We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\"odinger equation (NLSE) with semi-smooth nonlinearity $ f(\rho) = \rho^\sigma$, where $\rho=|\psi|^2$ is the density with $\psi$ the wave function and $\sigma>0$ is the exponent of the semi-smooth nonlinearity. Under the assumption of $ H^2 $-solution of the NLSE, we prove error bounds at $ O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma}) $ and $ O(\tau + h^{2}) $ in $ L^2 $-norm for $0<\sigma\leq\frac{1}{2}$ and $\sigma\geq\frac{1}{2}$, respectively, and an error bound at $ O(\tau^\frac{1}{2} + h) $ in $ H^1 $-norm for $\sigma\geq \frac{1}{2}$, where $h$ and $\tau$ are the mesh size and time step size, respectively. In addition, when $\frac{1}{2}<\sigma<1$ and under the assumption of $ H^3 $-solution of the NLSE, we show an error bound at $ O(\tau^{\sigma} + h^{2\sigma}) $ in $ H^1 $-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional $ L^2 $-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of $ 0 < \sigma \leq \frac{1}{2}$, and to establish an $ l^\infty $-conditional $ H^1 $-stability to obtain the $ l^\infty $-bound of the numerical solution by using the mathematical induction and the error estimates for the case of $ \sigma \ge \frac{1}{2}$; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.
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We propose MNPCA, a novel non-linear generalization of (2D)$^2${PCA}, a classical linear method for the simultaneous dimension reduction of both rows and columns of a set of matrix-valued data. MNPCA is based on optimizing over separate non-linear mappings on the left and right singular spaces of the observations, essentially amounting to the decoupling of the two sides of the matrices. We develop a comprehensive theoretical framework for MNPCA by viewing it as an eigenproblem in reproducing kernel Hilbert spaces. We study the resulting estimators on both population and sample levels, deriving their convergence rates and formulating a coordinate representation to allow the method to be used in practice. Simulations and a real data example demonstrate MNPCA's good performance over its competitors.
We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.
We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.
The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection-diffusion-reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.
We consider rather general structural equation models (SEMs) between a target and its covariates in several shifted environments. Given $k\in\N$ shifts we consider the set of shifts that are at most $\gamma$-times as strong as a given weighted linear combination of these $k$ shifts and the worst (quadratic) risk over this entire space. This worst risk has a nice decomposition which we refer to as the "worst risk decomposition". Then we find an explicit arg-min solution that minimizes the worst risk and consider its corresponding plug-in estimator which is the main object of this paper. This plug-in estimator is (almost surely) consistent and we first prove a concentration in measure result for it. The solution to the worst risk minimizer is rather reminiscent of the corresponding ordinary least squares solution in that it is product of a vector and an inverse of a Grammian matrix. Due to this, the central moments of the plug-in estimator is not well-defined in general, but we instead consider these moments conditioned on the Grammian inverse being bounded by some given constant. We also study conditional variance of the estimator with respect to a natural filtration for the incoming data. Similarly we consider the conditional covariance matrix with respect to this filtration and prove a bound for the determinant of this matrix. This SEM model generalizes the linear models that have been studied previously for instance in the setting of casual inference or anchor regression but the concentration in measure result and the moment bounds are new even in the linear setting.
An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, $C^0$ interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in $H^{-2}(\Omega)$.
We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.
This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent.
We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.
We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.
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