Related papers: Further improving quantum algorithms for nonlinear…
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic.…
Nonlinearity presents a significant challenge in problems involving dynamical systems, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. In this paper, we introduce the Koopman…
We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
Large-scale quantum devices provide insights beyond the reach of classical simulations. However, for a reliable and verifiable quantum simulation, the building blocks of the quantum device require exquisite benchmarking. This benchmarking…
Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct…
We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is…
This paper presents a Carleman-Fourier linearization method for nonlinear dynamical systems with periodic vector fields involving multiple fundamental frequencies. By employing Fourier basis functions, the nonlinear dynamical system is…
Classical optimization algorithms in machine learning often take a long time to compute when applied to a multi-dimensional problem and require a huge amount of CPU and GPU resource. Quantum parallelism has a potential to speed up machine…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…
We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare…
We outline refined versions of two major quantum algorithms for performing principal component analysis and solving linear equations. Our methods are exponentially faster than their classical counterparts and even previous quantum…
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…
Though quantum algorithm acts as an important role in quantum computation science, not only for providing a great vision for solving classically unsolvable problems, but also due to the fact that it gives a potential way of understanding…
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase…
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a…
We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite…
We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…