arXiv subjects

Manas Rachh

Publications and source records attributed to Manas Rachh.

44 records · Page 3Linked to original sources

arXiv2017

On the solution of Stokes equation on regions with corners

In Stokes flow, the stream function associated with the velocity of the fluid satisfies the biharmonic equation. The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. The problem was first examined by Lord Rayleigh in 1920; in 1973, the existence of infinite oscillations in the domain Green's function was proven in the case of the right angle by S.~Osher. In this paper, we observe that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form $\sum_{j} ( c_{j} t^{μ_{j}} \sin{(β_{j} \log{(t)})} + d_{j} t^{μ_{j}} \cos{(β_{j} \log{(t)})} )$, where $t$ is the distance from the corner and the parameters $μ_{j},β_{j}$ are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples.

math-ph↗

arXiv2017

Integral equation formulation of the biharmonic Dirichlet problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman-Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.

math.NA↗

arXiv2017

Optimal Jittered Sampling for two Points in the Unit Square

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.

math.NA↗

arXiv2017

Fast algorithms for Quadrature by Expansion I: Globally valid expansions

The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion, a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples. Keywords: Layer Potentials; Singular Integrals; Quadrature; High-order accuracy; Integral equations; Helmholtz equation; Fast multipole method.

math.NA↗

arXiv2017

On the location of maximal of solutions of Schrödinger's equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant $c>0$, such that if $Ω\subset \mathbb{R}^2$ is simply connected, $u:Ω\rightarrow \mathbb{R}$ vanishes on the boundary $\partial Ω$, and $|u|$ assumes a maximum in $x_0 \in Ω$, then $$ \inf_{y \in \partial Ω}{ \| x_0 - y\|} \geq c \left\| \frac{Δu}{u} \right\|^{-1/2}_{L^{\infty}(Ω)}.$$ It was conjectured by Pólya \& Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency $λ$ contains a disk of size $\sim λ^{-1/2}$. Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance $\sim λ^{-1/2}$ from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and Georgiev \& Mukherjee): if $u$ solves $-Δu = Vu$ on $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim \|V\|_{L^{\infty}(Ω)}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $Ω$ in the sense that $|B \cap Ω| \geq 0.99 |B|.$

math.AP↗

arXiv2016

On the Diffusion Geometry of Graph Laplacians and Applications

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.

math.SP↗

arXiv2016

An integral equation formulation for rigid bodies in Stokes flow in three dimensions

We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in $\mathcal{O}(n)$ time, where $n$ denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.

math.NA↗

arXiv2015

Integral equation methods for elastance and mobility problems in two dimensions

We present new integral representations in two dimensions for the elastance problem in electrostatics and the mobility problem in Stokes flow. These representations lead to resonance-free Fredholm integral equations of the second kind and well conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.

math.NA↗