## Primary tabs

## Theoretical Neuroscience Journal Club

#### Computational principles of memory

## Analysis and its Applications Seminar

#### Eigenvectors for the continuous spectrum

### Abstract

The spectral theorem says that every self-adjoint operator acting in a Hilbert space has a complete set of eigenvectors. When there is continuous spectrum these eigenvectors need not be in the original Hilbert space. They are, however, constrained to belong to a somewhat larger Hilbert space. This talk will describe how this larger space depends on the operator. In particular, it is possible to characterize this space in situations where there is no explicit formula for the eigenvectors.## Quantitative Biology Colloquium

#### The role of outcome preferences in optimizing heterogenous disease control strategies

### Abstract

As infectious diseases spread, they do not observe city, state, regional or national boundaries. As such, underlying susceptible population has a patchy structure which suggests metapopulation approach to epidemic modeling. When the patches of the metapopulation are managed by different public health authorities, it is natural to consider heterogeneous disease control strategies. For deterministic models, the basic reproduction number, R_0, is typically a sharp threshold separating the extinction or persistence of the disease. When comparing two control strategies, the one which minimizes R_0 is optimal. Stochastic epidemic models are necessary to account for inherent randomness in the initial phase of an outbreak. In this case, the disease is considered persistent if the probability of extinction is less than 1. This probability is often approximated by branching process techniques. It has been shown that R_0 is also a threshold in the case of branching processes. The probability of extinction is a hitting probability. In this talk, a case is made to consider other hitting probabilities to measure the effectiveness of control strategies when outcome preferences are biased by public health authorities. A technique to approximate these probabilities is presented. Results are compared to standard techniques.## Modeling and Computation Seminar

#### Multilevel Monte Carlo for spiking neuronal networks

### Abstract

Large network dynamics is the main concern in many computational neuroscience studies. A common task in simulating such dynamics is to estimate dynamical statistics like firing rates and correlations elicited by stimuli. These computational tasks are potentially expensive for classical Monte Carlo method. One possible solution is multilevel Monte Carlo method(MLMC). MLMC has been widely used to reduce variance and accelerate estimation for stochastic differential equations. For spiking neuron networks, however, it was unknown whether MLMC would be effective. Our work reveals that the applicability of MLMC heavily relies on the type of dynamics. In this study, we focused on networks of leaky-integrate-and-fire neuron and investigated the utility of MLMC for such networks. By analyzing an associated Fokker-Planck equation and by numerical tests, we found that 1. MLMC is effective for single cells under broad conditions. Thus, by induction, it could be extended to all feed-forward networks. 2. When applying MLMC to randomly-connected recurrent networks, MLMC turns out effective for systems operating in a homogeneous, "mean-field"-like regime in which cells are only weakly correlated. On the contrary, for networks operating in partially-synchronous regimes, the behavior of MLMC is poor, since any small numerical deviation could possibly lead to totally different dynamics of the network.## Brown Bag Seminar

#### RF Gradient Index Devices and the Application of Mathematical Optics

### Abstract

The introduction of metamaterials has enabled the design of a wide array of novel radio frequency (RF) devices. In this talk, the techniques of Mathematical Optics will be utilized for synthesis and analysis of dielectric-only gradient index metamaterial devices. The application of a WKB approximation to Maxwell’s equations will be shown to yield a Lagrangian formulation of optics. This formalism will be applied to the design of a gradient index zero order Quasi-Bessel beam lens. The synthesis of the lens requires solution of an integral equation of Abel’s type. Ray tracing of the resulting index distribution is difficult due to the presence of singularities, but may be accomplished by the clever application of numerical methods.## Applied Math Colloquium

#### Convergence of Hybrid LSQR AND RSVD Algorithms for Ill Posed Least Squares Problems

### Abstract

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. Traditionally LSQR iterative methods have been used to _nd the subspace for the solution. The subspace system inherits the ill-conditioning of the original problem and hybrid methods are used to impose regularization at the subspace level. Modern techniques employ a randomized singular value decomposition (RSVD) to find the subspace for the solution. In this case the subspace system inherits thedominant properties of the original problem, rather than the full condition of the original problem. But hybrid methods in which regularization is applied at the subspace level are still required. Our work is focused on connecting the regularized subspace problem with the regularized full space problem. This requires carefully connecting the singular value subspace for the full problem with that obtained at the subspace level. We examine and contrast hybrid LSQR and hybrid RSVD techniques, through the approximation of the relevant singular subspaces for the original problem, hence examining the convergence of regularized parameters for the hybrid solutions. On average the RSVD provides a good approximation of a truncated singular value decomposition (TSVD). Therefore analysis of the performance of the hybrid RSVD connects back to analysis of TSVD techniques for solving the least squares problem. We illustrate basis vectors obtained by standard SVD, RSVD and Krylov methods as a way to understand how the approaches differ or are equivalent. Numerical examples from image restoration and inversion of three dimensional magnetic data are presented to support the analysis in the context of solving practical least squares applications.