Related papers: Quantum Realization of the Finite Element Method
Partial differential equations (PDEs) are fundamental across numerous scientific fields. As these problems scale to high dimensions, classical numerical schemes introduce severe computational bottlenecks, known as the curse of…
In previous research, quantum resources were concretely estimated for solving Elliptic Curve Discrete Logarithm Problem(ECDLP). In [1], the quantum algorithm was optimized for the binary elliptic curves and the main optimization target was…
We present a variational quantum algorithm for structural mechanical problems, specifically addressing crack opening simulations that traditionally require extensive computational resources. Our approach provides an alternative solution for…
Quantum linear system algorithms (QLSAs) for gate-based quantum computing can provide exponential speedups for solving linear systems but face challenges when applied to finite element problems due to the growth of the condition number with…
We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices…
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the…
We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique…
We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the…
We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to…
In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform…
Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis…
Modeling hydrological fracture networks is a hallmark challenge in computational earth sciences. Accurately predicting critical features of fracture systems, e.g. percolation, can require solving large linear systems far beyond current or…
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes…
We design a variational quantum algorithm to solve multi-dimensional Poisson equations with mixed boundary conditions that are typically required in various fields of computational science. Employing an objective function that is formulated…
We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing mid-circuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
This work studies quantum algorithms to solve high-dimensional stochastic differential equations (SDEs) $\mathrm{d} \mathbf{X}_t = A(t) \mathbf{X}_t \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$. Aiming for a speed-up in the dimension $N$ of…
Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system…
Boundary element methods (BEM) reduce a partial differential equation in a domain to an integral equation on the domain's boundary. They are particularly attractive for solving problems on unbounded domains, but handling the dense matrices…
In this paper we propose two variants of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the new preconditioners, we use the simplest coarse solver…