As part of the Duke University Department of Mathematics, the Program in Applied Mathematics hosts this ongoing series of seminars. The presentations cover a broad range of topics including numerical analysis, ordinary and partial differential equations, nonlinear systems, scientific computing, dynamical systems theory, mathematical biology, pattern formation, and complex physical systems.

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As a convenience, some selected seminars and presentations can be viewed live via the web. Further, we have video archives of past talks, which are also publicly available for you to view at any time.

- Wednesday, March 29, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
*Entanglement and computational complexity for 1D quantum many-body systems*

Thomas Barthel (Duke University, Physics Department)- The Hilbert space dimension of quantum-many body systems grows exponentially with the system size. Fortunately, nature does usually not explore this monstrous number of degrees of freedom and we have a chance to describe quantum systems with much smaller sets of effective degrees of freedom. A very precise description for systems with one spatial dimension is based on so-called matrix product states (MPS). With such a reduced parametrization, the computation cost, needed to achieve a certain accuracy, is determined by entanglement properties (quantum non-locality) in the system.

I will give a short introduction to the notion of entanglement entropies and their scaling behavior in typical many-body systems. I will then employ entanglement entropies to bound the required computation costs in MPS simulations. This will lead us to the amazing conclusion that 1D quantum many-body systems can usually be simulated efficiently on classical computers, both for zero and finite temperatures, and for both gapless and critical systems.

In these considerations, we will encounter a number of mathematical concepts such as the theorem of typical sequences (central limit theorem), concentration of measure (Levy's lemma), singular value decomposition, path integrals, and conformal invariance.

- The Hilbert space dimension of quantum-many body systems grows exponentially with the system size. Fortunately, nature does usually not explore this monstrous number of degrees of freedom and we have a chance to describe quantum systems with much smaller sets of effective degrees of freedom. A very precise description for systems with one spatial dimension is based on so-called matrix product states (MPS). With such a reduced parametrization, the computation cost, needed to achieve a certain accuracy, is determined by entanglement properties (quantum non-locality) in the system.
- Wednesday, April 5, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
*Stability of stationary inverse transport equation in diffusion scaling*

Qin Li (University of Wisconsin, Madison)- We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameter—the Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

- Wednesday, April 12, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
*A PDE System Modeling Biological Network Formation*

Peter Markowich (KAUST)- Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid).

- Wednesday, April 19, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
*Domain Decomposition Methods for the solution of Helmholtz transmission problems*

Catalin Turc (NJIT)- We present several versions of non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for (a) multiple scattering configurations, (b) bounded composite scatterers with piecewise constant material properties, and (c) layered media. We show that DDM solvers give rise to important computational savings over other existing solvers, especially in the challenging high-frequency regime.

- Wednesday, April 26, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
*TBA*

Peter Mucha (University of North Carolina, Chapel Hill)

All seminars take place on Mondays at 4:30 pm in Room 119 Physics Building unless otherwise noted.

Tea and refreshments are served before the seminars at 4:00 pm in Physics 101.

Related Seminars

Past speakers in the Duke Applied Mathematics seminars (1997+)

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