Inverse problems is a field of mathematics comprised of many areas including analysis, modeling, PDE's and scientific computation. Inverse problems arise in abundance in engineering, biology, physics, geophysics and more. This seminar addresses fundamental topics in inverse problems in a variety of applications.

Regular meeting times & location: Thursdays at 2 pm in Weber 223

Abstract: Identifying safe ventilation patterns for patients with acute respiratory distress syndrome remains challenging because of the delicate balance between gas exchange and selecting ventilator settings to prevent further ventilator induced lung injury (VILI). Accordingly, we can use computational models (macro-scale) to help forecast the development of VILI. In particular, we used a particle swarm optimization algorithm to select subject-specific model parameters and identify injury cost functions capable of forecasting VILI. We can also use micro-scale models to assist in developing our understanding of the fluid-structure interactions that cause altered regional aeration and forces in an injured lung. Simulations using this model suggest that flooded alveoli increase septal strain in their neighbors and that recruitment of these injured alveoli occurs over a longer duration as the size of a flooded region increases.

Figure 1. Max septal strain during inflation causes an avalanche of reopening in a flooded lung region.

Feb. 7 | Algebraic Geometry in Two Applications: RF Emitter Localization and Initial Orbit Determination from Space-Based Sensors | Dr Alan Lovell, Space Vehicles Directorate, Air Force Research Laboratory |

March 7 | Sparse-data tomographic imaging in practice | Samuli Siltanen, University of Helsinki, Finland |

March 28 | Deblurring Images with Mathematical Models | Malena Espanol, The University of Akron |

April 25 | Differential imaging of evolution in elastic backgrounds with unknown microstructure | Fatemeh Pourahmadian, CU Boulder |

May 2 | Pulmonary Biomechanics of an Injured Lung: A Look at Micro and Macro-Scale Models | Michelle Mellenthin, CU Denver |

Oct. 4 | Attend Applied Math Seminar at 3 pm, Weber 223: The Role of the Push-Forward Measure in Solving Inverse Problems: An Interactive Talk Using Jupyter Notebooks | Troy Butler, Univ. CO Denver |

Oct. 11 | Chemical-specific contrast in laser microscopy: inverse problems and nonlinear mappings | Jesse Wilson, Dept. of ECE and SBME, CSU |

Oct. 25 | Hyperspectral Synthetic-Aperture Radar | Andrew Horman, Matrix Research |

Nov. 1 | no seminar - Math Day | |

Nov. 15 | Coherent imaging with incoherent light and super-resolution imaging with spatio-temporally modulated illumination light | Randy Bartels, Dept. of ECE and SBME, CSU |

Nov. 29 | Control and Inverse Problems for Differential Equations on Graphs | Sergei Avdonin, University of Alaska Fairbanks |

Abstract: This talk explores two estimation problems in the engineering realm, both of which can be formulated as a polynomial system. The first involves location of an RF transmitter using time-difference-of-arrival (TDOA) measurments that do not share any common receiver location. Localization requires three independent TDOA measurements, and previous work shows that if the three measurements all share a common receiver location, the problem reduces to solving a univariate quadratic polynomial (referred to as homogeneous TDOA). However, if this is not the case, they system consists of three coupled quadratic polynomials in as many variables (referred to as heterogeneous TDOA). The heterogeneous scenario allows for localization using two moving receivers, such as orbiting satellites.

The second problem involves determining the orbit of a space object by processing line-of-sight measurements to the object from an observer spacecraft. The measurement equations are cast as a set of polynomials equal in number to the states governing the relative motion between observer and object (normally 6), with the initial conditions of the states as variables.

For both problems, the polynomial roots can be solved using analytical methods such as the Macaulay resultant, or numerical techniques such as Bertini. Improved conditioning, through variable and equation scaling, is also explored to increase the likelihood of finding accurate solutions.

Abstract: Tomographic reconstruction from comprehensive projection data is a mildly ill-posed inverse problem which is already well-understood. By comprehensive projection data we mean a large number of radiographs taken from essentially all directions around the target. However, many practical imaging situations allow only sparsely collected data. Reasons for this may include desire to reduce radiation dose or data collection time, or mechanical restrictions on imaging directions. Tomographic reconstruction from sparsely collected data is a severely ill-posed problem, and it needs to be regularized by complementing the measurement information by {\em a priori} knowledge about the target. The shearlet transform provides a flexible and computational efficient way of enforcing piecewise smoothness in the attenuation coefficient. This is relevant in many practical applications of tomography. Shearlet-sparsity-promoting regularization is discussed and illustrated with both static and dynamic tomography examples. In particular, the examination of bone samples for detecting osteoarthritis can be speeded up by a factor of 20 using shearlet-based methods.

Abstract: When we use a camera, we want the recorded image to be an accurate representation of the scene that we see. However, in some situations such as photographing a moving object, what we obtain can be a blurred image. In image deblurring, we seek to recover the original, sharp image by using a mathematical model of the blurring process. In this lecture, we will see a brief introduction to the basic image deblurring problem and some mathematical tools to address it.

Abstract: Major components of nuclear power plants e.g., reactors, fuel cells and containment vessels are comprised of highly heterogeneous composites that (a) their topology and properties at micro- and meso- scales are uncertain or in most cases unknown, and (b) their deterioration due to various chemo-physical processes such ascorrosion, irradiation, thermal cycling, etc. are not yet fully understood. These processes are responsible for the continuous microstructural evolution, leading to an inevitable development of micro/macro cracks and other anomalies, that will gradually result in the loss of structural integrity and diminished functional performance such as radiation shielding. This talk is focused on timely detection of degradation in such materials -i.e., anomalies at the microstructure scale, and active spatiotemporal tracking of their evolution. In this vein, a fast waveform tomography solution will be introduced for 3D reconstruction of evolving regions in a complex elastic background by way of ultrasonic waves. To this end, sequential sets of boundary measurements are leveraged within the framework of active sensing to carefully design a non-iterative indicator functional that is insensitive to the (unknown) stationary scatterers of the background domain e.g., the time-invariant interfaces and inhomogeneities. This differential imaging functional is rooted in the recently developed generalized linear sampling method whose affiliated cost functional is rigorously revised so that itâ€™s minimizer carries pertinent invariant properties. The performance of this new damage indicator will be illustrated through a set of numerical experiments.

Abstract: Identifying safe ventilation patterns for patients with acute respiratory distress syndrome remains challenging because of the delicate balance between gas exchange and selecting ventilator settings to prevent further ventilator induced lung injury (VILI). Accordingly, we can use computational models (macro-scale) to help forecast the development of VILI. In particular, we used a particle swarm optimization algorithm to select subject-specific model parameters and identify injury cost functions capable of forecasting VILI. We can also use micro-scale models to assist in developing our understanding of the fluid-structure interactions that cause altered regional aeration and forces in an injured lung. Simulations using this model suggest that flooded alveoli increase septal strain in their neighbors and that recruitment of these injured alveoli occurs over a longer duration as the size of a flooded region increases.

Figure 1. Max septal strain during inflation causes an avalanche of reopening in a flooded lung region.

[Spring 2015] [Fall 2014]