More Like this

1. The Brown measure of the free multiplicative Brownian motion

   https://doi.org/10.1007/s00440-022-01142-z  

   Driver, Bruce K. ; Hall, Brian ; Kemp, Todd ( October 2022 , Probability Theory and Related Fields)

   The free multiplicative Brownian motion$$b_{t}$$$b_t$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$$\mathrm{GL}(N;C),$in the sense of$$*$$$\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$$b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$${\Sigma }_t$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$$W_t$on$$\overline{\Sigma }_{t},$$${\overline{\Sigma }}_t,$which is strictly positive and real analytic on$$\Sigma _{t}$$${\Sigma }_t$. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$$\begin{array}{c}{W}_t(r,\theta )=\frac{1}{r^2}{w}_t\left(\theta \right),\end{array}$where$$w_{t}$$$w_t$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$$u_t$is a “shadow” of the Brown measure of$$b_{t}$$$b_t$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

    

   more » « less
2. A simple analytic model for predicting the wicking velocity in micropillar arrays

   https://doi.org/10.1038/s41598-019-56361-7  

   Krishnan, Siva Rama ; Bal, John ; Putnam, Shawn A. ( December 2019 , Scientific Reports)

   Hemiwicking is the phenomena where a liquid wets a textured surface beyond its intrinsic wetting length due to capillary action and imbibition. In this work, we derive a simple analytical model for hemiwicking in micropillar arrays. The model is based on the combined effects of capillary action dictated by interfacial and intermolecular pressures gradients within the curved liquid meniscus and fluid drag from the pillars at ultra-low Reynolds numbers$${\boldsymbol{(}}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{7}}}{\boldsymbol{\lesssim }}{\bf{Re}}{\boldsymbol{\lesssim }}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{3}}}{\boldsymbol{)}}$$$\left({10}^{-7}\lesssim \mathrm{Re}\lesssim {10}^{-3}\right)$. Fluid drag is conceptualized via a critical Reynolds number:$${\bf{Re}}{\boldsymbol{=}}\frac{{{\bf{v}}}_{{\bf{0}}}{{\bf{x}}}_{{\bf{0}}}}{{\boldsymbol{\nu }}}$$$\mathrm{Re}=\frac{v_0{x}_0}{\nu }$, wherev₀corresponds to the maximum wetting speed on a flat, dry surface andx₀is the extension length of the liquid meniscus that drives the bulk fluid toward the adsorbed thin-film region. The model is validated with wicking experiments on different hemiwicking surfaces in conjunction withv₀andx₀measurements using Water$${\boldsymbol{(}}{{\bf{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{25}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{28}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$$\left(v_0\approx 2m/s,25\mu m\lesssim x_0\lesssim 28\mu m\right)$, viscous FC-70$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{0.3}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{18.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\boldsymbol{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{38.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$$\left(v_0\approx 0.3m/s,18.6\mu m\lesssim x_0\lesssim 38.6\mu m\right)$and lower viscosity Ethanol$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{1.2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{11.8}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{33.3}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$$\left(v_0\approx 1.2m/s,11.8\mu m\lesssim x_0\lesssim 33.3\mu m\right)$.

    

   more » « less
3. On the rigorous derivation of the incompressible Euler equation from Newton’s second law

   https://doi.org/10.1007/s11005-023-01630-w  

   Rosenzweig, Matthew ( January 2023 , Letters in Mathematical Physics)

   A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system ofNparticles interacting in$${\mathbb {T}}^d$$$T^d$,$$d\ge 2$$$d\ge 2$, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$$\lambda$in front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$$N^{-\theta }$for some$$\theta \in (0,1)$$$\theta \in (0,1)$, in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$$\lambda \sim N^{-1}$. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$$\theta \in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$N\rightarrow \infty $$$N\to \infty$. Han-Kwan and Iacobelli asked if their range for$$\theta $$$\theta$was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$$N\to \infty$for$$\theta \in (1-\frac{2}{d},1)$$$\theta \in (1-\frac{2}{d},1)$. Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta $$$\theta$. Additionally, we show that for$$\theta \le 1-\frac{2}{d}$$$\theta \le 1-\frac{2}{d}$, one cannot, in general, expect convergence in the modulated energy notion of distance.

    

   more » « less
4. Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

   https://doi.org/10.1007/s10107-022-01875-8  

   Zhang, Shixuan ; Sun, Xu Andy ( August 2022 , Mathematical Programming)

   In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$$(T+1)$-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$$\epsilon$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$$O\left({\left(\frac{2LT}{\epsilon }\right)}^d\right)$, and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$$O\left({\left(\frac{\mathrm{LT}}{4\epsilon }\right)}^d\right)$for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$$O\left({\left(\frac{\mathrm{LT}}{8\epsilon }\right)}^{d/2-1}\right)$for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$$\left(T\epsilon \right)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

    

   more » « less
5. Multidimensional WENO-AO Reconstructions Using a Simplified Smoothness Indicator and Applications to Conservation Laws

   https://doi.org/10.1007/s10915-023-02319-x  

   Huang, Chieh-Sen ; Arbogast, Todd ; Tian, Chenyu ( August 2023 , Journal of Scientific Computing)

   Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator$$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}-1}\sum _{j} ({\bar{u}}_{j} - {\bar{u}}_{m})^2$$${\sigma }^{\text{S}}=\frac{1}{N_{\text{S}}-1}{\sum }_j{({\overline{u}}_j-{\overline{u}}_m)}^2$, where$$N_{\textrm{S}}$$$N_{\text{S}}$is the number of mesh elements in the stencil,$${\bar{u}}_j$$${\overline{u}}_j$is the local function average over mesh elementj, and indexmgives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved.

    

   more » « less