Related papers: Learning based numerical methods for Helmholtz equ…
We exploit the similarities between Tikhonov regularization and Bayesian hierarchical models to propose a regularization scheme that acts like a distributed Tikhonov regularization where the amount of regularization varies from component to…
This paper develops some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only…
We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions…
This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone…
We present a wavenumber-explicit convergence analysis of the hp finite element method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber $k$. Our analysis covers the heterogeneous…
By learning the mappings between infinite function spaces using carefully designed neural networks, the operator learning methodology has exhibited significantly more efficiency than traditional methods in solving complex problems such as…
In this paper we investigate the problem of learning an unknown bounded function. We be emphasize special cases where it is possible to provide very simple (in terms of computation) estimates enjoying in addition the property of being…
The numerical waveforms for the extreme mass-ratio inspirals (EMRIs) require a huge amount of homogeneous solutions of the Teukolsky equation in the frequency domain. The calculation accuracy and efficiency of the homogeneous solutions are…
In numerous contexts, high-resolution solutions to partial differential equations are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain…
We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase…
Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain…
Two approaches to solution of the two-dimensional Helmholtz equation with a "wave number" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on…
This paper is devoted to the efficient numerical solution of the Helmholtz equation in a two- or three-dimensional rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and…
Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the…
In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into…
Effective hyper-parameter tuning is essential to guarantee the performance that neural networks have come to be known for. In this work, a principled approach to choosing the learning rate is proposed for shallow feedforward neural…
Optimizing deep neural networks is largely thought to be an empirical process, requiring manual tuning of several hyper-parameters, such as learning rate, weight decay, and dropout rate. Arguably, the learning rate is the most important of…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…