/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Structure and nanostructure in ionic liquids.

Until recently, “room-temperature” (<100–150 °C) liquid-state electrochemistry was mostly electrochemistry of diluted electrolytes(1)–(4) where dissolved salt ions were surrounded by a considerable amount of solvent molecules. Highly concentrated liquid electrolytes were mostly considered in the narrow (albeit important) niche of high-temperature electrochemistry of molten inorganic salts(5-9) and in the even narrower niche of “first-generation” room temperature ionic liquids, RTILs (such as chloro-aluminates and alkylammonium nitrates).(10-14) The situation has changed dramatically in the 2000s after the discovery of new moisture- and temperature-stable RTILs.(15, 16) These days, the “later generation” RTILs attracted wide attention within the electrochemical community.(17-31) Indeed, RTILs, as a class of compounds, possess a unique combination of properties (high charge density, electrochemical stability, low/negligible volatility, tunable polarity, etc.) that make them very attractive substances from fundamental and application points of view.(32-38) Most importantly, they can mix with each other in “cocktails” of one’s choice to acquire the desired properties (e.g., wider temperature range of the liquid phase(39, 40)) and can serve as almost “universal” solvents.(37, 41, 42) It is worth noting here one of the advantages of RTILs as compared to their high-temperature molten salt (HTMS)(43) “sister-systems”.(44) In RTILs the dissolved molecules are not imbedded in a harsh high temperature environment which could be destructive for many classes of fragile (organic) molecules.

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Ionic Liquids at Electrified Interfaces

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

The double layer structure of two ionic liquids (ILs), 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate ([Py1,4]FAP) and 1-ethyl-3-methylimidazolium tris(pentafluoroethyl)trifluorophosphate ([EMIm]FAP) at the polarized Au(111) electrode interface is probed using Atomic Force Microscopy force measurements. The force-separation profiles suggest a multilayered morphology is present at the electrified Au(111)−IL interface, with more near surface layers detected at higher potentials. At the (slightly negative) open circuit potential, multiple ion layers are present, and the innermost layer, in contact with the Au(111) surface, is enriched in the cation due to electrostatic adsorption. Upon applying negative electrode potentials (−1.0 V, −2.0 V), stronger IL near surface structure is detected: both the number of ion layers and the force required to rupture these layers increases. Positive electrode potentials (+1.0 V, +2.0 V) also enhance IL near surface structure, but not as much as negat...

Double Layer Structure of Ionic Liquids at the Au(111) Electrode Interface: An Atomic Force Microscopy Investigation

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as T...

A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

We discuss recent positive experiences applying convex feasibility algorithms of Douglas---Rachford type to highly combinatorial and far from convex problems.

/pdf/recent-results-on-douglas-rachford-methods-for-combinatorial-qjtegr5092.pdf

Recent Results on Douglas---Rachford Methods for Combinatorial Optimization Problems

In this paper, we present two Douglas–Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas–Rachford scheme, are promising.

A Cyclic Douglas–Rachford Iteration Scheme

We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practical verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state of the art on simulated and experimental data validates our approach and points toward directions for further improvement.

/pdf/proximal-heterogeneous-block-implicit-explicit-method-and-15vsmms4e7.pdf

Matthew K. Tam

Papers

Double Layer Structure of Ionic Liquids at the Au(111) Electrode Interface: An Atomic Force Microscopy Investigation

A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

Recent Results on Douglas---Rachford Methods for Combinatorial Optimization Problems

A Cyclic Douglas–Rachford Iteration Scheme

Proximal Heterogeneous Block Implicit-Explicit Method and Application to Blind Ptychographic Diffraction Imaging