類別:

高電荷細長體附近之泊松--玻爾茲曼方程的邊界層解

2026-08-04T08:16:48Z

標題: 高電荷細長體附近之泊松--玻爾茲曼方程的邊界層解; Boundary Layer Solutions to Poisson-Boltzmann Equations near a Highly Charged Slender Body 作者: 秦安; An Qin 摘要: 帶電細長體浸泡於電解質溶液中時，會在其表面附近產生高度局部化的電場，並形成薄的電雙層結構。本論文研究一個高電荷直圓柱附近的非線性泊松–玻爾茲曼方程之圓柱狄利克雷問題。其中圓柱半徑為√ϵδ，且0 < ϵ ≪ 1。在圓柱內部，電位滿足具有常數電荷密度ρ/ϵ的泊松方程；在外部電解質區域，電位滿足非線性泊松–玻爾茲曼方程，其中離子電荷函數f 為光滑函數，並滿足f(0) = 0, f′(u) ≤ −C0 < 0. 利用徑向對稱性與伸縮變數R =r√ϵ, ϕϵ(R) = Φ(√ϵR),外部問題可化為徑向邊界層方程ϕ′′ϵ +1/R ϕ′ϵ + f(ϕϵ) = 0, δ < R < Rϵ,ϕϵ(δ) = p, ϕϵ(Rϵ) = 0,其中Rϵ = r0/√ϵ。本論文證明此徑向邊值問題的適定性，並推導解的符號性質、徑向單調性，以及其在(δ,∞) 上的極限邊界層輪廓。此外，我們證明指數衰減估計|ϕϵ(R)| ≤ |p|e−√C0(R−δ),此估計顯示物理變數中的電位集中於圓柱表面附近厚度為O(√ϵ) 的薄層內。內部問題可被顯式求解，其伸縮變數下的電位為ϕin(R) = −ρ/4 R^2 + p +ρ/4 δ^2.最後，數值結果展示了外部邊界層輪廓，並說明當pρ > 0 時，圓柱表面附近會出現反離子累積現象。此圓柱模型可作為未來研究曲線細長體、Stern 層跳躍條件與曲率修正效應的主導階近似框架。; Charged slender bodies immersed in electrolytes generate localized electric fields and form thin electric double layers near their surfaces. In this thesis, we study a cylindrical Dirichlet problem for a nonlinear Poisson–Boltzmann equation near a highly charged straight cylinder of radius √ϵδ, where 0 < ϵ ≪ 1. Inside the cylinder, the potential satisfies a Poisson equation with constant charge density ρ/ϵ; outside, it satisfies a nonlinear Poisson–Boltzmann equation with a smooth ionic charge function f such that f(0) = 0, f′(u) ≤ −C0 < 0. Using radial symmetry and the stretched variable R = r√ϵ, ϕϵ(R) = Φ(√ϵR), the exterior problem is reduced to ϕ′′ϵ +1/R ϕ′ϵ + f(ϕϵ) = 0, δ < R < Rϵ, ϕϵ(δ) = p, ϕϵ(Rϵ) = 0, where Rϵ = r0/√ϵ. We prove well-posedness of this radial boundary value problem, derive its sign and monotonicity properties, and identify the limiting boundary-layer profile on (δ,∞). We also establish the exponential decay estimate |ϕϵ(R)| ≤ |p|e^−√C0(R−δ), which shows that the physical potential is localized within an O(√ϵ) layer near the cylindrical surface. The interior problem is solved explicitly: ϕin(R) = −ρ/4 R^2 + p +ρ/4 δ^2. Finally, numerical solutions illustrate the boundary-layer profile and the counterion accumulation phenomenon when pρ > 0. This cylindrical model serves as a leading-order framework for future extensions involving curved filaments, Stern-layer jump conditions, and curvature-induced corrections.

高維度時間序列並帶有測量誤差模型之模型選擇

2026-07-22T12:07:58Z

標題: 高維度時間序列並帶有測量誤差模型之模型選擇; Model Selection for High-Dimensional Time Series Models with Measurement Errors 作者: Hsueh-Han Huang; 黃學涵摘要: We use a fast stepwise regression method, called orthogonal greedy algorithm (OGA) to select variables for high-dimensional time series model with measurement errors. Under a weak sparsity condition, we derive a convergence rate of OGA, which is expressed in terms of the number of iterations, the sample size and the order of the moment imposed on the error process. Under a strong sparsity condition, we develop a consistent model selection procedure using OGA and a high-dimensional information criterion.

高斯隨機投影下的快速近似奇異值分解

2026-07-22T12:44:47Z

標題: 高斯隨機投影下的快速近似奇異值分解; Fast Approximation for SVD via Gaussian Random Projections 作者: Sheng-Yao Huang; 黃聖堯摘要: 奇異值分解 (SVD) 是一個有名的矩陣分解的工具，但在矩陣的大小過大時將會計算得很久。Rokhlin et al. [1] 對快速 SVD 近似提供一個隨機化算法 (稱作 rSVD)。方法是首先先用高斯隨機投影將矩陣的行 (column) 或列 (row) 做一個縮減，然後再對這個叫低維度的子空間做 SVD。Chen et al. [2] 證明了 rSVD 的一致性 (consistency)，本篇論文對 rSVD 的一致性給一個新的證明，證明方法為從矩陣角度高斯分配去做。Chen et al. [2] 還提出了一個根據高斯隨機投影的迭代法，此方法叫做 iSVD。除了一致性的證明外，還給了一個對圖片做低維度的估計當作例子。從例子的結果來看，可以發現到 iSVD 的計算時間比 SVD 少了許多，但出來的結果卻很相似。最後給了一個 iSVD 的python code，code 根據 Kolmogorov-Nagumo-type average 來完成。; Singular value decomposition (SVD) is a popular tool for dimension re-duction. When the size of matrix is large, the computing load is heavy. Rokhlin et al [1] proposed a randomized algorithm for fast SVD approxi-mation (abbreviated as rSVD). Often Gaussian random projection is used to reduce the number of columns or rows, and next SVD is carried out in this lower-dimensional subspace. Chen et al. [2] proved the consistency of rSVD. In this paper, we give the rSVD consistency a new proof. Our new proof is based on matrix angular Gaussian distribution and is more instructive. Chen et al. [2] further proposed an integration method based on multiple random Gaussian projections, called iSVD. In addition to the new proof for consis-tency, we also provide an iSVD example for image low-rank approximation. From this example, we can see that the runtime of iSVD is less than the run-time of SVD without sacrificing much of accuracy. Finally, we provide a python code for iSVD, it is based on Kolmogorov-Nagumo-type average.

高效能周道積分法模擬表面電漿特徵值問題

2026-07-22T12:32:55Z

標題: 高效能周道積分法模擬表面電漿特徵值問題; An Efficient Contour Integral Based Eigensolver for Surface Plasmon Simulations 作者: Weichien Liao; 廖為謙摘要: 數值模擬是研究表面電漿特性的重要方法。本文以馬克士威方程式建模，將此方程用 K.S.Yee 提出的時域有限差分法進行離散化 (discretization)，接著將離散後的方程做適當的相似變換 (similarity transformation)，使原始的問題轉換為特徵值問題 (eigenvalue problem)。此特徵值問題的矩陣為非共軛對稱 (non-Hermitian) 的方陣，並且其特徵值分佈呈現高度叢集性。因此為克服現有方法在解此問題所需特定區域特徵值的困難，而發展了一個高效的周道積分法應用於求解此問題。此方法結合了周道積分、快速矩陣向量乘法以及高效的線性系統求解。由數據結果可驗證本文提出之方法能高效求解線性系統以及特徵值。; Numerical simulations play a significant role for studying the properties of surface plasmon. The surface plasmon problem is first modelled by the Maxwell equations, and the equations is then discretized by the widely-used Yee’s scheme. After applying certain similarity transformations to the discretized system, the original simulation problem becomes a clustered non-Hermitian eigenvalue problem. An efficient contour integral (CI) based eigensolver is developed to overcome the difficulties of applying current existing methods to solve eigenvalues in particular designated regions for this problem. This efficient method combines the contour integral, the fast matrix-vector multiplication and efficient linear system solving. The numerical results can show the efficiency of solving linear systems and eigenvalues with the efficient CI eigensolver.