Multiresolution Approximation of a Bayesian Inverse Problem using Second-Generation Wavelets

Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. It is common practice to decompose the unknown field into some basis and infer the decomposition parameters instead of directly inferring the unknown. Given the multiscale nature of permeability fields, wavelets are a natural choice for parameterizing them. This study uses a Bayesian approach to incorporate the statistical sparsity that characterizes discrete wavelet coefficients. First, we impose a prior distribution incorporating the hierarchical structure of the wavelet coefficient and smoothness of reconstruction via scale-dependent hyperparameters. Then, Sequential Monte Carlo (SMC) method adaptively explores the posterior density on different scales, followed by model selection based on Bayes Factors. Finally, the permeability field is reconstructed from the coefficients using a multiresolution approach based on second-generation wavelets. Here, observations from the pressure sensor grid network are computed via Multilevel Adaptive Wavelet Collocation Method (AWCM). Results highlight the importance of prior modeling on parameter estimation in the inverse problem

Stabilized Conservative Level Set Method with Adaptive Wavelet-Based Mesh Refinement

This study investigates one of the well-known shortcomings of the conservative level set method, namely the ill-defined normal vector. A stabilized formulation is proposed which does not rely on the unit normal vector anymore. Instead, the proposed stabilized conservative level set, SCLS, utilizes a modified normal vector, magnitude of which is unit in the interfacial region of width ε and approaches zero in the far field from the interface. Respective adjustments have been applied on the reinitialization equation to comply with the proposed normal vector. This methodology is general and robust and it is not topology dependent. Since the information in the interfacial region are of a higher interest in comparison to the far field data, SCLS is specially well suited to be used with adaptive mesh refinement, AMR. In this research a general AMR-type approach named Adaptive Wavelet Collocation Method has been utilized, which makes use of wavelets to adapt on meshes in the interfacial region. A number of benchmark numerical problems have been performed to investigate the performance of the stabilized conservative level set approach. SCLS methodology is intended to be a base for simulation of interfacial phenomena, especially multiphase flows

Meta-learning biologically plausible plasticity rules with random feedback pathways

The biological plausibility of backpropagation and its relationship with synaptic plasticity remain open questions. The authors propose a meta-learning approach to discover interpretable plasticity rules to train neural networks under biological constraints. The meta-learned rules boost the learning efficiency via bio-inspired synaptic plasticity