Inverse problems have a rich history in imaging, addressing challenging problems such as the reconstruction of images from sparse, corrupted observations in a variety of complex applications (e.g., computed tomography). When analyzing results, uncertainty quantification (UQ) plays a key role in understanding the reliability of predicted outcomes due to changes in the input variables. This project will address inverse problems and UQ in the context of imaging applications, including recent approaches to tackle ill-posedness and assess the reliability of results under model uncertainties.

The proposed research collaboration will focus on numerical modeling of nonlinear optical phenomena, with particular interest on developing efficient numerical methods that naturally capture the inherent physical aspects of the problem under consideration. Nonlinear optics is the area of optics that studies the interaction of light with matter in the regime where the response of the material system depends in a nonlinear manner on the amplitude of the applied optical excitation. As a consequence, when light of one or more frequencies impinges on a sample, light at a different or several different frequencies emerges (a phenomenon called frequency mixing). A simple example is the second-harmonic generation that is obtained when the wavelength of the light that emerges is exactly half that of the incident beam. Such phenomena, although essentially known since the beginning of optics, became particularly important as lasers came into widespread use. Indeed, these modifications of the optical properties of a given material are only obtained with sufficiently intense (and coherent) light. Nonlinear optical phenomena are nowadays at the heart of optical communications systems, optical sensing, optical imaging, the design of new laser sources etc. As a support to experiments, numerical modelling of nonlinear optical phenomena is essential. Numerous challenges need to be overcome to find appropriate and relevant models and provide efficient numerical computing environments for a given realistic scenario.

Microswimmers are ubiquitous in nature and understanding their behavior is essential in biological fields and industrial applications. Due to their size and speeds, the fluid motion generated by these swimmers can be modeled using the Stokes equations, a linear PDE. The method of regularized Stokeslets and several variants and improvements have been developed over the last two decades for microscopic viscous flows. Recent developments related to these methods include regularized Stokeslet segments, a method of images for regularized Stokeslets near an infinite solid plane or outside a solid sphere, and evaluation techniques for the boundary integrals that arise in applications. In this project, the participants will have opportunities to (1) extend some of these methods to address novel applications and to (2) model applications and implement the methods. We expect the directions of the project will develop naturally based on the interests of the participants. Potential projects include comparisons of different numerical methods applied to test problems that provide insight into particular applications, modeling flows in confined domains, and improving on current numerical methods. The project will begin with a tutorial and hands-on activities on regularized Stokeslets, including discussions about their applications and limitations. Participants can expect some early reading assignments on background material.

Convolutional filters are powerful tools that have proven useful in multiple areas, such as data compression, shock filtering, post-processing, and machine learning. The popularity of this approach has given rise to the need for effective and efficient design of convolution kernels that are both scheme and grid agnostic. In this project, the goal is to increase efficiency, effectiveness and flexibility of robust convolutional filters. This project will focus on identifying the essential properties in convolutional filter design in the mesh-free context. This will entail: (1) An investigation of grid dependence of the data; (2) Optimising the underlying evaluation of the convolutional kernel; (3) Exploring the affect on stability and conservation of the resulting filter operator.

The development of mixed-precision algorithms for solving ordinary differential equations (ODEs) using Runge--Kutta methods [1,2] was proposed in [3]. The approach builds on the work in [6,5,4] which studies additive Runge--Kutta methods and perturbations of such methods. Using this rigorous approach, he developed novel mixed precision and mixed model Runge--Kutta additive (MP-ARK) methods that reduce the cost of the computationally expensive implicit stages in the Runge--Kutta methods by employing a cheaper low-precision or low-order model computation. The explicit stages are then performed using a high-precision or high- order model computation, providing a correction to the implicit stages. The structure of the MP-ARK is designed to further suppress the errors coming from either the low precision or low order model by introducing inexpensive explicit high-precision correction terms, or by designing novel methods that internally suppress the low- precision perturbations.

In this research project we will learn about additive Runge--Kutta methods and their perturbations, and about ODE linear and nonlinear stability theory.
Our research will focus on several questions for mixed model computations, including:
1) What stability properties do the low- and high-order models have to satisfy so
that the mixed model is stable?
2) How do the correction terms impact the stability of the mixed model?
3) Can we devise a different type of correction that will improve the stability of the
corrected mixed model method?

No specialized prior background is required: we will assume that the participants have used some numerical methods for ordinary differential equations (possibly even Runge-Kutta methods), can identify the difference between implicit and explicit methods, and are familiar with the concept of a fixed point iteration. However, these can be quickly picked up before or as the project begins.

An ingredient in many science and engineering applications, such as solving optimal design and control problems, is to efficiently and accurately simulate many realizations of underlying mathematical models, especially differential equations depending on a set of model parameters. For example, in control, one seeks an optimal strategy to drive a system to a prescribed configuration, and this will be sought via optimizing suitably chosen objective functions of the state variables (i.e., the solutions of the parametric equations) and hence of the model parameters. Many queries of high-fidelity solutions of the underlying models can be computationally prohibitive. Reduced order modeling (ROM) techniques have been developed to provide efficient and reliable surrogate solvers to address such challenges.

We will consider systems governed by kinetic models (e.g., Fokker-Planck, Vlasov-type, Boltzmann equation) and investigate ROMs with applications in control and design problem using such kinetic models which are characterized by high dimensions, multi-scale, and possible nonlinearity. We will address the following aspects or questions. (1) ROMs will be designed by leveraging specific model structures (e.g., bilinear, quadratic, asymptotic preservation). (2) Despite their high computational cost, high-fidelity full-order models (FOMs) often provide training data for building efficient ROM surrogate solvers, and we will examine the interplay between the choice of FOMs and the properties of ROMs. (3) In the optimization step of control and design problems, it is crucial to work with a quality initial guess for a given objective function or to design objective functions with “good” landscapes. When ROMs are used as a fast surrogate solver, what challenges and opportunities will be brought to these components?

The tasks above will provide a starting point. The team's various skill sets and perspectives will work together to refine, shape, and advance this collaborative pro

Recently, low rank tensor methods experienced progress in handling high-dimensional multi-scale partial differential equation (PDE) models. Leveraging the inherent low rank structure of solutions in various multi-scale scenarios, these methods have demonstrated computational savings in storage and CPU/GPU time. Despite this, several open challenges persist in the field.

- Establishing Rigorous Analysis: The accuracy, stability, and convergence of algorithms necessitate investigation. Addressing these aspects will bolster the reliability and applicability of low rank tensor methods in diverse multi-scale PDE contexts.
- Preserving Intrinsic Physics Structures: Ensuring the preservation of the underlying physical structures and model asymptotics is crucial. By doing so, low rank methods can effectively capture the essential features of the system and enhance the predictive power of the models.
- Multi-scale Multi-Physics Applications: A challenge lies in extending low rank algorithms to accommodate challenging applications that couple multiple physics across disparate scales, such as those encountered in high energy density, fusion, and space plasmas. Such systems often support stiff parasitic scales, which may be dynamically irrelevant but require short lengths and time scales for numerical stability. Developing efficient, low-rank compatible implicit solvers and specialized models is vital for broadening the applicability of low rank tensor methods.
- High-Performance Computing (HPC) Considerations: Efficient utilization of HPC resources is critical for reaching the full potential of low rank methods. Integrating these algorithms with HPC architectures will enable large-scale simulations and widen the scope of their applicability.

We'll explore these directions, identify common areas of interest, and share insights leading to a deeper understanding of low rank tensor methods and their applications in high-dimensional multi-scale PDE models.