Arbitrarily high order discontinuous Galerkin method in time combined with discontinuous Galerkin method with Lagrange multiplier (DGLM) in space is proposed to approximate the solution to hyperbolic conservation laws with boundary conditions. Stability of the approximate solution is proved in a broken $L^2(L^2)$ norm and also in an $l^\infty(L^2)$ norm. Error estimates of ${\mathcal{O}}(h^{r+\frac12}+k_n^{q+\frac12})$ with $P_r(E)$ and $P_q(J_n)$ elements $(r, q\ge \frac{d+1}2)$ are derived in a broken $L^2(L^2)$ norm, where $h$ and $k_n$ are the maximum diameters of the elements and the time step of $J_n$, respectively, $J_n$ is the time interval, and $d$ is the dimension of the spatial domain. An explanation on algorithmic aspects is given. $P_0$ time and space subcell limiting processes are applied to resolve the shocks. It is numerically shown that the high order DG-DGLM method is well-suited for long time integrations. Several numerical experiments for advection, shallow water, and compressible Euler equations are presented to show the performance of the high order DG-DGLM with $P_0$ time and space subcell limiting processes.

Compressed sensing has been a very successful high-dimensional signal acquisition and recovery technique that relies on linear operations. However, the actual measurements of signals have to be quantized before storing or processing. 1(One)-bit compressed sensing is a heavily quantized version of compressed sensing, where each linear measurement of a signal is reduced to just one bit: the sign of the measurement. Once enough of such measurements are collected, the recovery problem in 1-bit compressed sensing aims to find the original signal with as much accuracy as possible. 

For recovery of sparse vectors, a popular reconstruction method from 1-bit measurements is the binary iterative hard thresholding (BIHT) algorithm. The algorithm is a simple projected sub-gradient descent method, and is known to converge well empirically, despite the nonconvexity of the problem. The convergence property of BIHT was not theoretically justified, except with an exorbitantly large number of measurements (i.e., a number of measurement greater than max{k^10,24^48,k^3.5/ϵ}, where k is the sparsity, ϵ denotes the approximation error, and even this expression hides other factors). In this paper we show that the BIHT algorithm converges with only Õ(k/ϵ) measurements. Note that, this dependence on k and ϵ is optimal for any recovery method in 1-bit compressed sensing. With this result, to the best of our knowledge, BIHT is the only practical and efficient (polynomial time) algorithm that requires the optimal number of measurements in all parameters (both k and ϵ). This is also an example of a gradient descent algorithm converging to the correct solution for a nonconvex problem, under suitable structural conditions.

Joint work with Nami Matsumoto.

A numerical semigroup Λ is a subset of the nonnegative integers which contains 0, has finite complement, and is closed under addition. We characterize Λ by a number of invariants: the genus g = |N0 \ Λ|, the multiplicity m = min(Λ \ {0}), and the Frobenius number f = max(N0 \ Λ). Recently, Eliahou and Fromentin introduced the notion of depth q = ⌈(f+1)/q⌉. In 1990, Backelin showed that the number of numerical semigroups with Frobenius number f approaches Ci · 2^(f/2) for constants C0 and C1 depending on the parity of f. In this talk, we use Kunz words and graph homomorphisms to generalize Backelin’s result to numerical semigroups of arbitrary Frobenius number, multiplicity, and depth, in particular showing that there are ⌊(q+1)^2/4⌋^(f/(2q-2)+o(f)) semigroups with Frobenius number f and depth q.

In this talk, I will introduce some recent joint work with A. Avila and D. Damanik in showing positivity and large deviations of the Lyapunov exponent for Schrodinger operators with potentials generated by hyperbolic transformations. Specifically, we consider the base dynamics which is a subshift of finite type with an ergodic measure admitting a bounded distortion property and which has a fixed point. We show that if the potentials are locally constant or globally fiber bunched, then the set of zero Lyapunov exponent is finite. Moreover, we have a uniform large deviation estimate away from this finite set.