Manas Rachh

Mo Zhou, Jiequn Han, Manas Rachh et al.

We consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the $L^2$ distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment. We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research.

Florian Pausinger, Manas Rachh, Stefan Steinerberger

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.

Guillaume Bal, Jeremy Hoskins, Solomon Quinn et al.

This paper concerns a boundary integral formulation for the two-dimensional massive Dirac equation. The mass term is assumed to jump across a one-dimensional interface, which models a transition between two insulating materials. This jump induces surface waves that propagate outward along the interface but decay exponentially in the transverse direction. After providing a derivation of our integral equation, we prove that it has a unique solution for almost all choices of parameters using holomorphic perturbation theory. We then extend these results to a Dirac equation with two interfaces. Finally, we implement a fast numerical method for solving our boundary integral equations and present several numerical examples of solutions and scattering effects.

Jeremy G. Hoskins, Alan E. Lindsay, Manas Rachh

In this paper we present a new boundary integral equation formulation for the solution of the elastostatic traction boundary value problem in two and three dimensions. The approach relies on the introduction of new layer potentials, called string kernels, which are based on modifications of the Boussinesq-Cerruti family of half-space solutions. We prove that the resulting integral equations are second-kind integral equations, and show that they are well-behaved in the incompressible limit. We illustrate the performance of the method with several numerical examples.

Zijian Wang, Jeremy G. Hoskins, Manas Rachh et al.

In this paper, we study the scale-invariant quantity \[\mathcal{G}(Ω)=\frac{\|\partial_n u_1\|_{L^\infty(\partialΩ)}}{λ_1},\]where $u_1$ is the first $L^2$-normalized Dirichlet Laplace eigenfunction of a Euclidean domain $Ω$ and $λ_1$ is its eigenvalue. This is related to the peak boundary heat flux in the long time limit. For convex domains we prove that $\|\partial_n u_1\|_{L^\infty(\partialΩ)}$ is upper-bounded by a (domain-independent) constant multiple of $λ_1$. Using layer potentials, we derive shape-derivative formulae for efficient gradient computations. When combined with high-order Nyström discretization, a fast boundary integral equation solver, and eigenvalue rootfinding, this allows us to numerically optimize $\mathcal{G}$ over a class of rounded polygonal discretized domains. Based on extensive numerical experiments, we then conjecture that, over the set of convex domains, $\mathcal{G}$ is maximized by the semidisk, with the peak flux at the center of the diameter. To lend analytical support to this conjecture, we prove that the semidisk is a critical point of $\mathcal{G}$ under infinitesimal perturbations of its circular arc.

Travis Askham, Manas Rachh

The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier-Stokes equations. As the Stokes operator is a fourth-order operator, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high frequency "pollution" when using a Green's function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.

George C. Linderman, Manas Rachh, Jeremy G. Hoskins et al.

t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for dimensionality reduction and visualization that has become widely popular in recent years. Efficient implementations of t-SNE are available, but they scale poorly to datasets with hundreds of thousands to millions of high dimensional data-points. We present Fast Fourier Transform-accelerated Interpolation-based t-SNE (FIt-SNE), which dramatically accelerates the computation of t-SNE. The most time-consuming step of t-SNE is a convolution that we accelerate by interpolating onto an equispaced grid and subsequently using the fast Fourier transform to perform the convolution. We also optimize the computation of input similarities in high dimensions using multi-threaded approximate nearest neighbors. We further present a modification to t-SNE called "late exaggeration," which allows for easier identification of clusters in t-SNE embeddings. Finally, for datasets that cannot be loaded into the memory, we present out-of-core randomized principal component analysis (oocPCA), so that the top principal components of a dataset can be computed without ever fully loading the matrix, hence allowing for t-SNE of large datasets to be computed on resource-limited machines.

Xiuyuan Cheng, Manas Rachh, Stefan Steinerberger

We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i \in V$, we define the diffusion distance to a set $B \subset V$ as the smallest number of steps $d_{B}(i) \in \mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p_{ij}$ to visit $B$ within $d_{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $\mathcal{L}u = λu$ on $V$ and $$ B = \left\{ i \in V: - \varepsilon \leq u(i) \leq \varepsilon \right\} \neq \emptyset,$$ then, for all $i \in V$, $$ d_{B}(i) \log{\left( \frac{1}{|1-λ|} \right) } \geq \log{\left( \frac{ |u(i)| }{\|u\|_{L^{\infty}}} \right)} - \log{\left(\frac{1}{2} + \varepsilon\right)}.$$ $d_B(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.

L. F. Ricketson, A. J. Cerfon, M. Rachh et al.

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.