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Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $\Omega$. One of the requirements leading to approximation robustness is to place…
State-of-the-art embeddings often capture distinct yet complementary discriminative features: For instance, one image embedding model may excel at distinguishing fine-grained textures, while another focuses on object-level structure.…
We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing…
The accurate and efficient evaluation of potentials is of great importance for the numerical solution of partial differential equations. When the integration domain of the potential is irregular and is discretized by an unstructured mesh,…
Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of…
This letter describes an incremental multimodal surface mapping methodology, which represents the environment as a continuous probabilistic model. This model enables high-resolution reconstruction while simultaneously compressing spatial…
We introduce a new structured kernel interpolation (SKI) framework, which generalises and unifies inducing point methods for scalable Gaussian processes (GPs). SKI methods produce kernel approximations for fast computations through kernel…
Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness…
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…
A multiscale QM/classical approach is presented, that is able to model the optical properties of complex nanostructures composed of a molecular system adsorbed on metal nanoparticles. The latter are described by a combined…
In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational…
The finite element method (FEM) has several computational steps to numerically solve a particular problem, to which many efforts have been directed to accelerate the solution stage of the linear system of equations. However, the finite…
In the present thesis, a computational framework for the analysis of the deformation and damage phenomena occurring at the micro-scale of polycrystalline materials is presented. Micro-mechanics studies are commonly performed using the…
This paper introduces an interpolation-based method, called the reconstruction approach, for nonparametric regression. Based on the fact that interpolation usually has negligible errors compared to statistical estimation, the reconstruction…
In this study, we present a large-scale earth surface reconstruction pipeline for linear-array charge-coupled device (CCD) satellite imagery. While mainstream satellite image-based reconstruction approaches perform exceptionally well, the…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density,…
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to…
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…
Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent…