Applied Mathematics Commons^™

Decision Making Under Interval Uncertainty (And Beyond), Vladik Kreinovich

Departmental Technical Reports (CS)

To make a decision, we must find out the user's preference, and help the user select an alternative which is the best -- according to these preferences. Traditional utility-based decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is often unable to select one of these alternatives. In this chapter, we show how we can extend the utility-based decision theory to such realistic (interval) cases.

Should Voting Be Mandatory? Democratic Decision Making From The Economic Viewpoint, Olga Kosheleva, Vladik Kreinovich, Boakun Li

Departmental Technical Reports (CS)

Many decisions are made by voting. At first glance, the more people participate in the voting process, the more democratic -- and hence, better -- the decision. In this spirit, to encourage everyone's participation, several countries make voting mandatory. But does mandatory voting really make decisions better for the society? In this paper, we show that from the viewpoint of decision making theory, it is better to allow undecided voters not to participate in the voting process. We also show that the voting process would be even better -- for the society as a whole -- if we allow partial …

Ubiquity Of Data And Model Fusion: From Geophysics And Environmental Sciences To Estimating Individual Risk During An Epidemic, Omar Ochoa, Aline Jaimes, Christian Servin, Craig Tweedie, Aaron Velasco, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to combine the results of measuring a local value of a certain quantity with results of measuring average values of this same quantity. For example, in geosciences, we need to combine the seismic models (which describe density at different locations and depths) with gravity models which describe density averaged over certain regions. Similarly, in estimating the risk of an epidemic to an individual, we need to combine probabilities describe the risk to people of the corresponding age group, to people of the corresponding geographical region, etc. In this paper, we provide general techniques for …

How To Define Relative Approximation Error Of An Interval Estimate: A Proposal, Vladik Kreinovich

Departmental Technical Reports (CS)

The traditional definition of a relative approximation error of an estimate X as the ratio |X - x|/|x| does not work when the actual value x is 0. To avoid this problem, we propose a new definition |X - x|/|X|. We show how this definition can be naturally extended to the case when instead of a numerical estimate X, we have an interval estimate [x], i.e., an interval that is guaranteed to contain the actual (unknown) value x.

Derivation Of Hill's Equation From Scale Invariance, Andres Ortiz^, Vladik Kreinovich*

COURI Symposium Abstracts, Spring 2012

No abstract provided.

Generating Minimal Pair-Wise Covering Test Suites, Luis C. Gutierrez ^, Martine Ceberio *

COURI Symposium Abstracts, Spring 2012

Software is ubiquitous and needs to be reliable. Software testing therefore plays an important role in software development. Proper testing a software system informs about its quality and reliability so as to prevent unexpected behavior during system execution. One of the methods to prevent failures consists in testing a system under different input values, but when all possible input values are tested, an impractical number of test cases might result. In software testing, pair-wise testing is a combinatorial technique which uses combination of pair input values to generate test cases. Using pair-wise testing dramatically reduces the number of test cases, …

Development Of New Mathematical Methods For Post-Pareto Optimality, Victor Manuel Carrillo

Open Access Theses & Dissertations

Many real-world applications of multi-objective optimization involve a large number of objectives. A multi-objective optimization task involving multiple conflicting objectives ideally demands finding a multi-dimensional Pareto-optimal front. Although the classical methods have dealt with finding one preferred solution with the help of a decision-maker, evolutionary multi-objective optimization (EMO) methods have been attempted to find a representative set of solutions in the Pareto-optimal front. Multiple objective evolutionary algorithms (MOEAs), which are biologically-inspired optimization methods, have become popular approaches to solve problems with multiple objective functions. With the use of MOEAs, multiple objective optimization becomes a two-part problem. First, the multiple objective …

Granular Computing For Assessment Of Mild Traumatic Brain Injury, Melaku Ayenew Bogale

Open Access Theses & Dissertations

Mild traumatic brain injury (mTBI) is one of the most common neurological disorders. It is a serious public health problem in the United States. Although, penetrating (open) brain injuries that result in extended period of loss of consciousness (LOC) usually gets attention and well taken care of by the emergency departments, mild traumatic brain injury with no visible sign of damage, may be undetected or misdiagnosed. The clinical assessments and evaluations are mostly based on subjective cognitive and behavioral tests. Many people after suffering mTBI complain about decreased balance, coordination and stability even though the clinical evaluations show no sign …

Constrained Optimal Control For A Multi-Group Discrete Time Influenza Model, Paula Andrea Gonzalez Parra

Open Access Theses & Dissertations

During the last decades, mathematical epidemiological models have been used to understand the dynamics of infectious diseases and guide public health policy. In particular, several continuous models have been considered to study single in uenza outbreaks and the impact of dierent control policies. In this dissertation, a discrete time model is introduced in order to study optimal control strategies for in uenza transmission; since epidemiological data is collected on discrete units of time, a discrete formulation is more ecient. From a mathematical point of view, continuous time model are easier to analyze, however, the numerical solution of discrete-time models is …

Constrained Optimization Schemes For Geophysical Inversion Of Seismic Data, Uram Anibal Sosa Aguirre

Open Access Theses & Dissertations

Many experimental techniques in geophysics advance the understanding of Earth processes by estimating and interpreting Earth structure (e.g., velocity and/or density structure). These techniques use dierent types of geophysical data which can be collected and analyzed separately, sometimes resulting in inconsistent models of the Earth depending on data quality, methods and assumptions made. This dissertation presents two approaches for geophysical inversion of seismic data based on constrained optimization. In one approach we expand a one dimensional (1-D) joint inversion least-squares (LSQ) algorithm by introducing a constrained optimization methodology. Then we use the 1-D inversion results to produce 3-D Earth velocity …

Analytical And Numerical Solution To The Partial Differential Equation Arising In Financial Modeling, Pavel Bezdek

Open Access Theses & Dissertations

In this work we will present a self-contained introduction to the option pricing problem. We will introduce some basic ideas from the probability theory and stochastic differential equations. Later we will move to the partial differential equations since the option pricing problem arising in financial mathematics when asset is driven by a stochastic volatility process and assumed presence of transaction cost leads to solving non-linear partial dif- ferential equation. We will also present the complete process from deriving the desired partial differential equation to the proof of existence of a solution and also the numerical simulations. Using techniques form stochastic …

Quantum Mechanical Model Of The Hydrogen Atom, Andres Ortiz^, Snehanshu Saha*

COURI Symposium Abstracts, Spring 2011

The quantum mechanical wave function of an object under a certain potential is determined by the Schrödinger equation. It is known that the latter equation can be solved analytically only for a finite number of distinct potentials, being a Coulomb potential one of them. For an electron in a hydrogen atom, we analytically find the angular parts of its wave function by the usual method. Under the appropriate change of variable, the radial part of the equation is turned into a Legendre differential equation whose solutions are Legendre polynomials. Using finite difference method and shooting method taking into account physically …

Software Development For A Three-Dimensional Gravity Inversion And Application To Study Of The Border Ranges Fault System, South-Central Alaska, Rolando Cardenas

Open Access Theses & Dissertations

The Border Ranges Fault System (BRFS) bounds the Cook Inlet and Susitna Basins, an important petroleum province within south-central Alaska. A primary goal in the research is to test several plausible models of structure along the Border Ranges Fault System using a novel three-dimensional inversion utilizing gravity and magnetic data, constrained with other geophysical, borehole and surface geological information. This research involves the development of inversion modeling software using a Borland C++ compiler as part of the Rapid Application Development (RAD) Studio. The novel inversion approach directly models known geology, and "a priori" uncertainties on the geologic model to allow …

Solving The Partial Differential Equation Of Vibrations With Interval Parameters Using The Interval Finite Difference Method, Brenda G. Medina

Open Access Theses & Dissertations

Accuracy and efficiency are among the main factors that drive today's innovative disciplines. As technology rapidly advances, efficiency takes on new meanings but what about accuracy? How accurate is accurate? Human error, uncertainties in measurement, and rounding errors are just some causes of inaccuracy. Interval Computations is an area that allows for such issues to be taken into account; for each measurement attained (for example), an interval can be built by considering the error associated with the measurement, and such an interval can be utilized in the mathematical computations of interest.

We consider the partial differential equation (PDE) of vibrations …

A Sparse Representation Technique For Classification Problems, Reinaldo Sanchez Arias

Open Access Theses & Dissertations

In pattern recognition and machine learning, a classification problem refers to finding an algorithm for assigning a given input data into one of several categories. Many natural signals are sparse or compressible in the sense that they have short representations when expressed in a suitable basis. Motivated by the recent successful development of algorithms for sparse signal recovery, we apply the selective nature of sparse representation to perform classification. In order to find such sparse linear representation, we implement an l₁-minimization algorithm. This methodology overcomes the lack of robustness with respect to outliers. In contrast to other classification …

On Constrained Optimization Schemes For Joint Inversion Of Geophysical Datasets, Uram Anibal Sosa Aguirre

Open Access Theses & Dissertations

In the area of geological sciences, there exist several experimental techniques used to advance in the understanding of the Earth. We implement a joint inversion least-squares (LSQ) algorithm to characterize one dimensional Earth's structure by using seismic shear wave velocities as a model parameter. We use two geophysical datasets sensitive to shear velocities, namely Receiver Function and Surface Wave dispersion velocity observations, with a choice of an optimization method: Truncated Singular Value Decomposition (TSVD) or Primal-Dual Interior-Point (PDIP). The TSVD and the PDIP methods solve a regularized unconstrained and a constrained minimization problem, respectively. Both techniques include bounds into the …

Digital Image Processing Based On Sparse Representation And Convex Programming, Carlos Andres Ramirez

Open Access Theses & Dissertations

Sparse representation models have been of central interest in recent years due to important achievements in computational harmonic analysis, such as wavelet transformations, and the most recent sampling theory, compressed sensing. Numerous applications based on sparse models have been studied in the last decade leading to promising results. These applications include areas in seismology, image processing, wireless sensor networks, computed tomography and magnetic resonance imaging just to mention a few.

In this work, we propose to extend such applications in the area of image processing, particularly for the image segmentation problem, and examine algorithms involved in sparse modeling from both …

Multi-Objective Network Reliability Optimization Using Evolutionary Algorithms, Franciso Oswaldo Aguirre

Open Access Theses & Dissertations

This work presents a new multiple objective evolutionary algorithm to solve three well known network reliability allocation problems considering different conflicting objectives to be optimized simultaneously. The new algorithm is applied in the design of a telecommunication network that is formed for several stations or nodes interconnected by telecommunication links or paths. The problem presented in this work involves finding which links to activate in order to obtain connectivity in the nodes. The number of nodes that need to be connected depends of the case that is being evaluated. The three network reliability problems considered are: all-terminal, k-terminal, and two-terminal. …