spaces and its convergence and rates of convergence are well-known. In order to adapt the method to the feature of the sought solution and the nature of the underlying problems, one needs to consider its extension for solving inverse problems in Banach spaces with a strong convex regularization term incorporated into the algorithm design. This leads to the mirror descent method which has been studied in optimization but not quite well-understood for inverse problems. Although its convergence property has been established in our early papers, how to derive the rates of convergence has been a challenging open question due to the appearance of non-Hilbertian structure of the underlying space and the non-quadratic feature of the regularization term. By interpreting the mirror descent method as a dual gradient method, recently we made progress toward deriving the convergence rates which I will address in the talk. 

In many cases, the ill-posed inverse problems may be formulated as a system consisting of a number of equations.  Solving such a problem of large size by the mirror descent method using the whole information at each iteration step can be very expensive, due to the huge amount of memory and excessive computational load per iteration. To solve such large-scale ill-posed systems efficiently, recently we developed a stochastic mirror descent method which uses only a small portion of equations randomly selected at each iteration step. The method scales very well with the problem size and has the capability of capturing features of sought solutions. We will discuss the convergence property of the method if time permits. 



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Topic: Mathematics and Computational Sciences Seminar Series 
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