Related papers: Quantum Realization of the Finite Element Method
Analytical and practical evidence indicates the advantage of quantum computing solutions over classical alternatives. Quantum-based heuristics relying on the variational quantum eigensolver (VQE) and the quantum approximate optimization…
Optimization problems become fundamentally challenging as the number of variables increases. Because the volume of the search space grows exponentially, classical algorithms frequently fail to locate the global minimum of non-convex…
The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations.…
In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction…
Quantum computers are known to provide an exponential advantage over classical computers for the solution of linear differential equations in high-dimensional spaces. Here, we present a quantum algorithm for the solution of nonlinear…
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending…
The implementation of the finite element method for linear elliptic equations requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be…
We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a $2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow $\Gamma$ to…
For the solution of partial differential equations (PDEs), we show that the quantum Fourier transform (QFT) can enable the design of quantum circuits that are particularly simple, both conceptually and with regard to hardware requirements.…
Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this…
Quantum machine learning algorithms have emerged to be a promising alternative to their classical counterparts as they leverage the power of quantum computers. Such algorithms have been developed to solve problems like electronic structure…
The paper presents a two-dimensional geometrically nonlinear formulation of a beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which…
We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in…
Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing…
Quantum algorithms provide an exponential speedup for solving certain classes of linear systems, including those that model geologic fracture flow. However, this revolutionary gain in efficiency does not come without difficulty. Quantum…
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Pad\'e approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce…
Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum…
Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of…