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In this paper, we introduce low-complexity multidimensional discrete cosine transform (DCT) approximations. Three dimensional DCT (3D DCT) approximations are formalized in terms of high-order tensor theory. The formulation is extended to…
High-order CFD is gathering a broadening interest as a future industrial tool, with one such approach being Flux Reconstruction (FR). However, due to the need to mesh complex geometries if FR is to displace current, lower order methods, FR…
In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a…
A semi-implicit finite difference time domain (FDTD) numerical Maxwell solver is developed for full electromagnetic Particle-in-Cell (PIC) codes for the simulations of plasma-based acceleration. The solver projects the volumetric Yee…
The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…
This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in $R^d$, $d=2,3$. This construction is quite close to…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
The thesis focuses on various techniques to find an alternate approximation method that could be universally used for a wide range of CFD problems but with low computational cost and low runtime. Various techniques have been explored within…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…
We present an accurate, efficient and massively parallel finite-element code, DFT-FE, for large-scale ab-initio calculations (reaching $\sim 100,000$ electrons) using Kohn-Sham density functional theory (DFT). DFT-FE is based on a local…
Adapting Foundation Models (FMs) for downstream tasks through Federated Learning (FL) emerges a promising strategy for protecting data privacy and valuable FMs. Existing methods fine-tune FM by allocating sub-FM to clients in FL, however,…
The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode…
We study an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes using the CutFEM method. Our main contribution is the reconstruction of conservative fluxes from the CutFEM solution and their use in a…
We present a previously unexplored forward-mode differentiation method for Maxwell's equations, with applications in the field of sensitivity analysis. This approach yields exact gradients and is similar to the popular adjoint variable…
Discrete cosine transform (DCT) and other Fourier-related transforms have broad applications in scientific computing. However, off-the-shelf high-performance multi-dimensional DCT (MD DCT) libraries are not readily available in parallel…
We develop a cut finite element method (CutFEM) for convection-diffusion problems posed on mixed-dimensional domains, i.e., unions of manifolds of different dimensions arranged in a hierarchical structure where lower-dimensional components…
In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…
In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the…
Communication efficiency is a widely recognised research problem in Federated Learning (FL), with recent work focused on developing techniques for efficient compression, distribution and aggregation of model parameters between clients and…
We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the…