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Davoud Mirzaei

Publications and source records attributed to Davoud Mirzaei.

15 recordsLinked to original sources
arXiv2026

Scale-uniform inverse inequalities for scaled kernel spaces

Inverse inequalities are an important tool in the stability and convergence analysis of kernel approximation methods. In a multiscale setting, however, the trial space changes with the kernel scale $\delta$, and inverse estimates for a fixed kernel are not sufficient. The constants must remain controlled as $\delta\to0$. In this paper, we study scale-uniform inverse inequalities for spaces generated by scaled positive definite kernels whose native spaces are Sobolev spaces. We first establish inverse estimates in scale-dependent Sobolev spaces. On bounded domains, this leads to a scale-uniform inverse inequality in standard Sobolev norms with $L^2$ as the weaker norm. The estimate requires only that the separation distance of the centers be bounded above by a fixed multiple of the kernel scale. This allows the kernel scale to decrease more slowly than the separation distance, as is relevant in multiscale refinement. We then establish more general scale-uniform Bernstein inequalities on the whole space. Our result covers a broad range of weaker and stronger Sobolev indices under the same relation between the separation distance and the kernel scale. The proof works directly with the Fourier representation of functions in the kernel space and combines a low-high frequency decomposition with frame estimates for separated exponential polynomials. This avoids the additional scale factors that arise from separate comparisons of scaled and standard Sobolev norms.

math.NA↗
arXiv2026

The least squares RBF-PU method for linear elasticity in the diaphragm geometry

The human diaphragm is vital for respiration and its simulation can aid in further understanding complications arising in individuals that have been subjected to mechanical ventilation. The diaphragm geometry is very thin and hence its numerical simulation poses a challenge for most numerical methods. In this paper, we compute the deformation of real diaphragm geometries which are modelled as linearly elastic and are subject to realistic boundary conditions. The method used is the unfitted least-squares radial basis function partition of unity method, which is well suited to solve problems on thin geometries given its unfitted nature and the shape and refinement of the cylindrical patches that are used. We derive a theoretical proof showing that the method converges with high order to the solution of this problem.

math.NA↗
arXiv2025

LREI: A fast numerical solver for quantum Landau-Lifshitz equations

We develop LREI (Low-Rank Eigenmode Integration), a memory- and time-efficient scheme for solving quantum Landau-Lifshitz (q-LL) and quantum Landau-Lifshitz-Gilbert (q-LLG) equations, which govern spin dynamics in open quantum systems. Although system size grows exponentially with the number of spins, our approach exploits the low-rank structure of the density matrix and the sparsity of Hamiltonians to avoid full matrix computations. By representing density matrices via low-rank factors and applying Krylov subspace methods for partial eigendecompositions, we reduce the per-step complexity of Runge-Kutta and Adams-Bashforth schemes from $\mathcal{O}(N^3)$ to $\mathcal{O}(r^2N)$, where $N = 2^n$ is the Hilbert space dimension for $n$ spins and $r \ll N$ the effective rank. Similarly, memory costs shrink from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$, since no full $N\times N$ matrices are formed. A key advance is handling the invariant subspace of zero eigenvalues. By using Householder reflectors built for the dominant eigenspace, we perform the solution entirely without large matrices. For example, a time step of a twenty-spin system, with density matrix size over one million, now takes only seconds on a standard laptop. Both Runge-Kutta and Adams-Bashforth methods are reformulated to preserve physical properties of the density matrix throughout evolution. This low-rank algorithm enables simulations of much larger spin systems, which were previously infeasible, providing a powerful tool for comparing q-LL and q-LLG dynamics, testing each model validity, and probing how quantum features such as correlations and entanglement evolve across different regimes of system size and damping.

quant-ph↗
arXiv2025

Numerical solution of quantum Landau-Lifshitz-Gilbert equation

The classical Landau-Lifshitz-Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems, yet its classical nature limits its applicability to inherently quantum phenomena such as entanglement and nonlocal correlations. Inspired by the need to incorporate quantum effects into spin dynamics, recently a quantum generalization of the LLG equation is proposed [Phys. Rev. Lett. 133, 266704 (2024)] which captures essential quantum behavior in many-body systems. In this work, we develop a robust numerical methodology tailored to this quantum LLG framework that not only handles the complexity of quantum many-body systems but also preserves the intrinsic mathematical structures and physical properties dictated by the equation. We apply the proposed method to a class of many-body quantum spin systems, which host topological states of matter, and demonstrate rich quantum behavior, including the emergence of long-time entangled states. This approach opens a pathway toward reliable simulations of quantum magnetism beyond classical approximations, potentially leading to new discoveries.

quant-ph↗
arXiv2025

Stochastic Simulation and Monte Carlo Method

These lecture notes are intended to cover some introductory topics in stochastic simulation for scientific computing courses offered by the IT department at Uppsala University, as taught by the author. Basic concepts in probability theory are provided in the Appendix A, which you may review before starting the upcoming sections or refer to as needed throughout the text.

math.NA↗
arXiv2024

Numerical Linear Algebra: Least Squares, QR and SVD

These lecture notes focus on some numerical linear algebra algorithms in scientific computing. We assume that students are familiar with elementary linear algebra concepts such as vector spaces, systems of equations, matrices, norms, eigenvalues, and eigenvectors. In the numerical part, we do not pursue Gaussian elimination and other LU factorization algorithms for square systems. Instead, we mainly focus on overdetermined systems, least squares solutions, orthogonal factorizations, and some applications to data analysis and other areas.

math.HO↗
arXiv2024

Fundamentals of Computing

This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.

math.HO↗
arXiv2021

Error and Stability Estimates of a Least-Squares Variational Kernel-Based Method for Second Order Elliptic PDEs

We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds.

math.NA↗
arXiv2020

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method is benefited from a direct discretization approach and is called the `direct RBF partition of unity (D-RBF-PU)' method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular 2D and 3D domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.

math.NA↗
arXiv2015

Analysis of moving least squares approximation revisited

In this article the error estimation of the moving least squares approximation is provided for functions in fractional order Sobolev spaces. The analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details. An application to Galerkin method for partial differential equations is also supplied.

math.NA↗
arXiv2014

A new low-cost meshfree method for two and three dimensional problems in elasticity

In this paper, we continue the development of the Direct Meshless Local Petrov-Galerkin (DMLPG) method for elasto-static problems. This method is based on the generalized moving least squares approximation. The computational efficiency is the most significant advantage of the new method in comparison with the original MLPG. Although, the "Petrov-Galerkin" strategy is used to build the primary local weak forms, the role of trial space is ignored and direct approximations for local weak forms and boundary conditions are performed to construct the final stiffness matrix. In this modification the numerical integrations are performed over polynomials instead of complicated MLS shape functions. In this paper, DMLPG is applied for two and three dimensional problems in elasticity. Some variations of the new method are developed and their efficiencies are reported. Finally, we will conclude that DMLPG can replace the original MLPG in many situations.

math.NA↗
arXiv2013

Solving Heat Conduction Problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method

As an improvement of the Meshless Local Petrov-Galerkin (MLPG), the Direct Meshless Local Petrov-Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low-degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.

math.NA↗
arXiv2013

Direct meshless local Petrov-Galerkin (DMLPG) method: A generalized MLS approximation

The Meshless Local Petrov{Galerkin (MLPG) method is one of the popular meshless methods that has been used very successfully to solve several types of boundary value problems since the late nineties. In this paper, using a generalized moving least squares (GMLS) approximation, a new direct MLPG technique, called DMLPG, is presented. Following the principle of meshless methods to express everything "entirely in terms of nodes", the generalized MLS recovers test functionals directly from values at nodes, without any detour via shape functions. This leads to a cheaper and even more accurate scheme. In particular, the complete absence of shape functions allows numerical integrations in the weak forms of the problem to be done over low{degree polynomials instead of complicated shape functions. Hence, the standard MLS shape function subroutines are not called at all. Numerical examples illustrate the superiority of the new technique over the classical MLPG. On the theoretical side, this paper discusses stability and convergence for the new discretizations that replace those of the standard MLPG. However, it does not treat stability, convergence, or error estimation for the MLPG as a whole. This should be taken from the literature on MLPG.

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