Related papers: Learning based numerical methods for Helmholtz equ…
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations.…
We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the…
Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers…
We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave…
In these notes we propose and analyze an inertial type method for obtaining stable approximate solutions to nonlinear ill-posed operator equations. The method is based on the Levenberg-Marquardt (LM) iteration. The main obtained results…
The inverse-free extreme learning machine (ELM) algorithm proposed in [4] was based on an inverse-free algorithm to compute the regularized pseudo-inverse, which was deduced from an inverse-free recursive algorithm to update the inverse of…
Tikhonov regularization is one of the most commonly used methods of regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to…
We present an algorithm based on numerical techniques that have become standard for solving nonlinear integral equations: Newton's method, homotopy continuation, the multilevel method and random projection to solve the inversion problem…
Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing…
The high-frequency Helmholtz equation on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently, discretized by the higher-order finite element method (FEM) and the continuous…
In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep…
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions…
Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is…
In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential…
In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243-262] to the case of…
We propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the L2 norm least squares functional with the weak imposition of the continuity across the interior faces as well as…
The harmonic balance method is the most commonly used method for solving periodic solutions of nonlinear dynamic systems, but the high-order approximation of nonlinear terms requires sophisticated formula derivation, which limits its…
Based on the joint bidiagonalization process of a large matrix pair $\{A,L\}$, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: $\min\|Lx\| \…
We study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Applying these methods is equivalent to using neural networks to solve…