arXiv subjects

Qiqi Wang

Publications and source records attributed to Qiqi Wang.

At least 37 records · Page 2Linked to original sources

arXiv2021

Aerodynamic Risk Assessment using Parametric, Three-Dimensional Unstructured, High-Fidelity CFD and Adaptive Sampling

We demonstrate an adaptive sampling approach for computing the probability of a rare event for a set of three-dimensional airplane geometries under various flight conditions. We develop a fully automated method to generate parameterized airplanes geometries and create volumetric mesh for viscous CFD solution. With the automatic geometry and meshing, we perform the adaptive sampling procedure to compute the probability of the rare event. We show that the computational cost of our adaptive sampling approach is hundreds of times lower than a brute-force Monte Carlo method.

cs.CE↗

arXiv2021

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory-driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.

math.DS↗

arXiv2021

An ergodic averaging method to differentiate covariant Lyapunov vectors

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of CLVs along their own directions. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale-Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the CLV self-derivatives with a statistical linear response formula.

nlin.CD↗

arXiv2021

Multi-Group Multicast Beamforming: Optimal Structure and Efficient Algorithms

This paper considers the multi-group multicast beamforming optimization problem, for which the optimal solution has been unknown due to the non-convex and NP-hard nature of the problem. By utilizing the successive convex approximation numerical method and Lagrangian duality, we obtain the optimal multicast beamforming solution structure for both the quality-of-service (QoS) problem and the max-min fair (MMF) problem. The optimal structure brings valuable insights into multicast beamforming: We show that the notion of uplink-downlink duality can be generalized to the multicast beamforming problem. The optimal multicast beamformer is a weighted MMSE filter based on a group-channel direction: a generalized version of the optimal downlink multi-user unicast beamformer. We also show that there is an inherent low-dimensional structure in the optimal multicast beamforming solution independent of the number of transmit antennas, leading to efficient numerical algorithm design, especially for systems with large antenna arrays. We propose efficient algorithms to compute the multicast beamformer based on the optimal beamforming structure. Through asymptotic analysis, we characterize the asymptotic behavior of the multicast beamformers as the number of antennas grows, and in turn, provide simple closed-form approximate multicast beamformers for both the QoS and MMF problems. This approximation offers practical multicast beamforming solutions with a near-optimal performance at very low computational complexity for large-scale antenna systems.

eess.SP↗

arXiv2021

Differentiating densities on smooth manifolds

Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. We highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.

math.NA↗

arXiv2021

Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarithmic SRB density along the unstable manifold. We develop a recursive formula for the density gradient that can be efficiently computed along trajectories, and demonstrate its use in determining the differentiability of statistics. Our numerical procedure is illustrated on low-dimensional chaotic systems whose statistics exhibit both smooth and rough regions in parameter space.

nlin.CD↗

arXiv2020

Adjoint-based trailing edge shape optimization of a transonic turbine vane using large eddy simulations

The shape of the trailing edge of a gas turbine nozzle guide vane has a significant effect on the downstream stagnation pressure loss and heat transfer over the surface of the vane. Traditionally, adjoint-based design optimization methods for turbomachinery components have used low-fidelity simulations like Reynolds averaged Navier-Stokes. To reliably capture the complex flow phenomena involved in turbulent flow over a turbine vane, high-fidelity simulations like large eddy simulation (LES) are required. In this paper, an adjoint-based trailing edge shape optimization using LES is performed to reduce pressure loss and heat transfer over the surface of the vane. The chaotic dynamics of turbulence limits the effectiveness of the adjoint method for long-time averaged objective functions computed from LES. A viscosity stabilized unsteady adjoint method is used to obtain gradients of the design objective function with reasonable accuracy. A gradient utilizing Bayesian optimization is used to robustly handle noise in the objective function and gradient evaluations. The trailing edge shape is parameterized using a linear combination of $5$ convex designs. Results from the optimization, performed on the supercomputer Mira, are compared with optimal designs generated using derivative-free design optimization of the same problem.

physics.flu-dyn↗

arXiv2020

On the probability of finding a nonphysical solution through shadowing

This paper proves that shadowing solutions can be almost surely nonphysical. This finding invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior. This theoretical finding has implications for many applications in which chaotic mechanics plays an important role. It suggests, for example, that we can control the climate through subtle perturbations. It also suggests that numerical simulations of chaotic dynamics, such as turbulent flows, may fail to predict the true long-term or statistical behavior.

nlin.CD↗

arXiv2020

Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity

In this computational paper, we perform sensitivity analysis of long-time (or ensemble) averages in the chaotic regime using the shadowing algorithm. We introduce automatic differentiation to eliminate the tangent/adjoint equation solvers used in the shadowing algorithm. In a gradient-based optimization, we use the computed shadowing sensitivity to minimize different long-time averaged functionals of a chaotic time-delayed system by optimal parameter selection. In combined state and parameter estimation for data assimilation, we use the computed sensitivity to predict the optimal trajectory given information from a model and data from measurements beyond the predictability time. The algorithms are applied to a thermoacoustic model. Because the computational framework is rather general, the techniques presented in this paper may be used for sensitivity analysis of ensemble averages, parameter optimization and data assimilation of other chaotic problems, where shadowing methods are applicable.

math.DS↗

arXiv2020

Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive explanation of the key components of the S3 algorithm, including the density gradient function.

nlin.CD↗

arXiv2020

Hierarchical Jacobi Iteration for Structured Matrices on GPUs using Shared Memory

High fidelity scientific simulations modeling physical phenomena typically require solving large linear systems of equations which result from discretization of a partial differential equation (PDE) by some numerical method. This step often takes a vast amount of computational time to complete, and therefore presents a bottleneck in simulation work. Solving these linear systems efficiently requires the use of massively parallel hardware with high computational throughput, as well as the development of algorithms which respect the memory hierarchy of these hardware architectures to achieve high memory bandwidth. In this paper, we present an algorithm to accelerate Jacobi iteration for solving structured problems on graphics processing units (GPUs) using a hierarchical approach in which multiple iterations are performed within on-chip shared memory every cycle. A domain decomposition style procedure is adopted in which the problem domain is partitioned into subdomains whose data is copied to the shared memory of each GPU block. Jacobi iterations are performed internally within each block's shared memory, avoiding the need to perform expensive global memory accesses every step. We test our algorithm on the linear systems arising from discretization of Poisson's equation in 1D and 2D, and observe speedup in convergence using our shared memory approach compared to a traditional Jacobi implementation which only uses global memory on the GPU. We observe a x8 speedup in convergence in the 1D problem and a nearly x6 speedup in the 2D case from the use of shared memory compared to a conventional GPU approach.

cs.MS↗

arXiv2020

A computable realization of Ruelle's formula for linear response of statistics in chaotic systems

We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points. The reformulation is based on splitting the overall sensitivity into that to stable and unstable components of the perturbation. The unstable contribution to the sensitivity is regularized using ergodic properties and the hyperbolic structure of the dynamics. Numerical examples of uniformly hyperbolic attractors are used to validate the S3 formula against a naïve finite-difference calculation; sensitivities match closely, with far fewer sample points required by S3.

math.DS↗

arXiv2020

A BeiDou Signal Acquisition Approach Using Variable Length Data Accumulation based on Signal Delay and Multiplication

The secondary modulation with the NeumannHoffman code increases the possibility of bit sign transition. Unlike other GNSS signals, there is no pilot component for synchronization in BeiDou B1/B3 signals, which increases the complexity in acquisition. A previous study has shown that the delay and multiplication (DAM) method is able to eliminate the bit sign transition problem, but it only applies to pretty strong signals. In this paper, a DAM-based BeiDou signal acquisition approach, called variable length data accumulation (VLDA), is proposed to acquire weak satellite signals. Firstly, the performance of DAM method versus the different delays is analyzed. The DAM operation not only eliminates bit sign transition, but it also increases noise power. Secondly, long-term signal is periodically accumulated to improve signal intensity in order to acquire weak signals. While considering the Doppler frequency shift of ranging codes, the signal length must be compensated before accumulating long-term signal. Finally, the fast-Fourier-transform based parallel code phase algorithm are used for acquisition. The simulation results indicate that the proposed VLDA method has better acquisition sensitivity than traditional non-coherent integration method under the same calculation amount. The VLDA method only requires approximately 27.5% of calculations to achieve the same acquisition sensitivity (35 dBHz). What is more, the actual experimental results verify the feasibility of the VLDA method. It can be concluded that the proposed approach is an effective and feasible method for solving the bit sign transition problem.

eess.SP↗

arXiv2019

Feasibility analysis of ensemble sensitivity computation in turbulent flows

In chaotic systems, such as turbulent flows, the solutions to tangent and adjoint equations exhibit an unbounded growth in their norms. This behavior renders the instantaneous tangent and adjoint solutions unusable for sensitivity analysis. The Lea-Allen-Haine ensemble sensitivity (ES) estimates provide a way of computing meaningful sensitivities in chaotic systems by utilizing tangent/adjoint solutions over short trajectories. In this paper, we analyze the feasibility of ES computations under optimistic mathematical assumptions on the flow dynamics. Furthermore, we estimate upper bounds on the rate of convergence of the ES method in numerical simulations of turbulent flow. Even at the optimistic upper bound, the ES method is computationally intractable in each of the numerical examples considered.

physics.comp-ph↗

arXiv2019

Sensitivity computation of statistically stationary quantities in turbulent flows

It is well-known that linearized perturbation methods for sensitivity analysis, such as tangent or adjoint equation-based, finite difference and automatic differentiation are not suitable for turbulent flows. The reason is that turbulent flows exhibit chaotic dynamics, leading to the norm of an infinitesimal perturbation to the state growing exponentially in time. As a result, these conventional methods cannot be used to compute the derivatives of long-time averaged quantities to control or design inputs. The ensemble-based approaches and shadowing-based approaches to circumvent the problems of the conventional methods in chaotic systems, also suffer from computational impracticality and lack of consistency guarantees, respectively. We introduce the space-split sensitivity, or the S3 algorithm, which is a Monte-Carlo approach to the chaotic sensitivity computation problem. In this work, we derive the S3 algorithm under simplifying assumptions on the dynamics and present a numerical validation on a low-dimensional example of chaos.

nlin.CD↗

arXiv2019

Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)

We present the Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS) algorithm for computing sensitivities of long-time averaged quantities in chaotic dynamical systems. FD-NILSS does not require tangent solvers, and can be implemented with little modification to existing numerical simulation software. We also give a formula for solving the least-squares problem in FD-NILSS, which can be applied in NILSS as well. Finally, we apply FD-NILSS for sensitivity analysis of a chaotic flow over a 3-D cylinder at Reynolds number 525, where FD-NILSS computes accurate sensitivities and the computational cost is in the same order as the numerical simulation.

physics.comp-ph↗

arXiv2019

Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NILSS)

This paper develops the non-intrusive formulation of the Least-squares shadowing (LSS) method, for computing the sensitivity of long-time averaged objectives in chaotic dynamical systems. This non-intrusive formulation constrains the computation to only the unstable subspace, greatly reducing the cost of LSS for many problems; moreover, it reparametrizes the LSS problem, requiring only minor modifications to existing tangent solvers. NILSS is demonstrated on a chaotic flow over a backward-facing step simulated with a mesh of 12e3 cells.

physics.comp-ph↗

arXiv2019

Nanophotonic Media for Artificial Neural Inference

We show optical waves passing through a nanophotonic medium can perform artificial neural computing. Complex information, is encoded in the wave front of an input light. The medium transforms the wave front to realize sophisticated computing tasks such as image recognition. At the output, the optical energy is concentrated to well-defined locations, which for example can be interpreted as the identity of the object in the image. These computing media can be as small as tens of wavelengths and offer ultra-high computing density. They exploit sub-wavelength scatterers to realize complex input output mapping beyond the capabilities of traditional nanophotonic devices.

physics.optics↗