# Our Activities

## Upcoming Activities

### Masoud Shirazi, Engeneering Mechanics, VT

"Simultaneous Localization and Mapping (SLAM) Using Extended Kalman Filter (EKF)"

The problem of localizing of a moving robot in an unknown environment and simultaneously creating a map of that environment is attracted interest of researchers in robotics. This problem is challenging in the case that the robot does not have access to GPS data and only relies on its own sensor and dead reckoning approach to perceive the environment and localize itself. Due to imperfection of the dynamic model of the robot's movement and noisy measurement, the localization and mapping error will below up. In a localization problem, a priori map of the environment is used to correct the location of the robot. In a mapping problem, the trajectory of the robot is known and it is used to correct the map. In simultaneous localization and mapping, however, there is no a priori information about robot's trajectory or the environment. In this talk, we approach this problem using the classic Extended Kalman Filter(EKF).

### Masoud Shirazi, Engeneering Mechanics, VT

"Comparing the Effects of Intrinsic and Extrinsic Noise on the Vicsek Model in Three dimensions"

A group of simple individuals may show ordered, complex behavior through local interactions. This phenomenon is called collective behavior, which has been observed in a vast variety of natural systems such as fish schools or bird flocks. The Vicsek model is a well-established mathematical model to study collective behavior through interaction of individuals with their neighbors in the presence of noise. How noise is modeled can impact the collective behavior of the group. Extrinsic noise captures uncertainty imposed on individuals, such as noise in measurements, while intrinsic noise models uncertainty inherent to individuals, akin to free will. In this paper, the effects of intrinsic and extrinsic noise on characteristics of the transition between order and disorder in the Vicsek model in three dimensions are studied through numerical simulation.

## Past Activities

### Ben Beach (Math Department)

"HAPOD: Hierarchical Approximate POD"

Proper Orthogonal Decomposition (POD), combined with the Method of Snapshots, is a popular method for reduced order modeling of nonlinear PDEs. However, the computation of the POD bases can be very expensive computationally (roughly order O(m^2n)), and the issues are compounded when the snapshot matrix is too large to fit in memory. HAPOD (Himpe 2016) is a method for the approximation of POD that can save on memory, is very general and easily parallelizable, and can be computationally cheaper than full POD. However, depending on the decay of singular values and the selected input parameters, HAPOD may provide no benefits over full POD, and can actually be more expensive in some cases. This discussion will introduce HAPOD and start to explore the boundary where HAPOD begins to impart computational benefits.

### Jiaqi Zhang (Math Department)

"A High-Order and Interface-Preserving Discontinuous Galerkin Method for Level-Set Reinitialization"

We perform the reinitialization by solving the Hamilton-Jacobi (HJ) equation as a system of conservation laws by the discontinuous Galerkin method except in interface cells (cells crossed by the initial zero level set). An accurate and interface-preserving local projection is developed to obtain the gradients on interface cells, where the missing constant is determined according to the location of the initial zero level set, while in non-interface cells the constant is updated by the continuity of the solution in an order analogous to the Fast Marching method. A new limiter, which is performed on the second derivatives, is constructed to stabilize the solution. Numerical results demonstrate that our method is accurate and stable for nontrivial test cases and also preserves the initial zero level set very well. All the simulations are performed using the deal.II C++ open source library.

### Sarah Kadelka (Math Department)

"Bistable dynamics between primed and tolerant states following challenge with exdotoxin"

Biological experiments have shown different molecular dynamics after challenge and boosting with different doses of endotoxin, with high-high challenge leading to tolerance and low-high challenge leading to priming. To provide insight into the relationship between dose and dynamics, we developed a mathematical model of molecular interactions within a pathway. We analyzed the model using asymptotic stability and bifurcation techniques. Our model exhibits bistable dynamics between a tolerant and a primed state. We used the model to determine the feedback mechanisms needed for bistability and determined the relationship between our results and the experimental data.

### Taewon Cho (Math Department)

"Numerical methods for separable nonlinear inverse problems with constraint and low rank"

In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example image reconstruction or debarring requires to use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest we will add more variables such that we must solve nonlinear problems. It is quiet different than a linear inverse problem, and we need to use different methods to solve these kinds of problems.

### Selin Sariaydin (Math Department)

"Computing Reduced Order Models using Randomization"

Diffuse Optical Tomography (DOT) in medical image reconstruction presents huge computational challenges since we need to solve many large-scale discretized PDEs for each evaluation of the misfit or objective function. Moreover, in the nonlinear case each time the Jacobian is evaluated an additional set of systems must be solved. One can use reduced order models to reduce this cost. However, computing the reduced order model still requires solving many systems. We aim to reduce this cost by randomization.

### Masoud Shirazi (BEaM)

"Relationship of Environmental Structure to Echolocation Pulse Quality"

Acoustic sampling of bats requires recording adequate numbers of high-quality calls suitable for species identification. Recorded bat call quality and abundance can be influenced by factors such as microphone orientation and height, detector deployment, and environmental conditions (humidity, temperature, environmental structure, i.e. density of tree stems and foliage). However, no studies have assessed the impact of these factors on recorded call quality under controlled experimental conditions. Accordingly, we assessed the relationship of bat call quality to two environmental structural characteristics, basal area and clutter. In an anechoic chamber, we conducted an acoustic playback experiment wherein we recorded synthetic bat calls generated with a custom sonar emitter and passed through a gradient of basal area and clutter conditions at five azimuthal angles. This allowed us to compare known call quality to subsequently recorded call quality. We analyzed raw spectrograms as well as zero-crossing calls. For spectrograms, we measured root means square error with respect to the control condition and multi-dimensional spread based on PCA. For zero-crossing calls, we measured a suite of commonly used call parameters with Kaleidoscope software. We assessed the relationship between measured variables and structural conditions with a series of regression models. We found trends with reduction in call quality due to increasing amounts of basal area and clutter, but these were confounded by interactions of the two variables. Microphone angle also contributed to variation in call quality. Moreover, quality reduction differed among full spectrum and zero-crossing calls which could have implications for field sampling.

### Andrea Carracedo Rodriguez (Math Department)

"Interpolatory Model Reduction of Parameterized Bilinear Dynamical Systems"

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been already successfully extended to nonparametric bilinear dynamical systems. However, this is not the case for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce the conditions that projection subspaces need to satisfy in order to obtain parametric tangential interpolation of each subsystem transfer functions. These conditions also guarantee that the parameter gradient of each subsystem transfer function is matched tangentially by the parameter gradient of the corresponding reduced order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including extra vectors in the projection subspaces. As in the linear case, for two-sided projections, the basis construction does not require computing neither the gradient nor the Hessian to be matched.

### Kasia Świrydowicz (Math Department)

"Topology Optimization"

Structural designs in topology optimization often involve many load cases. We state the problem of finding optimal design as a minimization problem in which we minimize compliance (weighted average of the compliance under each load case). Finding optimal design requires many minimization steps, and in each step we often solve hundreds of linear systems, one per every load case. In this talk, we use three different strategies to speed up the computation of the optimal structural design. We use randomization to reduce the number of load cases, we apply recycled Conjugate Gradient, and we estimate the quadratic form arising in the compliance equation using properties of Krylov subspace. We demonstrate effectiveness of this method using examples.

### Angelo Marney (Math Department)

"Semi-analytical finite element method for elastic wave propagation"

Long range ultrasonic inspection is a commonly used technique for detecting damage in long structures. To aid in in choosing the correct frequency to excite the desired mode of propagation, a dispersion curve for a structure can be computed. In this presentation, I will introduce the semi-analytical finite element method and how it can be used to determine dispersion curves of structures. This finite element method is "semi-analytical" in sense that the displacement solution is assumed to have a particular form in the wave propagation direction, so numerically approximations of the displacement only needs to be done on a cross section of the structure. I will talk about the advantages and disadvantage of this method. I will also talk about current applications as well as research done in recent years.

### Sarah Kerrigan (Math Department)

"Knowledge about student understanding of Eigentheory: Information gained from multiple choice extended assessment"

Through the use of a new assessment tool, Multiple Choice Extended, we investigated student's understanding of eigentheory. The purpose of the study was two fold consisting of both analysis of student knowledge of eigentheory and a review of the assessment that was developed. This talk addresses results on student understanding from three of the six questions from the multiple choice extended assessment and some constraints and affordances of the different assessment forms.

### Michael Brennan (Math Department)

"A Parametric Model Order Reduction Method Using the Loewner Framework and the Iterative Rational Krylov Algorithm"

Linear dynamical systems play a large role in predicting the behavior of physical phenomena, and computationally efficient methods of forming solutions to these systems are of great interest. These systems often depend on a set of physical parameters, and it can be the case that solutions are required for many different sets of parameter values. Systems of this nature can also be of large dimension, and thus computationally burdensome to work with directly. In these cases, we look to form an approximate system of reduced dimension. Currently, methods for forming optimal reduced models of non-parametric systems are known, but it remains mysterious what the best method is in parametric cases. In this talk, we develop a reduction method for parametric systems that integrates the Iterative Rational Krylov Algorithm (IRKA) with the Loewner Framework that has comparable accuracy to reducing the system using optimal means after evaluating the parameters. The most significant development of this work is a modified Loewner framework that matches the end behavior of the original system in the frequency domain.

### Alan Lattimer (Jensen Hughes)

"Data and Analytics on a Global Scale"

The typical method for reducing and solving a time dependent partial differential equation (PDE) begins with discretizing the spatial domain and then using a method, e.g. finite elements, finite differences, or finite volumes, to create an N-dimensional system of ordinary differential equations (FOM ODE). This large FOM ODE is is often reduced input-independent model-reduction techniques, such as IRKA, to create a reduced-order ODE model (ROM ODE). This ROM ODE is then discretized in time to create the ROMr O∆E. Unfortunately, there is no guarantee that the H2 optimality conditions obtained from IRKA are preserved in the time-discretized model. In this talk, we will explore ways that this optimality can be preserved when discretizing the system in time.

### Tanya Coutray (Walmart)

"Data and Analytics on a Global Scale"

Tanya Coutray is a Senior Director of Data and Analytics in Walmart's Global Ethics and Compliance organization. She is an alumna of Virginia Tech, where she studied theoretical and applied mathematics. Tanya will give an overview of the types of analytics that are done at Walmart, the complexities of scope and scale, and the fascinating dynamics of a truly international presence. She will share her experience in technology, data and analytics, and how her degrees in mathematics from Virginia Tech positioned her in this career. Tanya's talk will be held Wednesday, October 19, at 3:30pm in War Memorial 124 and lunch will be held Thursday, October 20, at 12:00pm in McBryde 455.

### Mariette Wessels (Math Department)

"Google Internship(s): a mathematics student's perspective"

In this talk I will provide a brief overview of two software engineering internships at Google over the past two summers. The focus will be on how a background in mathematics can provide unique insights into software engineering (beyond the obvious applications where one uses mathematics directly), and to highlight some of the opportunities to use mathematics to solve the problems we face in software engineering. In addition, I will briefly discuss company culture, what makes working at Google different, and what some of the different roles of software engineers entail.

### Gregory Marx (Math Department)

"Free noncommutative function theory"

In this talk, we introduce free noncommutative functions ( e.g., polynomials with freely noncommuting variables ) and the relatively new notion of completely positive noncommutative kernels. We present a decomposition theorem for such kernels and relate this result to others in the literature.

### Ruchi Guo(Math Department)

"A Class of Immersed Finite Element Spaces Defined by the Actual Interface"

In this talk, we present a new class of immersed finite element (IFE) spaces defined by the actual interface for solving the second order elliptic interface problems on Cartesian meshes. Functions in these IFE spaces are locally piecewise polynomials and constructed with either linear polynomials, or bilinear polynomials, or rotate-Q1 polynomials, or Crouzeix-Raviart polynomials. We will discuss the unisolvence, boundedness and optimal approximation capability of these IFE spaces by a new unified framework.

### Robert Torrence (Math Department)

"An Investigation of Pharmaceutical Cost and Usage trends of Members of the Medicaid Expansion Population"

At the passing of the Affordable Care Act (ACA), states were required to expand Medicaid to individuals and families within 138% of the Federal Poverty Limit (FPL). However, due to a Supreme Court ruling, states may now opt whether or not to expand Medicaid in their state. Actuaries working in Medicaid at Anthem are interested the healthcare costs and behaviors of their members who are newly covered by Medicaid in states that have accepted the expansion. They performed a study comparing the costs, usage, and distribution of services utilized by members categorized as Expansion and Non-Expansion in 5 states that have accepted the expansion. Results discussed include the expansion populations' use of emergency services, and particularly the investigation of their rapid increase in Pharmaceutical costs over time.

### Klajdi Sinani (Math Department)

"Model Reduction for Systems with Nonlinear Frequency Dependence via a Structure-Preserving Algorithm"

Very large-scale dynamical systems, even linear time-invariant systems, can present significant computational difficulties when used in numerical simulation. Model reduction is one response to this challenge but standard methods often are restricted to systems that are presented as standard first-order realizations; in the frequency domain such systems will be linear in the frequency parameter. We consider here dynamical systems with a nonlinear frequency dependence; systems for which either a standard first-order realization is unknown or inconvenient to obtain. Such systems may nonetheless have realizations that reflect important structural features of the model and we may wish to retain this structure in any reduced model used as a surrogate. In this work, we present a structure-preserving model reduction algorithm for systems having quite general nonlinear frequency dependence. We take advantage of recent algorithms that produce high quality rational interpolants to transfer functions that only require transfer function evaluation, thus allowing for nonstandard realizations that are nonlinear in the frequency parameter. However, our final reduced model will have a structure that reflects the structure of the original system, and indeed, may not have a rational transfer function. We illustrate our approach on a benchmark problem that offers a transcendental transfer function.

### Stephanie Gamble (Math Department)

"Vibrational Energies of the Hydrogen Bonds of $H_3O_2^-$ and $H_5O_2^+$"

We approximate the vibrational energies of the symmetric and asymmetric stretches of the hydrogen bonds of the molecules $H_3O_2^-$ and $H_5O_2^+$ by applying an improvement to the standard time-independent Born-Oppenheimer approximation. These two molecules are symmetric around a central hydrogen which participates in hydrogen bonding. Unlike the standard Born-Oppenheimer approximation, this approximation appropriately scales the hydrogen nuclei differently than the heavier oxygen nuclei. This results in significantly more accurate approximations for the stretching vibrational energies, which we compare to experimental measurements.

### Masoud Shirazi(Biomedical Engineering and Mechanics Department)

"Obstacle avoidance and tracking using passive sonar inspired by eavesdropping among bats"

Proximity sensors like LIDAR and SONAR are widely used in multi-agent robotic systems. These sensors are called active" since they actively send a signal and receive information by analyzing the echo from the environment. Such sensing methods are analogous to echolocation used by bats and whales, which relies on self-generated sounds. A recent behavioral study of bats showed that, when two bats were flying close to each other competing for food, one of them flew without echolocating presumably to avoid jamming. This observation suggests that, when flying in silence, the bat eavesdrops on its conspecific's echolocation signals to perceive the environment. This free information available by passive listening is usually neglected in engineering applications for the sake of simplicity. However, strategies such as allowing agents to actively sense only intermittently can limit the complexity of this type of sensing by limiting acoustic clutter, and can increase system efficiency by conserving the total sensing energy budget. This paper studies a tracking problem with passive sensing using numerical simulation, where a follower robot tracks a leader while avoiding obstacles. We find that the robot is successfully able to avoid obstacles in the environment, including the leader, using only passive sensing.

### Jonathan Baker (Dept. of Mathematics)

“Actually, Nonnormal Coefficients Make Lyapunov Equations Easier to Solve”

The singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low-rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. I will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically, as the numerical range of the coefficient matrix extends farther into the right half-plane. I will also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role.

## Upcoming Activities

Please, check back later for upcoming activities.

## Past Activities

### Kihyo Moon(Dept. of Mathematics)

"An immersed discontinuous Galerkin method for acoustic wave propagation in inhomogeneous media"

We present an immersed discontinuous Galerkin finite element method on Cartesian meshes for two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed method uses the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid it uses a specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. We present several computational results that suggest that the proposed method has optimal convergence rates. Several computational examples are included with linear and curved interfaces.

### Dan Sweeney (Biomedical Engineering)

“Clinical Insights from Single Cell Electroporation”

The permeability of biological membranes may be increased through the generation of nanoscale defects using intense electric fields in a process called electroporation. Many current cancer treatments, including electrochemotherapy, gene electrotransfer, and irreversible electroporation, depend on the electroporation process to effecively treat patients. Therefore, it is imperative to understand how the electric field distribution and variation in time affects cells and their surrounding tissue components. We have shown that higher-frequency electrical pulses may electroporate cells in a manner similar to conventional pulsing schemes, but at the expense of a higher input energy. Our data provides insight into the mechanisms underlying clinical and benchtop observations of the differences between conventional and high-frequency bipolar pulsing schemes used in electroporation-based treatments and therapies.

### Drayton Munster (Dept. of Mathematics)

Optimization via Parametric Model Reduction with Stochastic Error Estimates

While parametric model reduction offers significant computational savings for evaluation heavy applications, such as parameter inversion and optimization, its use is tempered by concerns for the approximation error. For reduced order models with many parameters, error bounds are typically not available or only at a very high cost. In this work, we will explore inexpensive stochastic estimates for the approximation error and how these estimates can be used to guide the optimization process and update the reduced order model.

### Arielle Grim-McNally (Dept. of Mathematics)

Reusing and Recycling Preconditioners for Sequences of Linear Systems Combined with Krylov Subspace Recycling

Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. While solving a sequence of many linear systems, it is advantageous to recycle preconditioners, that is, update a previous preconditioner and reuse the updated version. In this talk, we introduce a simple and effective method for doing this. Although our approach can be used for matrices changing slowly in any way, we focus on the important case of sequences of the type $(s_k\textbf{E}(\textbf{p}) + \textbf{A}(\textbf{p}))\textbf{x}_k = \textbf{b}_k$. We update preconditioners by defining a map from a new matrix to a previous matrix, for example the first matrix in the sequence, and combine the preconditioner for this previous matrix with the map to define the new preconditioner. This approach has several advantages. The update is entirely independent from the original preconditioner, so it can be applied to any preconditioner. The possibly high cost of an initial preconditioner can be amortized over many linear solves. The cost of updating the preconditioner is more or less constant and independent of the original preconditioner. There is flexibility in balancing the quality of the map with the computational cost. Theoretical and numerical results will be discussed, as well as future work. Specifically, we will discuss combining recycling preconditioners with recycling Krylov subspaces from previous systems.

### Kasia Swirydowicz (Dept. of Mathematics)

Improving preconditioners for GPUs by fixed-point iteration

Krylov subspace solvers often need a preconditioner to achieve good convergence. Popular preconditioners based on incomplete factorizations, such ILU, due to the sequential nature of forward and backward substitution, might be very time-consuming on GPUs that rely on a high degree of fine-grain parallelism for high performance. Weaker preconditioners, such as the Sparse Approximate Inverse (SAI) and Jacobi preconditioner, are better suited for fine-grain parallelism, but they are not as effective in reducing the iteration count as ILU preconditioners. To make the preconditioner more effective, we replace the preconditioner by a of couple steps of a fixed point iteration derived from that preconditioner. This improves convergence, and a careful implementation also improves the computation over communication ratio. We derive such a method for SAI. As a part of the method, we need to compute matrix-vector products with powers of R (for example, $R^3 x$), where $R$ is the matrix residual of the SAI, and $x$ is vector. The issues of GPU device-side synchronization, and memory management emerging in the implementation of, for example, 'triple matvec' are presented during this talk.

### Tiger Wang (Dept. of Mathematics)

Korteweg stresses and shear bands

The PEC (partially extending strand convection) model of Larson is able to describe thixotropic yield stress behavior in the limit where the relaxation time is large. We discuss the development of shear bands in a Poiseuille flow which is started up from rest by an imposed pressure gradient. The shear band transition region is smoothed after introducing the Korteweg stresses. The eventual location is decided by equal area rule.

### Dr. Nitsan Ben-Gal (3M)

Mathematics at a 'Materials Company': How 3M Creates Advanced Technologies

3M is a global innovation company developing products and solutions for the Industrial, Health Care, Safety, Graphics, Electronics, Energy, and Consumer sectors. While famous for its materials and processing capabilities, 3M’s Software, Electronic & Mechanical Systems Laboratory (SEMS) works with all areas of the company to develop innovative products and solutions utilizing Mathematical, Computer Science, and Engineering expertise. In this talk I will give an overview of some technologies in development or recently deployed within the Advanced Technologies Group that utilize its mathematical and computational expertise, and how these interplay with 3M’s established businesses. I will also discuss my own experience as a mathematician both on the job market and at 3M.

### Dr. Eduardo Moreno (Institute of Cybernetics, Mathematics and Physics in Havana, Cuba)

Dr. Eduardo Moreno from the Institute of Cybernetics, Mathematics and Physics (ICIMAF) Havana, Cuba will present information about the Institute of Cybernetics, Mathematics and Physics and describe projects that this group is working on. This will include the guided waves method and dispersion curves, challenges and applications, as well as ultrasound doppler for coronary artery bypass graft evaluation.

### Dr. Clas Jacobson (United Technologies)

Systems Engineering: Imperatives, Definitions, Technology & Talent

This talk will present the “why, what and how” of systems engineering from an industrial perspective, from the view of product development for infrastructure systems and that are often safety or operations critical. These systems stay in operation for a long time and are part of “product families” aimed at different market segments. This context gives the “why” and “why now” for a change in the content, and the need to re-think what systems engineering is and how it is deployed in industry. The “what” and “how” relate to four thrusts that define systems engineering: (1) requirements, (2) architecture, (3) model based development, and (4) design flows. Within each of these four elements is a context of what output is needed to address product development needs (the link to the “why”) and what technology is available to make these areas actionable today (the “how”). Gaps and technology thrusts to fill these gaps will be presented. Some views of industrial readiness will be given, along with views of the role of mathematics and the type of work done in industry. Dr. Jacobson is a Senior Fellow at the United Technologies Research Center, where he works with UTC business units to ensure systems engineering and controls capabilities are available for product development. Prior to this role he was Chief Scientist for United Technologies Systems & Controls Engineering and, before that, at the United Technologies Research Center (UTRC) in management and technical positions starting in 1995. He has held positions at UTRC as Director of the Carrier Program Office and Director of the Systems Department responsible for systems engineering capabilities. Dr. Jacobson received his Ph.D degree in electrical engineering in 1986 from Rensselaer Polytechnic Institute. He was an Associate Professor at Northeastern University in Boston from 1986–1995.

### Dr. Jesse Chan (Dept. of Mathematics)

GPU-accelerated Bernstein-Bezier discontinuous Galerkin methods

The computationally intensive nature of time-explicit nodal discontinuous Galerkin methods is well-suited to implementation on Graphics Processing Units (GPUs). We evaluate the use of Bernstein-Bezier bases as an alternative to nodal polynomials for discontinuous Galerkin discretizations and show how to exploit properties of derivative and lift operators specific to Bernstein polynomials. Issues of efficiency and numerical stability are discussed in the context of a model wave propagation problem, and computational experiments comparing high-order nodal bases and high-order Bernstein bases are presented.

### Tanner Slagel (Dept. of Mathematics)

Combined Subsampling for large scale least squares problems

Solving large scale least squares problems can be a daunting task, especially when computational resources are limited and the size of the data is so large traditional methods are no longer feasible. We present a novel framework called combined subsampling for solving large scale least squares problems in a way that overcomes these computational limitations. Combined subsampling is an approach that utilizes stochastic programming to repeatedly sample data from the original least squares problem in order to make inferences about the solution. We provide both theoretical results and numerical investigations that show that our approach is preferred to traditional methods when solving very large least squares problems.