Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.Comment: 27 pages, 9 figures; Accepted by SCIENCE CHINA Mathematics on January 17, 202

Nonmonotone local minimax methods for finding multiple saddle points

In this paper, by designing a normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai--Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics on January 3, 202

Mesoscopic Ordering of Water Dynamics in Self-Assembled Materials Revealed by Transient VSFG Microscopy

We report, for the first time, observations of mesoscopically homogeneous but macroscopically heterogenous water dynamics in self-assembled materials by a new, spatially resolved infrared (IR) pump vibrational sum frequency generation (VSFG) probe microscope. Using this new technique, we spatially resolved dynamics of water bounded by host-guest, self-assembled sheets comprised of sodium dodecyl sulfate (SDS) and β-cyclodextrin (β-CD). We found that the strong hydrogen-bond interactions between β-CD and nearby water not only template nearby water networks to adopt the chirality of β-CD, but also allow resonant energy transfer from β-CD to nearby water. More interestingly, the resonant energy transfer dynamics are heterogeneous among domains, while remaining uniform within domains. This surprising result indicates that the water near self-assembled materials can be templated uniformly across micron domains. Because SDS@2β-CD is a synthetic analogue that parallels the morphology, rigidity and crystallinity of protein assemblies, similar mesoscopic ordering of water structure and dynamics could also exist in biological soft materials. The advancement of adding spatial resolution to ultrafast molecular vibrational spectroscopy opens a new way to probe mesoscopic molecular structure ordering and relaxation dynamics in biological systems, and hydro-responsive self-assembly materials for micro-optics and electronics. </b

Spatially dependent H-bond dynamics at interfaces of water/biomimetic self-assembled lattice materials

Understanding hydrogen-bond interactions in self-assembled lattice materials is crucial for preparing such materials, but the role of hydrogen bonds (H bonds) remains unclear. To gain insight into H-bond interactions at the materials' intrinsic spatial scale, we investigated ultrafast H-bond dynamics between water and biomimetic self-assembled lattice materials (composed of sodium dodecyl sulfate and β-cyclodextrin) in a spatially resolved manner. To accomplish this, we developed an infrared pump, vibrational sum-frequency generation (VSFG) probe hyperspectral microscope. With this hyperspectral imaging method, we were able to observe that the primary and secondary OH groups of β-cyclodextrin exhibit markedly different dynamics, suggesting distinct H-bond environments, despite being separated by only a few angstroms. We also observed another ultrafast dynamic reflecting a weakening and restoring of H bonds between bound water and the secondary OH of β-cyclodextrin, which exhibited spatial uniformity within self-assembled domains, but heterogeneity between domains. The restoration dynamics further suggest heterogeneous hydration among the self-assembly domains. The ultrafast nature and meso- and microscopic ordering of H-bond dynamics could contribute to the flexibility and crystallinity of the material--two critically important factors for crystalline lattice self-assemblies--shedding light on engineering intermolecular interactions for self-assembled lattice materials

Nanoscale optical pulse limiter enabled by refractory metallic quantum wells.

The past several decades have witnessed rapid development of high-intensity, ultrashort pulse lasers, enabling deeper laboratory investigation of nonlinear optics, plasma physics, and quantum science and technology than previously possible. Naturally, with their increasing use, the risk of accidental damage to optical detection systems rises commensurately. Thus, various optical limiting mechanisms and devices have been proposed. However, restricted by the weak optical nonlinearity of natural materials, state-of-the-art optical limiters rely on bulk liquid or solid media, operating in the transmission mode. Device miniaturization becomes complicated with these designs while maintaining superior integrability and controllability. Here, we demonstrate a reflection-mode pulse limiter (sub-100 nm) using nanoscale refractory films made of Al2O3/TiN/Al2O3 metallic quantum wells (MQWs), which provide large and ultrafast Kerr-type optical nonlinearities due to the quantum size effect of the MQW. Functional multilayers consisting of these MQWs could find important applications in nanophotonics, nonlinear optics, and meta-optics

The AAA-ATPase Yta4/ATAD1 interacts with the mitochondrial divisome to inhibit mitochondrial fission.

Mitochondria are in a constant balance of fusion and fission. Excessive fission or deficient fusion leads to mitochondrial fragmentation, causing mitochondrial dysfunction and physiological disorders. How the cell prevents excessive fission of mitochondria is not well understood. Here, we report that the fission yeast AAA-ATPase Yta4, which is the homolog of budding yeast Msp1 responsible for clearing mistargeted tail-anchored (TA) proteins on mitochondria, plays a critical role in preventing excessive mitochondrial fission. The absence of Yta4 leads to mild mitochondrial fragmentation in a Dnm1-dependent manner but severe mitochondrial fragmentation upon induction of mitochondrial depolarization. Overexpression of Yta4 delocalizes the receptor proteins of Dnm1, i.e., Fis1 (a TA protein) and Mdv1 (the bridging protein between Fis1 and Dnm1), from mitochondria and reduces the localization of Dnm1 to mitochondria. The effect of Yta4 overexpression on Fis1 and Mdv1, but not Dnm1, depends on the ATPase and translocase activities of Yta4. Moreover, Yta4 interacts with Dnm1, Mdv1, and Fis1. In addition, Yta4 competes with Dnm1 for binding Mdv1 and decreases the affinity of Dnm1 for GTP and inhibits Dnm1 assembly in vitro. These findings suggest a model, in which Yta4 inhibits mitochondrial fission by inhibiting the function of the mitochondrial divisome composed of Fis1, Mdv1, and Dnm1. Therefore, the present work reveals an uncharacterized molecular mechanism underlying the inhibition of mitochondrial fission

The individual numerical values used to generate the summary data graphs displayed in the main and supplementary figures.

The individual numerical values used to generate the summary data graphs displayed in the main and supplementary figures.</p

(related to Fig 1F).

Mitochondrial dynamics in WT cells. Images were acquired every half a minute with a spinning-disk microscope. (AVI)</p