Scholarly Works, Mathematics

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https://hdl.handle.net/10919/24285

Research articles, presentations, and other scholarship

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Increased risks of mosquito-borne disease emergence in temperate regions of South America
Estallo, Elizabet L.; Lopez, Maria Soledad; Luduena-Almeida, Francisco; Madelon, Magali I.; Layun, Federico; Robert, Michael A. (Elsevier, 2024-11-14)
Krylov Subspace Based FISTA‐Type Methods for Linear Discrete Ill‐Posed Problems
Buccini, Alessandro; Chen, Fei; Pasha, Mirjeta; Reichel, Lothar (Wiley, 2024-12-29)
Several iterative soft‐thresholding algorithms, such as FISTA, have been proposed in the literature for solving regularized linear discrete inverse problems that arise in various applications in science and engineering. These algorithms are easy to implement, but their rates of convergence may be slow. This paper describes novel approaches to reduce the computations required for each iteration by using Krylov subspace techniques. Specifically, we propose to impose sparsity on the coefficients in the representation of the computed solution in terms of a Krylov subspace basis. Several numerical examples from image deblurring and computerized tomography are used to illustrate the efficiency and accuracy of the proposed methods.
Advanced surrogate model for electron-scale turbulence in tokamak pedestals
Farcas, Ionut-Gabriel; Merlo, Gabriele; Jenko, Frank (Cambridge University Press, 2024-10-28)
We derive an advanced surrogate model for predicting turbulent transport at the edge of tokamaks driven by electron temperature gradient (ETG) modes. Our derivation is based on a recently developed sensitivity-driven sparse grid interpolation approach for uncertainty quantification and sensitivity analysis at scale, which informs the set of parameters that define the surrogate model as a scaling law. Our model reveals that ETG-driven electron heat flux is influenced by the safety factor q, electron beta βe and normalized electron Debye length λD, in addition to well-established parameters such as the electron temperature and density gradients. To assess the trustworthiness of our model’s predictions beyond training, we compute prediction intervals using bootstrapping. The surrogate model’s predictive power is tested across a wide range of parameter values, including within-distribution testing parameters (to verify our model) as well as out-of-bounds and out-of-distribution testing (to validate the proposed model). Overall, validation efforts show that our model competes well with, or can even outperform, existing scaling laws in predicting ETG-driven transport.
Scientific machine learning based reduced-order models for plasma turbulence simulations
Gahr, Constantin; Farcas, Ionut-Gabriel; Jenko, Frank (AIP Publishing, 2024-11-18)
This paper investigates non-intrusive Scientific Machine Learning (SciML) Reduced-Order Models (ROMs) for plasma turbulence simulations. In particular, we focus on Operator Inference (OpInf) to build low-cost physics-based ROMs from data for such simulations. As a representative example, we consider the (classical) Hasegawa-Wakatani (HW) equations used for modeling two-dimensional electrostatic drift-wave turbulence. For a comprehensive perspective of the potential of OpInf to construct predictive ROMs, we consider three setups for the HW equations by varying a key parameter, namely, the adiabaticity coefficient. These setups lead to the formation of complex and nonlinear dynamics, which makes the construction of predictive ROMs of any kind challenging. We generate the training datasets by performing direct numerical simulations of the HW equations and recording the computed state data and outputs over a time horizon of 100 time units in the turbulent phase. We then use these datasets to construct OpInf ROMs for predictions over 400 additional time units, that is, 400 % more than the training horizon. Our results show that the OpInf ROMs capture important statistical features of the turbulent dynamics and generalize beyond the training time horizon while reducing the computational effort of the high-fidelity simulation by up to five orders of magnitude. In the broader context of fusion research, this shows that non-intrusive SciML ROMs have the potential to drastically accelerate numerical studies, which can ultimately enable tasks such as the design of optimized fusion devices.
Global well-posedness and scattering results for nonlinear waves
Camliyurt, Guher; Kenig, Carlos E. (2024-10-23)
A fluid mechanical study of rotation-induced traumatic brain injury
Wang, Qifu; Zhang, Jiaqi; Bates, David; Feng, James J.; Yue, Pengtao; Wu, Qianhong (2025)
Traumatic brain injury (TBI) is a serious health issue. Studies have highlighted the severity of rotation induced TBI. However, the role of cerebrospinal fluid (CSF) in transmitting the impact from the skull to the soft brain matter remains unclear. Herein, we use experiments and computations to define and probe this role in a simplified setup. A spherical hydrogel ball, serving as a soft brain model, was subjected to controlled rotation within a water bath, emulating the CSF, filling a transparent cylinder. The cylinder and ball velocities, as well as the ball’s deformation over time, were measured. We found that the soft hydrogel ball is very sensitive to decelerating rotational impacts, experiencing significant deformation during the process. A finite-element code is written to simulate the process. The hydrogel ball is modelled as a poroelastic material infused with fluid and its coupling with the suspending fluid is handled by an arbitrary Lagrangian-Eulerian method. The results indicate that the density contrast, as well as the rotational velocity difference, between the hydrogel ball and the suspending fluid play a central role in the ball’s deformation due to centrifugal forces. This approach contributes a deeper understanding of brain injuries and may portend the development of preventive measures and improved treatment strategies.
On fusing active and passive acoustic sensing for simultaneous localization and mapping
Bradley, Aidan J.; Abaid, Nicole (2024)
Studies on the social behaviors of bats show that they have the ability to eavesdrop on the signals emitted by conspecifics in their vicinity. They can fuse this “passive” data with actively collected data from their own signals to get more information about their environment, allowing them to fly and hunt more efficiently and to avoid or cause jamming when competing for prey. Acoustic sensors are capable of similar feats but are generally used in only an active or passive capacity at one time. Is there a benefit to using both active and passive sensing simultaneously in the same array? In this work we define a family of models for active, passive, and fused sensing systems to measure range and bearing data from an environment defined by point-based landmarks. These measurements are used to solve the problem of simultaneous localization and mapping (SLAM) with extended Kalman filter (EKF) and FastSLAM 2.0 approaches. Our results show agreement with previous findings. Specifically, when active sensing is limited to a narrow angular range, fused sensing can perform just as accurately if not better, while also allowing the sensor to perceive more of the surrounding environment.
Combining Active and Passive Acoustic Sensing in Teams of Mobile Robots
Bradley, Aidan J.; Abaid, Nicole (2024-10-29)
Evidence suggests that bats are able to take advantage of both their own echolocation signals (active sensing) and the signals of conspecifics in their environment (passive sensing). This work follows a bioinspired approach to investigate whether we can enable robots to do the same. We have simulated a pair of vehicles that acoustically sense their environment both actively and passively. Our results show that, while the ability to fuse acoustic sensing techniques may not provide a significant improvement over active sensing alone, it is rarely worse and often allows for more information about the environment to be observed.
Reverse social contagion as a mechanism for regulating mass behaviors in highly integrated social systems
Porfiri, Maurizio; De Lellis, Pietro; Aung, Eighdi; Meneses, Santiago; Abaid, Nicole; Waters, Jane S.; Garnier, Simon (Oxford University Press, 2024-06-26)
Mass behavior is the rapid adoption of similar conduct by all group members, with potentially catastrophic outcomes such as mass panic. Yet, these negative consequences are rare in integrated social systems such as social insect colonies, thanks to mechanisms of social regulation. Here, we test the hypothesis that behavioral deactivation between active individuals is a powerful social regulator that reduces energetic spending in groups. Borrowing from scaling theories for human settlements and using behavioral data on harvester ants, we derive ties between the hypermetric scaling of the interaction network and the hypometric scaling of activity levels, both relative to the colony size. We use elements of economics theory and metabolic measurements collected with the behavioral data to link activity and metabolic scalings with group size. Our results support the idea that metabolic scaling across social systems is the product of different balances between their social regulation mechanisms.
Scalable Implicit Solvers with Dynamic Mesh Adaptation for a Relativistic Drift-Kinetic Fokker-Planck-Boltzmann Model
Rudi, Johann; Heldman, Max; Constantinescu, Emil M.; Tang, Qi; Tang, Xian-Zhu (2023-03-10)
In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker–Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. We develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.
Constraining Earth's nonlinear mantle viscosity using plate-boundary resolving global inversions
Hu, Jiashun; Rudi, Johann; Gurnis, Michael; Stadler, Georg (National Academy of Sciences, 2024-07-05)
Variable viscosity in Earth’s mantle exerts a fundamental control on mantle convection and plate tectonics, yet rigorously constraining the underlying parameters has remained a challenge. Inverse methods have not been sufficiently robust to handle the severe viscosity gradients and nonlinearities (arising from dislocation creep and plastic failure) while simultaneously resolving the megathrust and bending slabs globally. Using global plate motions as constraints, we overcome these challenges by combining a scalable nonlinear Stokes solver that resolves the key tectonic features with an adjoint-based Bayesian approach. Assuming plate cooling, variations in the thickness of continental lithosphere, slabs, and broad scale lower mantle structure as well as a constant grain size through the bulk of the upper mantle, a good fit to global plate motions is found with a nonlinear upper mantle stress exponent of 2.43 ± 0.25 (mean ± SD). A relatively low yield stress of 151 ± 19 MPa is required for slabs to bend during subduction and transmit a slab pull that generates asymmetrical subduction. The recovered long-term strength of megathrusts (plate interfaces) varies between different subduction zones, with South America having a larger strength and Vanuatu and Central America having lower values with important implications for the stresses driving megathrust earthquakes.
Modifying the Asynchronous Jacobi Method for Data Corruption Resilience
Vogl, Christopher J.; Atkins, Zachary R.; Fox, Alyson; Miedlar, Agnieszka; Ponce, Colin (Society for Industrial and Applied Mathematics, 2024-10-09)
Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.
From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
Aluffi, Paolo; Mihalcea, Leonardo C.; Schürmann, Jörg; Su, Changjian (American Mathematical Society, 2024-01-01)
The equivariant motivic Chern class of a Schubert cell in a complete flag manifold X = G/B is an element in the equivariant K-theory ring of X to which one adjoins a formal parameter y. In this paper we prove several folklore results about motivic Chern classes, including finding specializations at y = −1 and y = 0; the coefficient of the top power of y; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures on the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincar´e duals of CSM classes. In addition, we prove a ‘star duality’ for the motivic Chern classes, showing how they behave under the involution taking a vector bundle to its dual. We use the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar´e duals, and their Segre versions. We explain the functoriality properties needed to extend the results to partial flag manifolds G/P.
Scalable computation of energy functions for nonlinear balanced truncation
Kramer, Boris; Gugercin, Serkan; Borggaard, Jeff; Balicki, Linus (2024-07-01)
Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. A computational challenges that has so far prevented its deployment on large-scale systems is that the energy functions required for characterization of controllability and observability are solutions of various high-dimensional Hamilton-Jacobi-(Bellman) equations, which are computationally intractable in high dimensions. This work proposes a unifying and scalable approach to this challenge by considering a Taylor-series-based approximation to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of nonlinear balancing. The value of a formulation parameter provides either openloop balancing or a variety of closed-loop balancing options. To solve for the coefficients of Taylor-series approximations to the energy functions, the presented method derives a linear tensor system and heavily utilizes it to numerically solve structured linear systems with billions of unknowns. The strength and scalability of the algorithm is demonstrated on two semi-discretized partial differential equations, namely the Burgers and the Kuramoto-Sivashinsky equations.
Positivity of Peterson Schubert calculus
Goldin, Rebecca; Mihalcea, Leonardo; Singh, Rahul (Elsevier, 2024-10)
The Peterson variety is a subvariety of the flag manifold G/B equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersection multiplicities yields a Z-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find formulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt.
Motivic Chern Classes of Schubert Cells, Hecke Algebras, and Applications to Casselman's Problem
Aluffi, Paolo; Mihalcea, Leonardo C.; Schuermann, Joerg; Su, Changjian (Société Mathematique de France, 2024-04-02)
Motivic Chern classes are elements in the K-theory of an algebraic variety X, depending on an extra parameter y. They are determined by functoriality and a normalization property for smooth X. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds G=B. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of G, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Fehér, Rimányi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman’s basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman’s basis.
Parametric level-sets enhanced to improve reconstruction (PaLEnTIR)
Ozsar, Ege; Kilmer, Misha E.; de Sturler, Eric; Saibaba, Arvind; Miller, Eric L. (IOP, 2025-01-08)
We introduce PaLEnTIR, a significantly enhanced parametric level-set (PaLS) method addressing the restoration and reconstruction of piecewise constant objects. Our key contribution involves a unique PaLS formulation utilizing a single level-set function to restore scenes containing multi-contrast piecewise constant objects without requiring knowledge of the number of objects or their contrasts. Unlike standard PaLS methods employing radial basis functions (RBFs), our model integrates anisotropic basis functions (ABFs), thereby expanding its capacity to represent a wider class of shapes. Furthermore, PaLEnTIR streamlines the model by reducing redundancy and indeterminacy in the parameterization, resulting in improved numerical performance. We compare PaLEnTIR’s performance to state-ofthe art alternatives via a diverse collection of experiments encompassing denoising, deconvolution, sparse and limited angle of view X-ray computed tomography (2D and 3D), and nonlinear diffuse optical tomography (DOT) tasks using both real and
Sum of Squares Approximations to Energy Functions
Adjerid, Hamza; Borggaard, Jeffrey T. (IEEE, 2024-01-01)
Energy functions offer natural extensions of controllability and observability Gramians to nonlinear systems, enabling various applications such as computing reachable sets, optimizing actuator and sensor placement, performing balanced truncation, and designing feedback controllers. However, these extensions to nonlinear systems depend on solving Hamilton-Jacobi-Bellman (HJB) partial differential equations, which are infeasible for large-scale systems. Polynomial approximations are a viable alternative for modest-sized systems, but conventional polynomial approximations may yield negative values of the energy away from the origin. To address this issue, we explore polynomial approximations expressed as a sum of squares to ensure non-negative approximations. In this study, we focus on a reduced sum of squares polynomial where the coefficients are found through least-squares collocation-minimizing the HJB residual at sample points within a desired neighborhood of the origin. We validate the accuracy of these approximations through a case study with a closed-form solution and assess their effectiveness for controlling a ring of van der Pol oscillators with a Laplacian-like coupling term and discretized Burgers equation with source terms.
Algebraic hierarchical locally recoverable codes with nested affine subspace recovery
Haymaker, Kathryn; Malmskog, Beth; Matthews, Gretchen L. (Springer, 2024-10-24)
Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves.We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.
Pedagogical moves related to analogy that support a unified understanding of eigentheory concepts in a quantum mechanics class
Serbin, Kaitlyn Stephens; Wawro, Megan (American Physical Society, 2024-10-30)
It is beneficial for quantum mechanics students to have a unified understanding of eigentheory concepts, so they can recognize the shared structure of mathematized phenomena from the different quantum mechanical systems of spin, energy, or position and recognize those as instantiations of the same overarching concept. Quantum mechanics instructors should, therefore, provide opportunities for their class community to develop a shared unified understanding of eigentheory concepts. One such opportunity can arise by engaging students in analogizing eigentheory concepts in one context with those from another context. We investigate the pedagogical moves related to analogies that can be used by a quantum mechanics course instructor to support a class community in developing a shared unified understanding of eigenequations. We analyze classroom data to characterize an instructor's pedagogical moves as he engaged students in analogical reasoning. Some moves include posing tasks conducive to analogizing; preparing, soliciting, and scaffolding students' participation in analogical reasoning; using deictic gestures and inscriptions; juxtaposing symbols representing the analogized concepts; and explicitly highlighting the sameness of the analogized concepts. We exemplify these pedagogical moves with analytical descriptions of illustrative class episodes. We discuss how these pedagogical moves can support the class community's expansion of their common ground by fostering the development of the class's shared unified understanding of eigentheory concepts.

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