In this talk, we will discuss the capabilities of spectral method in solving one dimensional conservation laws. Firstly, I will talk about our simple bound-preserving sweeping procedure for conservative numerical approximation, which is a very general framework acting as a postprocessing step accommodating point-based or cell average-based discretization. In the second part of the talk, I will talk our ongoing construction of total variation bounded (TVB) spectral methods through the flux limiter developed in `Sulin Wang and Zhengfu Xu. Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws. Mathematics of Computation, 2018’.

The LIGO-Virgo-KAGRA (LVK) collaboration has published about 90 gravitational wave observations since the first detection in 2015, the majority of which have been from binary black holes. With such a statistically significant number of events, it is possible to infer astrophysical distribution of black hole masses and spins. Of particular interest are spins(mis-aligned with the orbital angular momentum) of the component black holes in a binary, as they can shed light on the binary formation mechanism. Spin-misalignments of the component black holes leave their imprints on the waveform. One such modulation is due to an asymmetry between the +m and -m multipoles of the harmonic decomposition of the gravitational wave signal from the binary. This is not included in most signal models used in LVK analysis to date. I will present the first frequency-domain model of the asymmetry in the dominant co-precessing-frame signal multipole throughout inspiral, merger and ringdown. This model has been incorporated into the latest iteration of PHENOM family of precessing-binary models (PhenomXO4a) that will be used for analysing data from the fourth observing run (O4) of the LVK. I will briefly describe the features captured in PhenomXO4a and the prospect of spin measurements using this model.

In the common practice of the method-of-lines approach for discretizing a time-dependent partial differential equation (PDE), people first apply spatial discretization to convert the PDE into an ordinary differential equation system. Subsequently, a time integrator is used to discretize the time variable. When a multi-stage Runge-Kutta (RK) method is used for time integration, by default, the same spatial operator is used at all RK stages. But what if one allows different spatial operators at different stages? In this talk, we present two of our recent explorations on blending different stage operators in RK discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. In our first work, we mix the DG operator with the local derivative operator, yielding an RKDG method featuring compact stencils and simple boundary treatment. In our second work, we mix the DG operators with polynomials of degrees k and k-1, and the resulting method may allow larger time step sizes and fewer floating-point operations per time step.

Standard stellar-mass binary black holes inspiral and merge in just a few seconds when observed in ground-based gravitational-wave detector data. However, future detectors will probe much lower frequencies, implying that sources of gravitational waves will spend even longer time in-band than those observed today. Such long-lived signals could result from the early inspiral of binary neutron stars, sub-solar mass primordial black holes, and (mini) extreme mass ratio inspirals, all of which could be visible in both future ground-and space-based detectors, such as Cosmic Explorer and LISA. However, data quality problems, such as gaps, glitches and non-stationary noise, and computational cost, will inhibit the observations of these systems if robust methods are not designed to handle these issues. In this talk, I will describe each of these sources individually, the problems with computation and data quality that we are likely to face in future detectors, and methods to actually perform robust searches for these systems.

In this talk I will discuss various approaches to modeling. I will start with introducing the Energetic variational approach, discuss modeling and numerical simulations of multi-component fluid flow, sintering process and flow through poroelastic medium. Then I will discuss modeling carcinogenesis and immune response in cancer patients using chemical reaction networks and mass-action law. Lastly I will discuss the opinion dynamics model. The models presented were made in collaborations with Chun Liu, James Brannick, Qingcheng Yang, James Adler, Xiaozhe Hu, Yiwei Wang, Leili Shahriyari, Bruce Boghosian, Christoph Borgers, Natasa Dragovic, Anna Haensch.

Boundary Value Problems (BVPs) are ubiquitous in engineering and scientific applications. One of the most robust and accurate methods for solving BVPs is the Boundary Integral Equation Method, which has the advantage of dimensionality reduction: all of the unknowns reside on the boundary surface instead of in the volume. A key challenge when solving integral equations is the accurate and robust discretization of the underlying singular and near-singular integral operators, which requires special quadrature methods. Accurate discretization of these integral operators is especially important in problems that involve close interactions of objects. In this talk, we present some recent advancements on singular and near-singular numerical integration based on the trapezoidal quadrature rule. This quadrature method achieves high-order accuracy by a local singular error correction derived from generalized Euler-Maclaurin formulas, which leads to a family of quadrature rules that are simple, accurate, robust, and easy to use.

Rapidly growing fields such as data science, uncertainty quantification, and machine learning rely on fast and accurate methods for inverse problems. Three emerging challenges on obtaining relevant solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. Tackling the immediate challenges that arise from growing model complexities (spatiotemporal measurements) and data-intensive studies (large-scale and high-dimensional measurements collected as time-series), state-of-the-art methods can easily exceed their limits of applicability. In this talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator change at different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. In the first part of the talk, to remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy. In the remainder of the talk, we focus on designing spatio-temporal Bayesian Besov priors for computing the MAP estimate in large-scale and dynamic inverse problems. Numerical examples from a wide range of applications, such as tomographic reconstruction, image deblurring, and dynamic tomography are used to illustrate the effectiveness of the described approaches.