arXiv subjects

Yilin Song

Publications and source records attributed to Yilin Song.

31 records · Page 2Linked to original sources

arXiv2025

On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$

In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three.

math.AP↗

arXiv2025

Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation $iu_t + Δ_Ωu = |u|^αu$ in the exterior domain $Ω$ of a smooth, compact and strictly convex obstacle in $\mathbb{R}^3$. It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^3))$ with $s_c := \frac{3}{2} - \frac{2}α \in (0, \frac{3}{2})$, then $u$ is global and scatters. In this paper, assuming that this conjecture holds, we prove that if $u$ is a solution to the nonlinear Schrödinger equation in exterior domain $Ω$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(Ω))$ with $s_c \in \left[\frac{1}{2}, \frac{3}{2}\right)$, then $u$ is global and scatters. The proof of the main results relies on the concentration-compactness/rigidity argument of Kenig and Merle [Invent. Math. {\bf 166} (2006)]. The main difficulty is to construct minimal counterexamples when the scaling and translation invariance breakdown on $Ω$. To achieve this, two key ingredients are required. First, we adopt the approach of Killip, Visan, and Zhang [Amer. J. Math. {\bf 138} (2016)] to derive the linear profile decomposition for the linear propagator $e^{itΔ_Ω}$ in $\dot{H}^{s_c}(Ω)$. The second ingredient is the embedding of the nonlinear profiles. More precisely, we need to demonstrate that nonlinear solutions in the limiting geometries, which exhibit global spacetime bounds, can be embedded back into $Ω$. Finally, to rule out the minimal counterexamples, we will establish long-time Strichartz estimates for the exterior domain NLS, along with spatially localized and frequency-localized Morawetz estimates.

math.AP↗

arXiv2024

Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on $\Bbb T^3$

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global.

math.AP↗

arXiv2024

On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement

In this article, we study the scattering theory for the two dimensional defocusing quintic nonlinear Schrödinger equation(NLS) with partial harmonic oscillator which is given by \begin{align}\label{NLS-abstract} \begin{cases}\tag{PHNLS} i\partial_tu+(\partial_{x_1}^2+\partial_{x_2}^2)u-x_2^2u=|u|^4u,&(t,x_1,x_2)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R},\\ u(0,x_1,x_2)=u_0(x_1,x_2). \end{cases} \end{align} First, we establish the linear profile decomposition for the Schrödinger operator $e^{it(\partial_{x_1}^2+\partial_{x_2}^2-x_2^2)}$ by utilizing the classical linear profile decomposition associated with the Schrödinger equation in $L^2(\mathbb{R})$. Then, applying the normal form technique, we approximate the nonlinear profiles using solutions of the new-type quintic dispersive continuous resonant (DCR) system. This allows us to employ the concentration-compactness/rigidity argument introduced by Kenig and Merle in our setting and prove scattering for equation (PHNLS) in the weighted Sobolev space. The second part of this paper is dedicated to proving the scattering theory for this mass-critical (DCR) system. Inspired by Dodson's seminal work [B. Dodson, Amer. J. Math. 138 (2016), 531-569], we develop long-time Strichartz estimates associated with the spectral projection operator $Π_n$, along with low-frequency localized Morawetz estimates, to address the challenges posed by the Galilean transformation and spatial translation.

math.AP↗

arXiv2024

Adjusting for Incomplete Baseline Covariates in Randomized Controlled Trials: A Cross-World Imputation Framework

In randomized controlled trials, adjusting for baseline covariates is often applied to improve the precision of treatment effect estimation. However, missingness in covariates is common. Recently, Zhao & Ding (2022) studied two simple strategies, the single imputation method and missingness indicator method (MIM), to deal with missing covariates, and showed that both methods can provide efficiency gain. To better understand and compare these two strategies, we propose and investigate a novel imputation framework termed cross-world imputation (CWI), which includes single imputation and MIM as special cases. Through the lens of CWI, we show that MIM implicitly searches for the optimal CWI values and thus achieves optimal efficiency. We also derive conditions under which the single imputation method, by searching for the optimal single imputation values, can achieve the same efficiency as the MIM.

stat.ME↗

arXiv2024

Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds

In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) $ i\partial_tu+Δ_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$.

math.AP↗

arXiv2024

Dynamics of the combined nonlinear Schrödinger equation with inverse-square potential

We consider the long-time dynamics of focusing energy-critical Schrödinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$ \begin{equation}\label{NLS-ab} \begin{cases} i\partial_tu-\mathcal{L}_au=-|u|^{\frac{4}{d-2}}u+|u|^{\frac{4}{d-1}}u, \quad (t,x)\in\mathbb{R}\times\mathbb{R}^d,\tag{CNLS$_a$},\\ u(0,x)=u_0(x)\in H^1_a(\mathbb{R}^d), \end{cases} \end{equation} where $\mathcal{L}_a=-Δ+a|x|^{-2}$ and the energy is below and equal to the threshold $m_a$, which is given by the ground state $W_a$ satisfying $\mathcal{L}_aW_a=|W_a|^{\frac{4}{d-2}}W_a$. When the energy is below the threshold, we utilize the concentration-compactness argument as well as the variatonal analysis to characterize the scattering and blow-up region. When the energy is equal to the threshold, we use the modulation analysis associated to the equation \eqref{NLS-ab} to classify the dynamics of $H_a^1$-solution. In both regimes of scattering results, we do not need the radial assumption in $d=4,5$. Our result generalizes the scattering results of [31-33] and [3] in the setting of standard combined NLS.

math.AP↗

arXiv2023

Online Map Vectorization for Autonomous Driving: A Rasterization Perspective

Vectorized high-definition (HD) map is essential for autonomous driving, providing detailed and precise environmental information for advanced perception and planning. However, current map vectorization methods often exhibit deviations, and the existing evaluation metric for map vectorization lacks sufficient sensitivity to detect these deviations. To address these limitations, we propose integrating the philosophy of rasterization into map vectorization. Specifically, we introduce a new rasterization-based evaluation metric, which has superior sensitivity and is better suited to real-world autonomous driving scenarios. Furthermore, we propose MapVR (Map Vectorization via Rasterization), a novel framework that applies differentiable rasterization to vectorized outputs and then performs precise and geometry-aware supervision on rasterized HD maps. Notably, MapVR designs tailored rasterization strategies for various geometric shapes, enabling effective adaptation to a wide range of map elements. Experiments show that incorporating rasterization into map vectorization greatly enhances performance with no extra computational cost during inference, leading to more accurate map perception and ultimately promoting safer autonomous driving.

cs.CV↗

arXiv2021

Remote Sensing Image Super-resolution and Object Detection: Benchmark and State of the Art

For the past two decades, there have been significant efforts to develop methods for object detection in Remote Sensing (RS) images. In most cases, the datasets for small object detection in remote sensing images are inadequate. Many researchers used scene classification datasets for object detection, which has its limitations; for example, the large-sized objects outnumber the small objects in object categories. Thus, they lack diversity; this further affects the detection performance of small object detectors in RS images. This paper reviews current datasets and object detection methods (deep learning-based) for remote sensing images. We also propose a large-scale, publicly available benchmark Remote Sensing Super-resolution Object Detection (RSSOD) dataset. The RSSOD dataset consists of 1,759 hand-annotated images with 22,091 instances of very high resolution (VHR) images with a spatial resolution of ~0.05 m. There are five classes with varying frequencies of labels per class. The image patches are extracted from satellite images, including real image distortions such as tangential scale distortion and skew distortion. We also propose a novel Multi-class Cyclic super-resolution Generative adversarial network with Residual feature aggregation (MCGR) and auxiliary YOLOv5 detector to benchmark image super-resolution-based object detection and compare with the existing state-of-the-art methods based on image super-resolution (SR). The proposed MCGR achieved state-of-the-art performance for image SR with an improvement of 1.2dB PSNR compared to the current state-of-the-art NLSN method. MCGR achieved best object detection mAPs of 0.758, 0.881, 0.841, and 0.983, respectively, for five-class, four-class, two-class, and single classes, respectively surpassing the performance of the state-of-the-art object detectors YOLOv5, EfficientDet, Faster RCNN, SSD, and RetinaNet.

cs.CV↗

arXiv2018

Diversity encouraged learning of unsupervised LSTM ensemble for neural activity video prediction

Being able to predict the neural signal in the near future from the current and previous observations has the potential to enable real-time responsive brain stimulation to suppress seizures. We have investigated how to use an auto-encoder model consisting of LSTM cells for such prediction. Recog- nizing that there exist multiple activity pattern clusters, we have further explored to train an ensemble of LSTM mod- els so that each model can specialize in modeling certain neural activities, without explicitly clustering the training data. We train the ensemble using an ensemble-awareness loss, which jointly solves the model assignment problem and the error minimization problem. During training, for each training sequence, only the model that has the lowest recon- struction and prediction error is updated. Intrinsically such a loss function enables each LTSM model to be adapted to a subset of the training sequences that share similar dynamic behavior. We demonstrate this can be trained in an end- to-end manner and achieve significant accuracy in neural activity prediction.

cs.CV↗

arXiv2018

Multi Resolution LSTM For Long Term Prediction In Neural Activity Video

Epileptic seizures are caused by abnormal, overly syn- chronized, electrical activity in the brain. The abnor- mal electrical activity manifests as waves, propagating across the brain. Accurate prediction of the propagation velocity and direction of these waves could enable real- time responsive brain stimulation to suppress or prevent the seizures entirely. However, this problem is very chal- lenging because the algorithm must be able to predict the neural signals in a sufficiently long time horizon to allow enough time for medical intervention. We consider how to accomplish long term prediction using a LSTM network. To alleviate the vanishing gradient problem, we propose two encoder-decoder-predictor structures, both using multi-resolution representation. The novel LSTM structure with multi-resolution layers could significantly outperform the single-resolution benchmark with similar number of parameters. To overcome the blurring effect associated with video prediction in the pixel domain using standard mean square error (MSE) loss, we use energy- based adversarial training to improve the long-term pre- diction. We demonstrate and analyze how a discriminative model with an encoder-decoder structure using 3D CNN model improves long term prediction.

cs.CV↗

arXiv2018

Pixel-wise object tracking

In this paper, we propose a novel pixel-wise visual object tracking framework that can track any anonymous object in a noisy background. The framework consists of two submodels, a global attention model and a local segmentation model. The global model generates a region of interests (ROI) that the object may lie in the new frame based on the past object segmentation maps, while the local model segments the new image in the ROI. Each model uses a LSTM structure to model the temporal dynamics of the motion and appearance, respectively. To circumvent the dependency of the training data between the two models, we use an iterative update strategy. Once the models are trained, there is no need to refine them to track specific objects, making our method efficient compared to online learning approaches. We demonstrate our real time pixel-wise object tracking framework on a challenging VOT dataset

cs.CV↗

arXiv2017

Unsupervised Learning of Spike Patterns for Seizure Detection and Wavefront Estimation of High Resolution Micro Electrocorticographic (μECoG) Data

For the past few years, we have developed flexible, active, multiplexed recording devices for high resolution recording over large, clinically relevant areas in the brain. While this technology has enabled a much higher-resolution view of the electrical activity of the brain, the analytical methods to process, categorize and respond to the huge volumes of seizure data produced by these devices have not yet been developed. In this work we proposed an unsupervised learning framework for spike analysis, which by itself reveals spike pattern. By applying advanced video processing techniques for separating a multi-channel recording into individual spike segments, unfolding the spike segments manifold and identifying natural clusters for spike patterns, we are able to find the common spike motion patterns. And we further explored using these patterns for more interesting and practical problems as seizure prediction and spike wavefront prediction. These methods have been applied to in-vivo feline seizure recordings and yielded promising results.

q-bio.NC↗