In chaotic advection generated by a class of time periodic cellular flows, the residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. We study the residual diffusion phenomenon computationally and analytically.

We make use of the Poincar e map of the advection-diffusion equation to bypass long time simulation and gain accuracy in computing effective diffusivity and learning adaptive basis. We observe a non-monotone relationship between residual diffusivity and the amount of chaos in the advection, though the overall trend is that sufficient chaos leads to higher residual diffusivity. The adaptive orthogonal basis with built-in sharp gradient structures is constructed by taking snapshots of solutions in time, preprocessing with deep neural network (DNN) if necessary and performing singular value decomposition of the matrix consisting of those snapshots as column vectors. The trained orthogonal adaptive basis makes possible low cost computation of the effective diffusivities at smaller molecular diffusivities. The testing errors decrease as the training occurs at smaller molecular diffusivities.

We also study the enhanced diffusivity in the so called elephant random walk model with stops by including symmetric random walk steps at small probability $\epsilon$. At any $\epsilon > 0$, the large time behavior transitions from sub-diffusive at $\epsilon = 0$ to diffusive in a wedge shaped parameter regime where the diffusivity is strictly above that in the un-perturbed model in the $\epsilon \downarrow 0$ limit. The perturbed model is shown to be solvable with the first two moments and their asymptotics calculated exactly in both one and two space dimensions. The model provides a discrete analytical setting of the residual diffusion phenomenon as molecular diffusivity tends to zero. On a related nonlinear case, we give theoretical proof that the turbulent flame speed as an effective burning velocity is decreasing with respect to the curvature diffusivity (Markstein number) for shear flows in the well-known G-equation model. Besides, we solve the selection problem of weak solutions when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.

Finally for the dimensionality reduction on DNNs, we propose BinaryRelax, a simple two-phase algorithm, for training DNNs with quantized weights. We relax the hard constraint that characterizes the quantization of weights into a continuous regularizer via Moreau envelope, which turns out to be the squared Euclidean distance to the set of quantized weights. The pseudo quantized weights are obtained by linearly interpolating between the float weights and their quantizations. A continuation strategy is adopted to push the weights towards the quantized state by gradually increasing the regularization parameter. We test BinaryRelax on the benchmark CIFAR and ImageNet color image datasets to demonstrate the superiority of the relaxed quantization approach and the improved accuracy over the state-of-the-art training methods. Moreover, we prove the convergence of BinaryRelax under an approximate orthogonality condition.