Fast and Robust Likelihood-Guided Diffusion Posterior Sampling with Amortized Variational Inference

Inverse spectral problems are studied for arbitrary order differential operators on a finite interval with jump conditions inside the interval. Properties of spectral characteristics are obtained, and uniqueness theorems are proved for this class of inverse problems.

In recent years, the problem of potential-induced degradation (PID) phenomenon has been deeply associated with solar power issues because it causes serious power attenuation of solar panels and results in lowering its power generation efficiency. Thus, effectively identifying the PID problem from insights of industry data analysis to reduce production costs and increase the performance of power generation is an interesting and important subject for the solar power industry. Moreover, by the traditional standard rule (IEC62804) and the condition of a 96 h testing time, the costs of testing time and assembling materials against PID are very high and must be improved. Given the above reasons, this study proposes a hybrid procedure to organizes four mathematical methods: the mini-module testing, solar cell testing, a settling time, and a neural network, which are named as Method-1–Method-4, respectively, to efficiently solve the PID problem. Consequently, there are four key outcomes from the empirical results for solar power application: (1) In Method-1 with a 96 h testing time, it was found that the large module with higher costs and the mini module with lower costs have a positive correlation; thus, we can replace the large-module testing by the effective mini module for lower cost on module materials. (2) In Method-2 with a 24 h testing time, it was also found that the mini module and the solar cell are positively correlated; this result provides evidence that we can conduct the PID test by the easier solar cell to lower the costs. (3) In Method-3, the settling time achieves an average accuracy of 94% for PID prediction with a 14 h testing time. (4) In Method-4, the experimental result provides an accuracy of 80% when identifying the PID problem with the mathematical neural network model and are obtained within a 2 h testing time. From the above results, these methods succeed in reducing cost of materials and testing time during the manufacturing process; thus, this study has an industrial application value. Concurrently, Method-3 and Method-4 are rarely seen in the limited literature review for identifying PID problem; therefore, this study also offers a novel contribution for technical application innovation.

This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

Parallel inverse-problem solver for time-domain optical tomography with perfect parallel scaling

Compressive sampling (CS) is a novel signal processing paradigm whereby the data compression is performed simultaneously with the sampling, by measuring some linear functionals of original signals in the analog domain. Once the signal is sparse sufficiently under some bases, it is strictly guaranteed to stably decompress/reconstruct the original one from significantly fewer measurements than that required by the sampling theorem, bringing considerable practical convenience. In the field of civil engineering, there are massive application scenarios for CS, as many civil engineering problems can be formulated as sparse inverse problems with linear measurements. In recent years, CS has gained extensive theoretical developments and many practical applications in civil engineering. Inevitable modelling and measurement uncertainties have motivated the Bayesian probabilistic perspective into the inverse problem of CS reconstruction. Furthermore, the advancement of deep learning techniques for efficient representation has also contributed to the elimination of the strict assumption of sparsity in CS. This paper reviews the advancements and applications of CS in civil engineering, focusing on challenges arising from data acquisition and analysis. The reviewed theories also have applicability to inverse problems in broader scientific fields.

We develop techniques to solve ill-posed inverse problems on the sphere by sparse regularization, exploiting sparsity in both axisymmetric and directional scale-discretized wavelet space. Denoising, inpainting, and deconvolution problems and combinations thereof, are considered as examples. Inverse problems are solved in both the analysis and synthesis settings, with a number of different sampling schemes. The most effective approach is that with the most restricted solution-space, which depends on the interplay between the adopted sampling scheme, the selection of the analysis/synthesis problem, and any weighting of the l1 norm appearing in the regularization problem. More efficient sampling schemes on the sphere improve reconstruction fidelity by restricting the solution-space and also by improving sparsity in wavelet space. We apply the technique to denoise Planck 353-GHz observations, improving the ability to extract the structure of Galactic dust emission, which is important for studying Galactic magnetism.

We propose a fast and efficient sampling strategy to build decision trees from a very large database, even when there are many continuous attributes which must be discretized at each step. Successive samples are used, one on each tree node. After a brief description of two fast sequential simple random sampling methods, we apply elements of statistical theory in order to determine the sample size that is sufficient at each step to obtain a decision tree as efficient as one built on the whole database. Applying the method to a simulated database (virtually infinite size), and to five usual benchmarks, confirms that when the database is large and contains many numerical attributes, our strategy of fast sampling on each node (with sample size about n = 300 or 500) speed up the mining process while maintaining the accuracy of the classifier.

Abstract. We study inverse spectral problems for arbitrary order ordinary differential equations on noncompact trees. As the main spectral characteristics we introduce and investigate the so-called Weyl matrices which are a generalization of the Weyl function for the classical Sturm–Liouville operator. We provide a procedure for constructing the solution of the inverse problem from the Weyl matrices and prove its uniqueness.

In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo (MCMC) sampling strategies have become the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms have been shown to be inefficient when sampling from high-dimensional posterior distributions or exhibit multimodality and/or strong parameter correlations. In such contexts, transitional MCMC (TMCMC) provides a more efficient alternative. Despite the recent applicability for Bayesian updating and model selection across a variety of disciplines, TMCMC may require a prohibitive number of tempering stages when the prior pdf is significantly different from the target posterior. Furthermore, the need to start with an initial set of samples from the prior distribution may present a challenge when dealing with implicit priors, e.g., based on feasible regions. Finally, TMCMC cannot be used for inverse problems with improper prior PDFs that represent a lack of prior knowledge on all or a subset of parameters. A generalization of TMCMC is proposed that alleviates such challenges and limitations toward providing a more robust and efficient tempering sampling strategy. We present convergence analysis, proving that the distance between the intermediate distributions and the target posterior distribution monotonically decreases as the algorithm proceeds. We also demonstrate the advantages of the proposed generalization through a series of test problems and an engineering application in the oil and gas industry.

Subsurface modeling is a key tool to describe, understand and quantify geological processes. As the subsurface is inaccessible and its observation is limited by acquisition methods, 3D models of the subsurface are usually built from the interpretation of sparse data with limited resolution. Therefore, uncertainties occur during the model building process, due to possible cognitive human biais, natural variability of geological objects and intrinsic uncertainties of data. In such context, the predictibility of models is limited by uncertainties, which must be assessed in order to reduce economical and human risks linked to the use of models. This work focuses more specifically on uncertainties about geological structures. In this context, a stochastic method is developed for generating structural models with various fault and horizon geometries as well as fault connections. Realistic geological objects are obtained using implicit modeling that represents a surface by an equipotential of a volumetric scalar field. Faults have also been described by a reduced set of uncertain parameters, which opens the way to the inversion of structural objects using geophysical or fluid flow data by baysian methods.

Consumer returns on live‐streaming channels, positively or negatively impacted by the interaction‐perception effect, are becoming an increasingly serious problem for manufacturers due to the value losses associated with returned products. This paper examines the influence of sample strategy on manufacturers and streamers and investigates how the strategy influences the streamer‐type selection. We model a manufacturer deploying an employee or influencer streamer to sell products, accounting for sample‐strategy adoption. We find that adopting the sample strategy impacts the choice of streamer types. Without the sample strategy, the manufacturer deploys the influencer streamer only at a low production cost and high value retained proportion. However, with the sample strategy, it prefers the influencer streamer at a low sample cost and high residual value. Counterintuitively, the choice of streamer types does not influence the implementation of the sample strategy. The manufacturer adopts the sample strategy at either a high residual value or high production cost.

The task of discriminating the neural pathways responsible for the activity recorded using a multi-contact nerve cuff electrode has recently been approached as an inverse problem of source localization, similar to EEG source localization. A major drawback of this method is that it requires a model of the nerve, and that the localization performance is highly dependent on the accuracy of this model. Using recordings from a 56-contact "matrix" cuff electrode placed on a rat sciatic nerve, we investigated a method that eliminates the need for a model, and uses instead an "experimental" leadfield constructed from a training set of experimental recordings. The resulting pathway-identification task is solved using an inverse problem framework. The experimental leadfield approach was able to identify the correct branch in cases in which a single fascicle was active with a success rate of 94.2%, but was not able to reliably identify combinations of fascicles. Nevertheless, the proposed methodology provides a framework for the study of multi-pathway discrimination, within which methods to improve performance can be investigated. Specifically, the influence of nerve anatomy and electrode design should be examined, and regularization approaches better suited to this novel inverse problem should be sought.

Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges by pretraining a neural network that acts as a surrogate conditional distribution that approximates the posterior distribution not only for one instance of the observed data but also for the distribution of the data pertaining to a specific inverse problem. When fed previously unseen data, the neural network — in our case, a conditional normalizing flow — provides the posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelity training data, which seldom exist in geophysical inverse problems because of the earth’s heterogeneous subsurface. In addition, the network is prone to errors if evaluated over data that are not drawn from the training data distribution. As such, we have aimed to increase the resilience of amortized variational inference in the presence of moderate data distribution shifts. We achieve this via a correction to the conditional normalizing flow’s latent distribution that improves the approximation to the posterior distribution for the data at hand. The correction involves relaxing the standard Gaussian assumption on the latent distribution and parameterizing it via a Gaussian distribution with an unknown mean and (diagonal) covariance. These unknowns are then estimated by minimizing the Kullback-Leibler divergence between the corrected and the (physics-based) true posterior distributions. Although generic and applicable to other inverse problems by means of a linearized seismic imaging example, we find that our correction step improves the robustness of amortized variational inference with respect to changes in the number of seismic sources, noise variance, and shifts in the prior distribution. This approach, given noisy seismic data simulated via the linearized Born modeling, provides a seismic image with limited artifacts and an assessment of its uncertainty at approximately the same cost as five reverse time migrations.

Investigating adaptive sampling strategies for optimal building energy performance using artificial neural networks and kriging surrogate models

In a probabilistic formulation, the solution to an inverse problem can be expressed an a posteriori probability density function (pdf) that combines the independent states of information provided by data and a priori information. Here, we define an a posteriori probability density function that defines the solution to a tomographic full waveform inverse problem, which provides a means of obtaining an uncertainty estimate. Unfortunately, no explicit formulation of the solution to this problem can be defined. Therefore, the a posteriori probability density function has to be sampled. The full waveform inverse problem is known to be computationally very hard and is, traditionally, considered out of reach for Monte Carlo sampling strategies. We show that by means of informative a priori information this problem become tractable for a sampling strategy anyways. We outline the theoretical framework for a full waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling, which allows for arbitrary complex a priori information to be incorporated. At the same time we show how temporally correlated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground penetrating radar full waveform data using multiplepoint based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full waveform inverse problem.