This paper has two major objectives. The first objective is to give a new, natural extension of the classical calculus of variations to a stochastic setting. Most significantly, it appears that this is the first time in the literature that a random objective functional is used rather than its mean. This is accomplished by using an appropriate class of variations.

Our second major objective is to give an efficient method of solution for these problems. To do this we derive the Euler equation in this setting which is the primary necessary condition for a critical point or extremal solution. We also derive transversality and corner conditions. These results, which are the three basic necessary conditions for an extremal, are sufficient to construct closed form solutions, when they exist. Our results hold for $n \ge 1$ dependent variables.

Of particular interest are several example problems. They illustrate random objective functionals that are functions of a Brownian path, pathwise extremals of these functionals, and the use of necessary conditions to find solutions in closed form. We also illustrate that our theory can generalize the usual stochastic control theory by considering individual paths and not the mean value of the objective functional.

Finally, these results illustrate steps towards a complete stochastic variational theory for random objective functionals by extending deterministic ideas of the first author. The three necessary conditions of this paper allow a general stochastic control theory with general equality and inequality constraints.

We introduce the operator-valued relative free entropy $\chi_\mb^{\ast}(X_1,X_2,\cdots,X_n:\mb)$ of a family of self-adjoint random variables $X_1,X_2,\cdots,X_n$ in a $\mb$-valued noncommutative probability space $(\ma,\emb,\mb)$. This notion extends D. Voiculescu's relative free entropy $\Phi^{\ast}$ which defined in a tracial W*-noncommutative probability space to a more general context. The free entropy of a semicircular variable with conditional expectation covariance can be computed by using the modular frames and then we point out the relation between the free entropy of a semicircular variable and the index of a conditional expectation. At last, we obtain an estimate of the free entropy dimension $\delta^\ast_{\mb,\tau}$.

In full waveform inversion, the lack of low frequency information in the inversion results has been a long standing problem. In this work, we show that by using mixed basis functions this problem can be resolved satisfactorily. Examples of full waveform inversion on layered systems, using surface reflection data from point sources, have shown excellent results nearly indistinguishable from the target model. Our method is robust against additive white noise (up to 20\% of the signal) and can resolve layers that are comparable to or smaller than a wavelength in thickness. Physical reason for the success of our approach is illustrated through a simple example.

In this paper we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

In this paper we present numerical methods for the nonlinear Schrödinger equations (NLS) in the semiclassical regimes: \[i \vep\; u_t^\vep=-\fl{\vep^2}{2}\;\btu u^\vep+V(\bx)u^\vep+ f(|u^\vep|^2)u^\vep, \qquad \bx\in{\Bbb R}^d,\] with nonzero far-field conditions. A time-splitting cosine-spectral (TS-Cosine) method is presented when the nonzero far-field conditions are or can be reduced to homogeneous Neumann conditions, a time-splitting Chebyshev-spectral (TS-Chebyshev) method is proposed for more general nonzero far-field conditions, and an efficient and accurate numerical method in which we use polar coordinates to properly match the nonzero far-field conditions is presented for computing dynamics of quantized vortex lattice of NLS in two dimensions (2D). All the methods are explicit, unconditionally stable and time reversible. Furthermore, TS-Cosine is time-transverse invariant and conserves the position density, where TS-Chebyshev can deal with more general nonzero far-field conditions. Extensive numerical tests are presented for linear constant/harmonic oscillator potential, defocusing nonlinearity of NLS to study the $\vep$-resolution of the methods. Our numerical tests suggest the following `optimal' $\vep$-resolution of the methods for obtaining `correct' physical observables in the semi-classical regimes: time step $k$-independent of $\vep$ and mesh size $h=O(\vep)$ for linear case; $k=O(\vep)$ and $h=O(\vep)$ for defocusing nonlinear case. The methods are applied to study numerically the semiclassical limits of NLS in 1D and the dynamics of quantized vortex lattice of NLS in 2D with nonzero far-field conditions.

Variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides a fast and accurate way for speed calculations. A variational principle based computation is carried out on a large ensemble of KPP random speeds through spatial, mean zero, stationary, Gaussian random shear flows inside two dimensional channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speed obeys the quadratic law. In the large rms amplitude regime, the enhancement follows the linear law. An asymptotic ensemble averaged speed formula is derived and agrees well with the numerics. Related theoretical results are presented with a brief outline of the ideas in the proofs. The ensemble averaged speed is found to increase sublinearly with enlarging channel widths, while the speed variance decreases. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities.

We develop an algorithm using two coupled parabolic equations for numerical simulation of wave propagation over long distances. The coupled parabolic equations are derived from a two mode wave decomposition. An iterative procedure is used in our numerical algorithm. The coupling between the two parabolic equations allow us to deal with fine scale inhomogeneities in the medium and capture the back scattering that are usually neglected in a one way parabolic approximation. We study stability issues of our numerical algorithm and present numerical examples.

An exact formula for the photonic band structure (and hence the dispersion relation) is found for a certain class of periodic media. These media have the special property that the spatially fluctuating part of their complex electrical permittivity and magnetic permeability, when Fourier transformed, vanishes in half of Fourier space. As a result the band structure is proved to be exactly the same as for a lossy homogeneous medium, at least when the frequency is not too high.

This paper reviews the efforts to construct (or prove the impossibility of) singular solutions of the Euler equations for three dimensional incompressible flow. A semi-analytic approach to this problem is formulated based on numerical computation of complex traveling wave solutions, followed by perturbation construction of a real solution, for axisymmetric flow with swirl. The perturbation analysis depends on small amplitude of the singularity in the traveling wave solution, for small swirl in the underlying background flow.

This paper first describes the rich and intricate structure of the \ql\ hyperbolic \eq s governing the planar motions of \icp\ \nl ly \el\ rods. It then shows that under a variety of constitutive restrictions, these \eq s have periodic travelling-wave \sol s coupling longitudinal and transverse extension, flexure, and shear.

We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schr\"odinger equation with an oscillatory potential. The problem therefore contains three scales, $\eps$ the oscillation length, $\theta$ the regularization parameter, $\delta$ the potential lattice.

We prove that the homogenized limit (as $\eps$ vanishes) of this equation is a scattering equation with discrete jumps. As $\delta$ vanishes, the discrete scattering kernel boils down to a standard regular scattering kernel. As $\theta$ vanishes we recover the quantum scattering operator with collisions preserving energy sphere.

We study the dynamics of wave propagation in nominally diffusive samples by solving the Bethe-Salpeter equation with recurrent scattering included in a frequency-dependent vertex function, which renormalizes the mean free path of the system. We calculate the renormalized time-dependent diffusion coefficient, $D(t)$, following pulsed excitation of the system. For cylindrical samples with reflecting side walls and open ends, we observe a crossover in dynamics in the transformation from a quasi-1D to a slab geometry implemented by varying the ratio of the radius, $R$, to the length, L. Immediately after the peak of the transmitted pulse, $D(t)$ falls linearly with a nonuniversal slope that approaches an asymptotic value for $R/L\gg 1$. The value of $D(t)$ extrapolated to $t=0$, depends only upon the dimensionless conductance $g$ for $R/L \ll 1$ and upon $kl_0$ and $L$ for $R/L \gg 1$, where $k$ is the wave vector and $l_0$ is the bare mean free path.