Applied Mathematics

The presentation will take us along the road to the ozone standard for the United States, announced in Mar 2008 by the US Environmental Protection Agency, and then the new proposal in 2014. That agency is responsible for monitoring that nation’s air quality standards under the Clean Air Act of 1970. I will describe how I, a Canadian statistician, came to serve on the US Clean Air Scientific Advisory Committee (CASAC) for Ozone that recommended the standard and my perspectives on the process of developing it. I will introduce the rich cast of players involved including the Committee, the EPA staff, “blackhats,” “whitehats,” “gunslingers,” politicians and an unrevealed character waiting in the wings who appeared onstage only as the 2008 standards had been formulated. And we will encounter a couple of tricky statistical problems that arose along with approaches, developed by the speaker and his coresearchers, which could be used to address them. The first was about how a computational model based on things like meteorology could be combined with statistical models to infer a certain unmeasurable but hugely important ozone level, the “policy related background level” generated by things like lightning, below which the ozone standard could not go. The second was about estimating the actual human exposure to ozone that may differ considerably from measurements taken at fixed site monitoring locations. Above all, the talk will be a narrative about the interaction between science and public policy - in an environment that harbors a lot of stakeholders with varying but legitimate perspectives, a lot of uncertainty in spite of the great body of knowledge about ozone and above all, a lot of potential risk to human health and welfare.

The main optimization problem in many applications in signal processing (e.g. in image reconstruction, MRI, seismic images, etc.) and statistics (e.g. model selection in regression methods), is the following sparse optimization problem. The goal is finding a sparse solution to the underdetermined linear system Ax = b, where A is an m x n matrix and b is an m-vector and m ≤ n [2]. The problem can be written as

min (over x) ||x||₁ subject to Ax = b.

There are several approaches to this problem that generally aim at approximate solutions, and often solve a simplified version of the original problem. For example passing from ℓ-norm to ℓ₁-norm yields an interesting convexification of the problem [1]. Moreover the equality Ax = b does not cover noisy cases in which Ax + r = b for some noise vector r

min (over x) ||x||₁ subject to ||Ax - b||₂ ≤ σ.

Extensive theoretical [6, 7] and practical studies [5, 8] have been carried on solving this problem and various succesfull methods adopting interior-point algorithms, gradient projections, etc. have been tested. The discrete nature of the original problem also suggests possibility of viewing the problem as a mixed-integer optimization problem [3]. However common methods for solving such mixed-integer optimization problems (e.g. Benders’ decomposition) iteratively generate hard binary optimization subproblems [4]. The exciting possibility that quantum computers may be able to perform certain computations faster than digital computers has recently spiked with the quantum hardware of D-Wave systems. The current implementations of quantum systems based on the principles of quantum adiabatic evolution, provide experimental resources for studying algorithms that reduce computationally hard problems to those that are native to the specific evolution carried by the system. In this project we will explore possibilities of designing optimization algorithms that use the power of quantum annealing in solving sparse recovery problems.

References
[1] Emmanuel J. Cand´es, Justin K. Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 59(8):1207–1223, 2006.
[2] Simon Foucart and Holger Rauhut. A Mathematical Introduction to Compressive Sensing. Birkh¨user Basel, 2013.
[3] N.B. Karahanoglu, H. Erdogan, and S.I. Birbil. A mixed integer linear programming formulation for the sparse recovery problem in compressed sensing. In Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, pages 5870–5874, May 2013.
[4] Duan Li and Xiaoling Sun. Nonlinear Integer Programming. International Series in Operations Research & Management Science. Springer, 2006.
[5] Ewout van den Berg and Michael P. Friedlander. SPGL1: A solver for large-scale sparse reconstruction, June 2007. http://www.cs.ubc.ca/labs/scl/spgl1.
[6] Ewout van den Berg and Michael P. Friedlander. Probing the pareto frontier for basis pursuit solutions. SIAM Journal on Scientific Computing, 31(2):890–912, 2008.
[7] Ewout van den Berg and Michael P. Friedlander. Sparse optimization with least-squares constraints. SIAM J. Optimization, 21(4):1201–1229, 2011.
[8] Ewout van den Berg, Michael P. Friedlander, Gilles Hennenfent, Felix J. Herrmann, Rayan Saab, and ¨Ozg¨ur Yilmaz. Algorithm 890: Sparco: A testing framework for sparse reconstruction. ACM Trans. Math. Softw., 35(4):29:1–29:16, February 2009.

The machine learning community has witnessed significant advances recently in the realm of image recognition [1,2]. Advances in computing power – primarily through the use of GPUs – has enabled a resurgence of neural networks with far more layers than was previously possible. For instance, the winning team, GoogLeNet [1,3], at the ImageNet 2014 competition triumphed with a 43.9% mean average precision, while the previous year’s winner, University of Amsterdam, won with 22.6% mean average precision.

Neural networks mimic the neurons in the brain. As in the human brain, multiple layers of computational “neurons” are designed to react to a variety of stimuli. For instance, a typical scheme to construct a neural network could involve building a layer of neurons that detects edges in an image. An additional layer could then be added which would be trained (optimized) to detect larger regions or shapes. The combination of these two layers could then identify and separate different objects present in a photograph. Adding further layers would allow the network to use the shapes to decipher the types of objects recorded in the image.

Goal of this project

An issue facing industries that deal with large numbers of digital photographs, such as magazines and retailers, is photo accuracy. Nearly all photos used in such contexts undergo some amount of editing (“Photoshopping”). Given the volume of photographs, mistakes occur [4]. Many of these images fall within a very narrow scope. An example would be the images used within a specific category of apparel on a retailer’s website. Detecting anomalies automatically in such cases would enable retailers such as Target to filter out mistakes before they enter production. By training a modern deep convolution neural network [1,5] on a collection of correct images within a narrow category, we would like to construct a network which will learn to recognize well-edited images. This amounts to learning a distribution of correct images so that poorly-edited images may be flagged as anomalies or outliers.

Keywords: neural networks, deep learning, image processing, machine learning

Prerequisites: Programming experience in Python. Experience in a Linux environment is a plus.

Developing a product to attain market maturity is a long process. Within this process it is important to make the right decisions, especially when conflicting goals arise. These are typically defined by ambitious product requirements in partly competing dimensions.

In the development of electric tools and accessories, such as those produced in a mechanical engineering enterprise like Hilti, lightweight construction, long-life cycle, robustness, high performance and low costs are typical targets. Regarding so called combis and breakers (see Figure 1) there exist corresponding key values, which help the customer to assess the product: single impact energy, weight, rated power, operator’s comfort, vibration level. The implication of these user oriented requirements present the most challenging task for development in generating a lightweight design having high performance and robustness. The task is especially demanding due to high dynamics inherent in the tools, causing very complex impact loads.

Combis and breakers achieve their huge performance from the so called electro-pneumatic hammering mechanism: ‘the heart of the tool.’ It consists of several pistons that achieve high velocities and a pneumatic spring which generates high pressures. The interaction of these different machine parts leads to very high stresses for the tool components, requiring a particularly robust design.

In this project we will perform a case study of a simulation based design, where we will pick up a real-life problem from industry, the so called chuck of a combi-hammer. As mentioned above, throughout its entire life time the chuck is frequently loaded by high impact. During our development process we have to ensure that the chuck subassembly reaches the lifetime target in a robust and reliable design. We will also practice team-based strategies to master all the unexpected difficulties arising in any complex collaborative development process.

The structure of the workshop will be a sequence of short introductory lectures, student’s research and intensive discussion phases.

In many practical situations encountered in industries, there is incomplete knowledge of material properties, boundary conditions, and sources for a given material/manufacturing process. However, process monitors such as thermocouples are typically used to measure temperature evolution in certain locations to bridge the resulting gaps. Spatial gradients of temperature are needed to predict required quantities such as internal stresses developed during the process. The temperature measurements are typically performed on a coarse grid. Computation of stresses needs temperatures on a much finer grid for a more precise estimation of spatial gradients. Usually bilinear and/or weighted interpolation techniques are used to improve the spatial resolution of temperatures. However, in the cases where there are strong exothermic and/or endothermic reactions occurring during the process, such interpolation techniques are error-prone. Using more thermocouples to measure temperature on finer grid would be an easy solution. However, such measurement is intrusive as well as perturbing in addition to increasing the cost of data acquisition. The mapping of temperatures from coarser grid to finer grid is an ill-posed problem. In addition to the spatial distribution temperatures, the thermocouple measurements also contain valuable temporal information in the form of derivatives. This additional information of temporal derivatives helps in improving the conditioning of the apparently ill-posed problem. The objective of this exercise is to develop an algorithm/procedure for mapping temperatures from coarse grid to a finer grid recognizing as well as the valuable temporal information in the corresponding derivatives.