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We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow…
This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed.…
Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and…
The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms…
In this paper, we are concerned with semiclassical states to the following fractional nonlinear elliptic equation, \begin{align*} \eps^{2s}(-\Delta)^s u + V(x) u=\mathcal{N}(|u|)u \quad \mbox{in} \,\,\, \R^N, \end{align*} where $0<s <1$,…
This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling H\"older regularity…
In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in…
One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural…
Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We…
Computing high-quality graph partitions is a challenging problem with numerous applications. In this paper, we present a novel meta-heuristic for the balanced graph partitioning problem. Our approach is based on integer linear programs that…
We study non-linear Schr\"odinger operators on graphs. We construct minimal nonnegative solutions to corresponding semi-linear elliptic equations and use them to introduce the notion of stochastic completeness at infinity in a non-linear…
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and…
We propose elliptical graphical models based on conditional uncorrelatedness as a general- ization of Gaussian graphical models by letting the population distribution be elliptical instead of normal, allowing the fitting of data with…
In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting and rewrite the latter as a preconditioner for the Schur…
Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve…
Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in…
Semi-linear elliptic Partial Differential Equations (PDEs) such as the non-linear Poisson Boltzmann Equation (nPBE) is highly relevant for non-linear electrostatics in computational biology and chemistry. It is of particular importance for…
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite…
For an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary…
Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation…