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Numerical simulations for engineering applications solve partial differential equations (PDE) to model various physical processes. Traditional PDE solvers are very accurate but computationally costly. On the other hand, Machine Learning…

This paper presents Mathematical Execution (ME), a new, unified approach for testing numerical code. The key idea is to (1) capture the desired testing objective via a representing function and (2) transform the automated testing problem to…

This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive…

The standard implementation of the Maximum Entropy Method (MEM) follows Bryan and deploys a Singular Value Decomposition (SVD) to limit the dimensionality of the underlying solution space apriori. Here we present arguments based on the…

The smooth particle mesh Ewald (SPME) method is the standard method for computing the electrostatic interactions in the molecular simulations. In this work, the multiple staggered mesh Ewald (MSME) method is proposed to boost the accuracy…

The paper is concerned with the development of efficient and accurate solution procedures for the isogeometric boundary element method (BEM) when applied to problems that contain inclusions that have elastic properties different to the…

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions…

Isogeometric approach applied to Boundary Element Methods is an emerging research area. In this context, the aim of the present contribution is that of investigating, from a numerical point of view, the Symmetric Galerkin Boundary Element…

A scene graph is a semantic representation that expresses the objects, attributes, and relationships between objects in a scene. Scene graphs play an important role in many cross modality tasks, as they are able to capture the interactions…

In computational engineering, ensuring the integrity and safety of structures in fields such as aerospace and civil engineering relies on accurate stress prediction. However, analytical methods are limited to simple test cases, and…

The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate…

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babu\v{s}ka and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]. We outline the numerical implementation of…

This paper develops a novel sequential Monte Carlo (SMC) approach for joint state and parameter estimation that can deal efficiently with abruptly changing parameters which is a common case when tracking maneuvering targets. The approach…

In this paper, we discuss multiscale methods for nonlinear problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are…

The method is proposed for the study of many-point boundary value problems for systems of nonlinear ODE, by reducing them to special equivalent integral equations, and allows us [in contrast with the known method [1]] to consider boundary…

This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic…

The smooth particle mesh Ewald (SPME) method is an FFT based method for the fast evaluation of electrostatic interactions under periodic boundary conditions. A highly optimized implementation of this method is available in GROMACS, a widely…

Deep learning solutions of the salient object detection problem have achieved great results in recent years. The majority of these models are based on encoders and decoders, with a different multi-feature combination. In this paper, we show…

Sparse mixture of experts (SMoE) have emerged as an effective approach for scaling large language models while keeping a constant computational cost. Regardless of several notable successes of SMoE, effective training such architecture…

We develop the theory of a new type of asymptotic expansions for functions of two variables the coefficients of which contain functions of one of the variables as well as functions of the quotient of these two variables. These combined…