278 research outputs found
An explicit and symmetric exponential wave integrator for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
We propose and analyze a novel symmetric exponential wave integrator (sEWI)
for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential
and typical power-type nonlinearity of the form ,
where is the density with the wave function and is the exponent of the nonlinearity. The sEWI is explicit and
stable under a time step size restriction independent of the mesh size. We
rigorously establish error estimates of the sEWI under various regularity
assumptions on potential and nonlinearity. For "good" potential and
nonlinearity (-potential and ), we establish an optimal
second-order error bound in -norm. For low regularity potential and
nonlinearity (-potential and ), we obtain a first-order
-norm error bound accompanied with a uniform -norm bound of the
numerical solution. Moreover, adopting a new technique of \textit{regularity
compensation oscillation} (RCO) to analyze error cancellation, for some
non-resonant time steps, the optimal second-order -norm error bound is
proved under a weaker assumption on the nonlinearity: . For
all the cases, we also present corresponding fractional order error bounds in
-norm, which is the natural norm in terms of energy. Extensive numerical
results are reported to confirm our error estimates and to demonstrate the
superiority of the sEWI, including much weaker regularity requirements on
potential and nonlinearity, and excellent long-time behavior with
near-conservation of mass and energy.Comment: 35 pages, 10 figure
On the Inversion of High Energy Proton
Inversion of the K-fold stochastic autoconvolution integral equation is an
elementary nonlinear problem, yet there are no de facto methods to solve it
with finite statistics. To fix this problem, we introduce a novel inverse
algorithm based on a combination of minimization of relative entropy, the Fast
Fourier Transform and a recursive version of Efron's bootstrap. This gives us
power to obtain new perspectives on non-perturbative high energy QCD, such as
probing the ab initio principles underlying the approximately negative binomial
distributions of observed charged particle final state multiplicities, related
to multiparton interactions, the fluctuating structure and profile of proton
and diffraction. As a proof-of-concept, we apply the algorithm to ALICE
proton-proton charged particle multiplicity measurements done at different
center-of-mass energies and fiducial pseudorapidity intervals at the LHC,
available on HEPData. A strong double peak structure emerges from the
inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios,
2D
The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition
We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In the general case in dimension d = 1, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in L^2) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions
Sharp asymptotics for Toeplitz determinants, fluctuations and the gaussian free field on a Riemann surface
We consider canonical determinantal random point processes with N particles
on a compact Riemann surface X defined with respect to the constant curvature
metric. In the higher genus (hyperbolic) cases these point processes may be
defined in terms of automorphic forms. We establish strong exponential
concentration of measure type properties involving Dirichlet norms of linear
statistics. This gives an optimal Central Limit Theorem (CLT), saying that the
fluctuations of the corresponding empirical measures converge, in the large N
limt, towards the Laplacian of the Gaussian free field on X in the strongest
possible sense. The CLT is also shown to be equivalent to a new sharp strong
Szeg\"o type theorem for Toeplitz determinants in this context. One of the
ingredients in the proofs are new Bergman kernel asymptotics providing
exponentially small error terms in a constant curvature setting.Comment: v1: 15 pages, v2: 21 pages, added new Bergman kernel asymptotics with
exponentially small error term
Diffusion Schr\"odinger Bridge with Applications to Score-Based Generative Modeling
Progressively applying Gaussian noise transforms complex data distributions
to approximately Gaussian. Reversing this dynamic defines a generative model.
When the forward noising process is given by a Stochastic Differential Equation
(SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the
associated reverse-time SDE may be estimated using score-matching. A limitation
of this approach is that the forward-time SDE must be run for a sufficiently
long time for the final distribution to be approximately Gaussian. In contrast,
solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized
optimal transport problem on path spaces, yields diffusions which generate
samples from the data distribution in finite time. We present Diffusion SB
(DSB), an original approximation of the Iterative Proportional Fitting (IPF)
procedure to solve the SB problem, and provide theoretical analysis along with
generative modeling experiments. The first DSB iteration recovers the
methodology proposed by Song et al. (2021), with the flexibility of using
shorter time intervals, as subsequent DSB iterations reduce the discrepancy
between the final-time marginal of the forward (resp. backward) SDE with
respect to the prior (resp. data) distribution. Beyond generative modeling, DSB
offers a widely applicable computational optimal transport tool as the
continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi,
2013).Comment: 58 pages, 18 figures (correction of Proposition 5
Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme
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