The so-called curse of dimensionality plagues high-dimensional computational algorithms. Conventionally, applied mathematicians have turned to linear algebra techniques to find low-rank matrix approximations which can be substituted gainfully into rate-limiting algorithmic operations.  And though this technique finds much success, the application of linear algebra to higher-order tensors is not so straightforward.  In this work, I investigate the use of multilinear algebra and higher-order decompositions to reduce the complexity of high-dimensional algorithms in quantum physi

Measures provide valuable insights into long-term and global behaviors across various dynamical systems. In this talk, I present our recent works employing measure theory and optimal transport to tackle challenges in dynamical system identification. First, we adopt a PDE-constrained optimization perspective to learn ODEs and SDEs from slowly sampled trajectories, enabling stable forward models and uncertainty quantification. We use optimal transport to align physical measures for parameter estimation, even when time-derivative data is unavailable.

We develop a static quantum embedding scheme that utilizes different levels of approximations to coupled cluster (CC) theory for an active fragment region and its environment. In this approach, we solve the local fragment problem using a high-level CC method and address the environment problem with a lower-level Møller–Plesset (MP) perturbative method combined with an efficient relaxation mechanism. We define a static renormalized interaction for the fragment problem with the quantities obtained from the low-level method.

I will discuss the problem of approximating a target matrix A with a structured matrix, given access to a limited number of adaptively chosen matrix-vector products with A. This general problem arises across computational science, both in algorithmic applications and, more recently, in Scientific Machine Learning (SciML), where it abstracts the central task of operator learning.

Wave turbulence, the longtime statistical behavior of a sea of weakly nonlinear dispersive waves, unlike the case of hydrodynamic turbulence, has, in many circumstances, a natural asymptotic closure. The closure gives a closed (kinetic) equation for the energy (or number) density clearly revealing the resonance mechanism by which waves of one wavelength and direction transfer energy to others throughout the spectrum. Equations for the higher order cumulants are all linear and satisfied by a frequency renormalization.

Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis, including rank-structured solvers and preconditioners based on "quantum-inspired" algorithms such as DMRG. However, the theory of QTT approximation is not fully understood.

In this talk, we describe the classical magneto-static approach to the theory of type-I superconductors. (See the complete abstract on the event flyer)

In a seminal article in 1929, P.A.M. Dirac wrote: "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation."