We study the high-order local discontinuous Galerkin (LDG) method for the $p$-Laplace equation. We reformulate our spatial discretization as an equivalent convex minimization problem and use a preconditioned gradient descent method as the nonlinear solver. For the first time, a weighted preconditioner that provides $hk$-independent convergence is applied in the LDG setting. For polynomial order $k \geqslant 1$, we rigorously establish the solvability of our scheme and provide a priori error estimates in a mesh-dependent energy norm. Our error estimates are under a different and non-equivalent distance from existing LDG results. For arbitrarily high-order polynomials under the assumption that the exact solution has enough regularity, the error estimates demonstrate the potential for high-order accuracy. Our numerical results exhibit the desired convergence speed facilitated by the preconditioner, and we observe best convergence rates in gradient variables in alignment with linear LDG, and optimal rates in the primal variable when $1 < p \leqslant 2$.
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By using the stochastic particle method, the truncated Euler-Maruyama (TEM) method is proposed for numerically solving McKean-Vlasov stochastic differential equations (MV-SDEs), possibly with both drift and diffusion coefficients having super-linear growth in the state variable. Firstly, the result of the propagation of chaos in the L^q (q\geq 2) sense is obtained under general assumptions. Then, the standard 1/2-order strong convergence rate in the L^q sense of the proposed method corresponding to the particle system is derived by utilizing the stopping time analysis technique. Furthermore, long-time dynamical properties of MV-SDEs, including the moment boundedness, stability, and the existence and uniqueness of the invariant probability measure, can be numerically realized by the TEM method. Additionally, it is proven that the numerical invariant measure converges to the underlying one of MV-SDEs in the L^2-Wasserstein metric. Finally, the conclusions obtained in this paper are verified through examples and numerical simulations.
The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation around this operator. The resolvent is found as $L^2 \to L^2$ operator using the Neumann series for the periodic fundamental solution of biharmonic operator. Results are based on some auxiliary Lemmas suggested by Bensoussan in 1986, Zhikov in 1991, Yu. Grabovsky and G. Milton in 1998, Pastukhova in 2016. Operators of the type considered in the paper appear in the study of the elastic properties of thin plates. The choice of the operator with constant coefficients is discussed separately and chosen in an optimal way w.r.t. the spectral radius and convergence of the Neumann series and uses the known bounds for ''homogenized'' coefficients. Similar ideas are usually applied for the construction of preconditioners for iterative solvers for finite dimensional problems resulting from discretized PDEs. The method presented is similar to Cholesky factorization transferred to elliptic operators (as in references mentioned above). Furthermore, the method can be applied to non-linear problems.
We study least-squares trace regression when the parameter is the sum of a $r$-low-rank matrix and a $s$-sparse matrix and a fraction $\epsilon$ of the labels is corrupted. For subgaussian distributions and feature-dependent noise, we highlight three needed design properties, each one derived from a different process inequality: a "product process inequality", "Chevet's inequality" and a "multiplier process inequality". These properties handle, simultaneously, additive decomposition, label contamination and design-noise interaction. They imply the near-optimality of a tractable estimator with respect to the effective dimensions $d_{eff,r}$ and $d_{eff,s}$ of the low-rank and sparse components, $\epsilon$ and the failure probability $\delta$. The near-optimal rate is $\mathsf{r}(n,d_{eff,r}) + \mathsf{r}(n,d_{eff,s}) + \sqrt{(1+\log(1/\delta))/n} + \epsilon\log(1/\epsilon)$, where $\mathsf{r}(n,d_{eff,r})+\mathsf{r}(n,d_{eff,s})$ is the optimal rate in average with no contamination. Our estimator is adaptive to $(s,r,\epsilon,\delta)$ and, for fixed absolute constant $c>0$, it attains the mentioned rate with probability $1-\delta$ uniformly over all $\delta\ge\exp(-cn)$. Without matrix decomposition, our analysis also entails optimal bounds for a robust estimator adapted to the noise variance. Our estimators are based on "sorted" versions of Huber's loss. We present simulations matching the theory. In particular, it reveals the superiority of "sorted" Huber's losses over the classical Huber's loss.
Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a $k$-dimensional and $l$-dimensional qudit pair, denoted $(k,l)$-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain $\mathsf{QMA}_1$-hard, in that $(2,5)$-QSAT is $\mathsf{QMA}_1$-complete. In contrast, $2$-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that $(3,d)$-QSAT on the 1D line with $d\in O(1)$ is also $\mathsf{QMA}_1$-hard. Finally, we initiate the study of 1D $(2,d)$-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from $\Omega(1/T^6)$ to $\Omega(1/T^2)$, for $T$ the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on $d'$-dimensional qudits, we show how to embed it into an effective null-space of a 1D $(3,d)$-QSAT instance, for $d\in O(1)$. Our approach may be viewed as a weaker notion of "simulation" (\`a la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based $\mathsf{QMA}_1$-hardness result, i.e. for frustration-free Hamiltonians.
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 contrast. Similar equations manifest in diverse applications like chemical diffusion, electrostatics, and Darcy flow. The neural network aims to establish the link between the complex thermal conductivity profiles and temperature distributions, as well as heat flux components within the microstructure, under fixed boundary conditions. A distinctive aspect is our independence from classical solvers like finite element methods for data. A noteworthy contribution lies in our novel approach to defining the loss function, based on the discretized weak form of the governing equation. This not only reduces the required order of derivatives but also eliminates the need for automatic differentiation in the construction of loss terms, accepting potential numerical errors from the chosen discretization method. As a result, the loss function in this work is an algebraic equation that significantly enhances training efficiency. We benchmark our methodology against the standard finite element method, demonstrating accurate yet faster predictions using the trained neural network for temperature and flux profiles. We also show higher accuracy by using the proposed method compared to purely data-driven approaches for unforeseen scenarios.
We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, $\epsilon$, while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.
In this paper, we conduct rigorous error analysis of the Lie-Totter time-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equation with a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodic boundary conditions on a $d$-dimensional torus $\mathbb T^d$. Different from existing works based on regularisation of the nonlinear term $ f(u)\approx f^\varepsilon(u)=u\ln (|u| + \varepsilon )^2,$ we directly discretize the LogSE with the understanding $f(0)=0.$ Remarkably, in the time-splitting scheme, the solution flow map of the nonlinear part: $g(u)= u {\rm e}^{-{\rm} i t \ln|u|^{2}}$ has a higher regularity than $f(u)$ (which is not differentiable at $u=0$ but H\"older continuous), where $g(u)$ is Lipschitz continuous and possesses a certain fractional Sobolev regularity with index $0<s<1$. Accordingly, we can derive the $L^2$-error estimate: $O\big((\tau^{s/2} + N^{-s})\ln\! N\big)$ of the proposed scheme for the LogSE with low regularity solution $u\in C((0,T]; H^s( \mathbb{T}^d)\cap L^\infty( \mathbb{T}^d)).$ Moreover, we can show that the estimate holds for $s=1$ with more delicate analysis of the nonlinear term and the associated solution flow maps. Furthermore, we provide ample numerical results to demonstrate such a fractional-order convergence for initial data with low regularity. This work is the first one devoted to the analysis of splitting scheme for the LogSE without regularisation in the low regularity setting, as far as we can tell.
We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu$ is large, the error is $O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.
A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.
Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using the Fourier transform in time, we derive the corresponding Eulerian short-time propagator in frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose the time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.
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