Related papers: Error analysis of a collocation method on graded m…
A convergence result for a discontinuous Galerkin multiscale method for a second order elliptic problem is presented. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$…
A new moving mesh scheme based on the Lagrange-Galerkin method for the approximation of the one-dimensional convection-diffusion equation is studied. The mesh movement, which is prescribed by a discretized dynamical system for the nodal…
Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability…
In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection- diffusion type is considered on the interval [0, 1]. The components of the solution of this…
In this work, we will give proper estimates for the discrete convolution complementary (DCC) kernels, which leads to the asymptotically compatible fractional Gr\"onwall inequality. The consequence can be applied in the analysis of the…
Kernel techniques are among the most popular and flexible approaches in data science allowing to represent probability measures without loss of information under mild conditions. The resulting mapping called mean embedding gives rise to a…
We consider the inference problem for parameters in stochastic differential equation models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to…
We consider a singularly perturbed convection-diffusion with exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost…
We consider a class of stochastic gradient optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimates which are uniform in time for the error between the solution of the numerical scheme,…
Motivated by applications in machine learning and statistics, we study distributed optimization problems over a network of processors, where the goal is to optimize a global objective composed of a sum of local functions. In these problems,…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from heterogeneous strain gradient elasticity and elasticity models…
We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Dur\'{a}n- and S-type we derive a-priori error estimates for the stationary problem. Using a…
In many statistical linear inverse problems, one needs to recover classes of similar curves from their noisy images under an operator that does not have a bounded inverse. Problems of this kind appear in many areas of application.…
A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville time-fractional derivative is developed and analyzed. This is the first paper to show that the L1 scheme for the model problem under consideration is second-order…
It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite…
Masked discrete diffusion models (MDMs) are a promising new approach to generative modelling, offering the ability for parallel token generation and therefore greater efficiency than autoregressive counterparts. However, achieving an…
This paper discusses the local linear smoothing to estimate the unknown first and second infinitesimal moments in second-order jump-diffusion model based on Gamma asymmetric kernels. Under the mild conditions, we obtain the weak consistency…
Various bias-correction methods such as EXTRA, gradient tracking methods, and exact diffusion have been proposed recently to solve distributed {\em deterministic} optimization problems. These methods employ constant step-sizes and converge…