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Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural networks (VPINNs). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on…
Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a…
To numerically solve the two-dimensional advection equation, we propose a family of fourth- and higher-order semi-Lagrangian finite volume (SLFV) methods that feature (1) fourth-, sixth-, and eighth-order convergence rates, (2)…
We have applied a collocation approach to obtain the numerical solution to the stationary Schr\"odinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to…
The high-precision solution of the radial Dirac equation is fundamental to relativistic quantum chemistry, essential for reliable pseudopotential generation and all-electron electronic structure methods. However, standard basis-set…
Fully kinetic simulations of the Vlasov equation require a careful numerical treatment of phase space advections to ensure accuracy and stability in six dimensions. To test the accuracy of full Vlasov codes, we have developed a surprisingly…
We propose and analyze an adaptive step-size variant of the Davis-Yin three operator splitting. This method can solve optimization problems composed by a sum of a smooth term for which we have access to its gradient and an arbitrary number…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse…
We construct one and two parameter deformations of the two dimensional Chebyshev polynomials with simple recurrence coefficients, following the algorithm in [3]. Using inverse scattering techniques, we compute the corresponding…
Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves.The aim of the present work is to…
In dimension 2, we introduce a distributional Jacobian determinant $\det DV_\beta(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto |Dv|^\beta(v_{x_1},-v_{x_2})$ for any $\beta>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $\beta…
The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a harmonic point source. Its solution consists of a set of discrete modes radiating into the upper…
A new gradient-based adaptive sampling method is proposed for design of experiments applications which balances space filling, local refinement, and error minimization objectives while reducing reliance on delicate tuning parameters. High…
A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and…
We develop a new approach to robust adaptive beamforming in the presence of signal steering vector errors. Since the signal steering vector is known imprecisely, its presumed (prior) value is used to find a more accurate estimate of the…
We present a novel computational scheme to solve acoustic wave transmission problems over composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. Our continuous problem is cast as a multiple traces time-domain…
We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications…