A Parallel-in-Time Preconditioner for the Schur Complement of Parabolic Optimal Control Problems

For optimal control problems constrained by a initial-valued parabolic PDE, we have to solve a large scale saddle point algebraic system consisting of considering the discrete space and time points all together. A popular strategy to handle such a system is the Krylov subspace method, for which an efficient preconditioner plays a crucial role. The matching-Schur-complement preconditioner has been extensively studied in literature and the implementation of this preconditioner lies in solving the underlying PDEs twice, sequentially in time. In this paper, we propose a new preconditioner for the Schur complement, which can be used parallel-in-time (PinT) via the so called diagonalization technique. We show that the eigenvalues of the preconditioned matrix are low and upper bounded by positive constants independent of matrix size and the regularization parameter. The uniform boundedness of the eigenvalues leads to an optimal linear convergence rate of conjugate gradient solver for the preconditioned Schur complement system. To the best of our knowledge, it is the first time to have an optimal convergence analysis for a PinT preconditioning technique of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner is robust with respect to the discretization step-sizes and the regularization parameter

A block α\alpha-circulant based preconditioned MINRES method for wave equations

In this work, we propose an absolute value block $\alpha$-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block $\alpha$-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block $\alpha$-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block $\alpha$-circulant preconditioner, which fills the gap between block $\alpha$-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen $\alpha$ the MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved

SOA pattern effect mitigation by neural network based pre-equalizer for 50G PON

Semiconductor optical amplifier (SOA) is widely used for power amplification in O-band, particularly for passive optical networks (PONs) which can greatly benefit its advantages of simple structure, low power consumption and integrability with photonics circuits. However, the annoying nonlinear pattern effect degrades system performance when the SOA is needed as a pre-amplifier in PONs. Conventional solutions for pattern effect mitigation are either based on optical filtering or gain clamping. They are not simple or sufficiently flexible for practical deployment. Neural network (NN) has been demonstrated for impairment compensation in optical communications thanks to its powerful nonlinear fitting ability. In this paper, for the first time, NN-based equalizer is proposed to mitigate the SOA pattern effect for 50G PON with intensity modulation and direct detection. The experimental results confirm that the NN-based equalizer can effectively mitigate the SOA nonlinear pattern effect and significantly improve the dynamic range of receiver, achieving 29-dB power budget with the FEC limit at 1e-2. Moreover, the well-trained NN model in the receiver side can be directly placed at the transmitter in the optical line terminal to pre-equalize the signal for transmission so as to simplify digital signal processing in the optical network unit

Spectral analysis for preconditioning of multi-dimensional Riesz fractional diffusion equations

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners

The Biological Behaviors of Rat Dermal Fibroblasts Can Be Inhibited by High Levels of MMP9

Aims. To explore the effects of the high expression of MMP9 on biological behaviors of fibroblasts. Methods. High glucose and hyperhomocysteine were used to induce MMP9 expression in skin fibroblasts. Cell proliferation was detected by flow cytometry and cell viability by CCK-8. ELISA assay was used to detect collagen (hydroxyproline) secretion. Scratch test was employed to evaluate horizontal migration of cells and transwell method to evaluate vertical migration of cells. Results. The mRNA and protein expressions of MMP9 and its protease activity were significantly higher in cells treated with high glucose and hyperhomocysteine than those in control group. At the same time, the S-phase cell ratio, proliferation index, cell viability, collagen (hydroxyproline) secretion, horizontal migration rate, and the number of vertical migration cells decreased in high-glucose and hyperhomocysteine-treated group. Tissue inhibitor of metalloproteinase 1 (TIMP1), which inhibits the activity of MMP9, recovered the above biological behaviors. Conclusions. High expression of MMP9 in skin fibroblasts could be induced by cultureing in high glucose and hyperhomocysteine medium, which inhibited cell biological behaviors. Inhibitions could be reversed by TIMP1. The findings suggested that MMP9 deters the healing of diabetic foot ulcers by inhibiting the biological behaviors of fibroblasts

Stress dependent gas-water relative permeability in gas hydrates: A theoretical model

        Research activities are currently being conducted to study multiphase flow in hydrate-bearing sediments (HBS). In this study, in view of the assumption that hydrates are evenly distributed in HBS with two major hydrate-growth patterns, i.e., pore filling hydrates (PF hydrates), wall coating hydrates (WC hydrates) and a combination of the two, a theoretical relative  permeability model is proposed for gas-water flow through HBS. Besides, in this proposed model, the change in pore structure (e.g., pore radius) of HBS due to effective stress is taken into account. Then, model validation is performed by comparing the predicted results from the derived model with that from the existing model and test data. By setting the value of hydrate saturation to zero, our derived model can be reducible to the existing model, which demonstrates that the existing model is a special case of our model. The results reveal that, under the same saturation, relative permeability to water Krw (or gas Krg) in PF hydrates is smaller than that in WC hydrates. Moreover, the morphological characteristics of relative permeability curve (relative permeability versus gas saturation) for WC hydrate and PF hydrate are different.Cited as: Lei, G., Liao, Q., Chen, W., Lin, Q., Zhang, L., Xue, L. Stress dependent gas-water relative permeability in gas hydrates: A theoretical model. Advances in Geo-Energy Research, 2020, 4(3): 326-338, doi: 10.46690/ager.2020.03.1

Audit and Improve Robustness of Private Neural Networks on Encrypted Data

Performing neural network inference on encrypted data without decryption is one popular method to enable privacy-preserving neural networks (PNet) as a service. Compared with regular neural networks deployed for machine-learning-as-a-service, PNet requires additional encoding, e.g., quantized-precision numbers, and polynomial activation. Encrypted input also introduces novel challenges such as adversarial robustness and security. To the best of our knowledge, we are the first to study questions including (i) Whether PNet is more robust against adversarial inputs than regular neural networks? (ii) How to design a robust PNet given the encrypted input without decryption? We propose PNet-Attack to generate black-box adversarial examples that can successfully attack PNet in both target and untarget manners. The attack results show that PNet robustness against adversarial inputs needs to be improved. This is not a trivial task because the PNet model owner does not have access to the plaintext of the input values, which prevents the application of existing detection and defense methods such as input tuning, model normalization, and adversarial training. To tackle this challenge, we propose a new fast and accurate noise insertion method, called RPNet, to design Robust and Private Neural Networks. Our comprehensive experiments show that PNet-Attack reduces at least $2.5\times$ queries than prior works. We theoretically analyze our RPNet methods and demonstrate that RPNet can decrease $\sim 91.88\%$ attack success rate.Comment: 10 pages, 10 figure

Experimental Simulation of Symmetry-Protected Higher-Order Exceptional Points with Single Photons

Exceptional points (EPs) of non-Hermitian (NH) systems have recently attracted increasing attention due to their rich phenomenology and intriguing applications. Compared to the predominantly studied second-order EPs, higher-order EPs have been assumed to play a much less prominent role because they generically require the tuning of more parameters. Here we experimentally simulate two-dimensional topological NH band structures using single-photon interferometry, and observe topologically stable third-order EPs obtained by tuning only two real parameters in the presence of symmetry. In particular, we explore how different symmetries stabilize qualitatively different third-order EPs: the parity-time symmetry leads to a generic cube-root dispersion, while a generalized chiral symmetry implies a square-root dispersion coexisting with a flat band. Additionally, we simulate fourfold degeneracies, composed of the non-defective twofold degeneracies and second-order EPs. Our work reveals the abundant and conceptually richer higher-order EPs protected by symmetries and offers a versatile platform for further research on topological NH systems.Comment: 15 pages, 9 figure