6 research outputs found
A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation
In this paper, we combine the Smolyak technique for multi-dimensional
interpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensional
oscillatory integration, to obtain a new Filon-Clenshaw-Curtis-Smolyak (FCCS)
rule for oscillatory integrals with linear phase over the dimensional cube
. By combining stability and convergence estimates for the FCC rule
with error estimates for the Smolyak interpolation operator, we obtain an error
estimate for the FCCS rule, consisting of the product of a Smolyak-type error
estimate multiplied by a term that decreases with
, where is the wavenumber and is
the number of oscillatory dimensions. If all dimensions are oscillatory, a
higher negative power of appears in the estimate. As an application, we
consider the forward problem of uncertainty quantification (UQ) for a
one-space-dimensional Helmholtz problem with wavenumber and a random
heterogeneous refractive index, depending in an affine way on i.i.d.
uniform random variables. After applying a classical hybrid
numerical-asymptotic approximation, expectations of functionals of the solution
of this problem can be formulated as a sum of oscillatory integrals over
, which we compute using the FCCS rule. We give numerical results for
the FCCS rule and the UQ algorithm showing that accuracy improves when both
and the order of the rule increase. We also give results for dimension-adaptive
sparse grid FCCS quadrature showing its efficiency as dimension increases
Meta-DSP: A Meta-Learning Approach for Data-Driven Nonlinear Compensation in High-Speed Optical Fiber Systems
Non-linear effects in long-haul, high-speed optical fiber systems
significantly hinder channel capacity. While the Digital Backward Propagation
algorithm (DBP) with adaptive filter (ADF) can mitigate these effects, it
suffers from an overwhelming computational complexity. Recent solutions have
incorporated deep neural networks in a data-driven strategy to alleviate this
complexity in the DBP model. However, these models are often limited to a
specific symbol rate and channel number, necessitating retraining for different
settings, their performance declines significantly under high-speed and
high-power conditions. We introduce Meta-DSP, a novel data-driven nonlinear
compensation model based on meta-learning that processes multi-modal data
across diverse transmission rates, power levels, and channel numbers. This not
only enhances signal quality but also substantially reduces the complexity of
the nonlinear processing algorithm. Our model delivers a 0.7 dB increase in the
Q-factor over Electronic Dispersion Compensation (EDC), and compared to DBP, it
curtails computational complexity by a factor of ten while retaining comparable
performance. From the perspective of the entire signal processing system, the
core idea of Meta-DSP can be employed in any segment of the overall
communication system to enhance the model's scalability and generalization
performance. Our research substantiates Meta-DSP's proficiency in addressing
the critical parameters defining optical communication networks
A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized multiscale data-driven stochastic basis functions that give an optimal approximation property of the solution operator. Our method consists of the offline and online stages. In the offline stage, we construct the localized multiscale data-driven stochastic basis functions by solving an optimization problem. In the online stage, using our basis functions, we can efficiently solve multiscale elliptic PDEs with random coefficients with relatively small computational costs. Therefore, our method is very efficient in solving target problems with many different force functions. The convergence analysis of the proposed method is also presented and has been verified by the numerical simulations