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At least 307 records · Page 17

Risk-averse optimization for resilience enhancement of complex engineering systems under uncertainties

With the growth of complexity and extent, large scale interconnected network systems, e.g., transportation networks or infrastructure networks, become more vulnerable to external disturbances. Hence, managing potential disruptive events during the design, operating, and recovery phase of an engineered system and therefore improving the system’s resilience is an important yet challenging task. Here, to ensure system resilience after the occurrence of failure events, this study proposes a mixed-integer linear programming (MILP) based restoration framework using heterogeneous dispatchable agents. The scenario-based stochastic optimization (SO) technique is adopted to deal with the inherent uncertainties imposed on the recovery process from nature. Moreover, different from conventional SO using deterministic equivalent formulations, the CVaR risk measure is implemented for this study because of the temporal sparsity of the decision making in applications such as the recovery from extreme events. The resulting restoration framework involves a large-scale MILP problem and thus an adequate decomposition technique i.e. modified Lagrangian dual decomposition, is also employed to achieve tractable computational complexity. Case study results based on the IEEE 37-bus test feeder demonstrate the benefits of using the proposed framework for resilience improvement as well as the advantages of adopting SO formulations.

42 ENGINEERING↗

A Stochastic Multi-Commodity Logistic Model for Disaster Preparation in Distribution Systems

In this paper, we propose a stochastic optimization approach for disaster preparation in distribution systems. For an upcoming storm, utilities should have a preparation plan that includes warehousing restoration supplies, securing staging sites (depots), and prepositioning crews and equipment. Pre-storm planning enables faster and more efficient post-disaster deploy ment of crews and equipment resources to damage locations. To assist utilities in making this important preparation, this paper develops a two-stage stochastic mixed integer linear program. The first stage determines the depots, number of crews in each site, and the amount of equipment. The second stage is the recourse action that deals with acquiring new equipment and assigning crews to repair damages in realized scenarios. The objective of the developed model is to minimize the costs of depots, crews, equipment, and penalty costs associated with delays in obtain ing equipment and restoration. We consider the uncertainties of damaged lines, number and type of equipment required, and expected repair times. The model is validated on modified IEEE 123-bus distribution test system.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Optimization and supervised machine learning methods for fitting numerical physics models without derivatives

Here, we address the calibration of a computationally expensive nuclear physics model for which derivative information with respect to the fit parameters is not readily available. Of particular interest is the performance of optimization-based training algorithms when dozens, rather than millions or more, of training data are available and when the expense of the model places limitations on the number of concurrent model evaluations that can be performed. As a case study, we consider the Fayans energy density functional model, which has characteristics similar to many model fitting and calibration problems in nuclear physics. We analyze hyperparameter tuning considerations and variability associated with stochastic optimization algorithms and illustrate considerations for tuning in different computational settings.

97 MATHEMATICS AND COMPUTING↗

Report on the Economic Optimization of a Demonstration Case for a Static N-R HES Configuration using RAVEN

The synthetic time history generation algorithm in RAVEN has been extensively tested and used to perform qualitative assessments of the statistical characteristics of the net demand vs. demand for different values of the penetration of wind generation. Reported results highlight the need to consider the impact of variable renewable on the electricity net demand profile, due to the increase in its volatility with increased wind penetration. This analysis provides a strong foundation for the modeling and simulation needs expressed under the "Nuclear-Renewable Hybrid Energy Systems" (NRHES) project. Following this confirmative analysis the entire stochastic optimization framework has been tested in a demo case. The case chosen is an optimization driven by profitability for a system composed by a nuclear plant and a hydrogen production industrial process. This type of economical analysis is aimed to assess the capability of a hybrid system to penetrate the current energy market. In this sense it differs from the approach suggested under NRHES, where a cost minimization approach is suggested. This analysis is instead aimed to show the flexibility of the developed framework, e.g. for the evaluation of retrofitting projects of already existing plants. The system considered is subject to variable prices of electricity, which are simulated using the algorithm for the production of synthetic time histories. The physical system is considered without inertia so that it can be replaced by dispatching rules based on the highest marginal profit. Two optimization problems have been considered, for the first the size of the nuclear power plant is fixed and the optimization variable is the capacity of the hydrogen production plant. For the second, both plant sizes are optimization parameters while the overall capital expenditure is caped. The economical figures of merits considered are the Internal Rate of Return (IRR) and/or Profitability Index (PI) for the first one (since no cap is placed on the investment size) and Net Present Value (NPV) and/or IRR for the second one where the capital expenditure is constrained. Economic input data concerning capital costs, size-scaling factors, and operational costs for both plants have been collected from literature in order to achieve a simulation as realistic as possible. More work will be needed to obtain even more realistic numbers, possibly from industrial partners, and considering market elasticity for the hydrogen market. In order to perform the optimization of the cases described above, the economical external RAVEN module (CashFlow) for financial analysis has been improved in flexibility and generality. Some improvements have also been made to the optimization algorithm for a more efficient treatment of the boundary constrains. In conclusion three major results have been obtained: a) collection of the economical parameters characterizing the nuclear power plants and hydrogen production plants, b) extension of the financial analysis module (CashFlow) to drive the calculation of all needed cash flow streams and c) the testing of the full framework in a nearly final configuration.

97 MATHEMATICS AND COMPUTING↗

Collaborative Project: Building improved optimized parameter estimation algorithms to improve methane and nitrogen fluxes in a climate model

Soils in natural and managed ecosystems and wetlands are well known sources of methane, nitrous oxides, and reactive nitrogen gases, but the magnitudes of gas flux to the atmosphere are still poorly constrained. Thus, the reasons for the large increases in atmospheric concentrations of methane and nitrous oxide since the preindustrial time period are not well understood. The low atmospheric concentrations of methane and nitrous oxide, despite being more potent greenhouse gases than carbon dioxide, complicate empirical studies to provide explanations. In addition to climate concerns, the emissions of reactive nitrogen gases from soils are important to the changing nitrogen balance in the earth system, subject to human management, and may change substantially in the future. Thus improved modeling of the emission fluxes of these species from the land surface is important. Currently, there are emission modules for methane and some nitrogen species in the Community Earth System Model’s Community Land Model (CLM-ME/N); however, there are large uncertainties and problems in the simulations, resulting in coarse estimates. In this proposal, we seek to improve these emission modules by combining state-of-the-art process modules for emissions, available data, and new optimization methods. In earth science problems, we often have substantial data and knowledge of processes in disparate systems, and thus we need to combine data and a general process level understanding into a model for projections of future climate that are as accurate as possible. The best methodologies for optimization of parameters in earth system models are still being developed. In this proposal we will develop and apply surrogate algorithms that a) were especially developed for computationally expensive simulations like CLM-ME/N models; b) were (in the earlier surrogate optimization Stochastic RBF) demonstrated to perform very well on computationally expensive complex partial differential equations in earth science with limited numbers of simulations; and, c) will be (as part of the proposed research) significantly improved both by adding asynchronous parallelism, early truncation of unsuccessful simulations, and the improvement of both serial and parallel performance by the use of derivative and sensitivity information from global and local surrogate approximations S(x) . The algorithm development and testing will be focused on the CLM-ME/N model application, but the methods are general and are expected to also perform well on optimization for parameter estimation of other climate models and other classes of continuous multimodal optimization problems arising from complex simulation models. In addition, this proposal will compile available datasets of emissions of methane, nitrous oxides and reactive nitrogen species and develop protocols for site level comparisons with the CLM-ME/N. Once the model parameters are optimized against site level data, the model will be simulated at the global level and compared to atmospheric concentration measurements for the current climate, and future emissions will be estimated using climate change as simulated by the CESM. This proposal combines experts in earth system modeling, optimization, computer science, and process level understanding of soil gas emissions in an interdisciplinary team in order to improve the modeling of methane and nitrogen gas emissions. This proposal thus meets the requirements of the SciDAC RFP, by integrating state-of-the-art computer science and earth system to build an improved earth system model.

54 ENVIRONMENTAL SCIENCES↗

A Scalable Solution Framework for Stochastic Transmission and Generation Planning Problems. Draft

Current commercial software tools for transmission and generation investment planning have limited stochastic modeling capabilities. Because of this limitation, electric power utilities generally rely on scenario planning heuristics to identify potentially robust and cost effective investment plans for a broad range of system, economic, and policy conditions. Several research studies have shown that stochastic models perform significantly better than deterministic or heuristic approaches, in terms of overall costs. However, there is a lack of practical solution approaches to solve such models. In this paper we propose a scalable decomposition algorithm to solve stochastic transmission and generation planning problems, respectively considering discrete and continuous decision variables for transmission and generation investments. Given stochasticity restricted to loads and wind, solar, and hydro power output, we develop a simple scenario reduction framework based on a clustering algorithm, to yield a more tractable model. The resulting stochastic optimization model is decomposed on a scenario basis and solved using a variant of the Progressive Hedging (PH) algorithm. We perform numerical experiments using a 240-bus network representation of the Western Electricity Coordinating Council in the US. Although convergence of PH to an optimal solution is not guaranteed for mixed-integer linear optimization models, we find that it is possible to obtain solutions with acceptable optimality gaps for practical applications. Our numerical simulations are performed both on a commodity workstation and on a high-performance cluster. The results indicate that large-scale problems can be solved to a high degree of accuracy in at most two hours of wall clock time.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Development of Multiresolution Capabilities for the Holistic Energy Resource Optimization Network (HERON) tool A progress update

INL researchers work on technoeconomic analyses for integrated energy systems (IES) using the Framework for Optimization of ResourCes and Economics (FORCE). Within FORCE, researchers use the Holistic Energy Resource Optimization Network (HERON) tool to conduct optimization of grid portfolios under uncertain market conditions. These optimizations determine optimal capacities for all IES components and strategies for resource dispatch which maximize some economic metric (e.g., net present value). Resource dispatch occurs on finer timescales (typically hours) and thus are asked to respond to a given time series (e.g. hourly load demand profiles for a grid, or pre-determined electricity prices). Volatile and complex bidding dynamics as well as poorly forecasted weather events within deregulated markets add uncertainty to the time series; FORCE can address this uncertainty by training a reduced order model on historical time series and generate unique synthetic time series which represent individual scenarios or realizations of the market. The IES configuration can be simulated under these different sampled realizations and a stochastic optimization is conducted which optimizes the expected value of the desired economic metric. The training of a synthetic time series generator is limited by the chosen time resolution; dynamics can occur on different time scales. Seasonal demand trends can dominate faster dynamical events (such as power outages from certain sectors or severe weather events) which might not get captured correctly by the trained model. In this report, we investigate different ways of addressing the training and generation of time series on multiple time scales using three main algorithms: wavelet decomposition, dynamic mode decomposition, and generative adversarial networks for time series. We demonstrate a time series analysis that yields information on not just the frequency space but also temporal space: where a fast Fourier transform can provide what frequencies dominate, the new algorithms can provide when the frequencies dominate as well. These analyses can help improve IES optimization by allowing researchers to couple simulations at different timescales when it is most needed - seasonal, day-ahead, and real time optimization - with greater computational efficiency. Future work will include implementation of a subset of the proposed algorithms into the FORCE toolset and application of these analyses into multiple timescale optimization.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Synthetic Electricity Market Data Generation and HERON Use Case Setup of Advanced Nuclear Reactors Coupled with Thermal Energy Storage Systems

This study evaluates and optimizes advanced nuclear reactors coupled with thermal energy storage (TES) systems in an Integrated Energy System (IES) architecture to enable advanced nuclear power plants (A NPP) to participate in multi-commodity markets, thus enhancing their economic competitiveness. Nuclear-TES coupling scenarios studied herein are designed attenuate the nuclear heat dynamics and defer energy delivery to a later time, enabling the nuclear reactor to continue operating at or near steady-state design conditions as usual while also enabling flexible generation. Three A-NPPs, namely, an advanced light-water reactor (A LWR), a high temperature gas-cooled reactor (HTGR) and a liquid-metal fast reactor (LMFR) were selected as the initial use cases for demonstrating the technoeconomic of thermally balanced energy storage coupling design for thermal power extraction. Each of the reactor technologies were evaluated in two different electricity markets. Stochastic optimization approach was adopted which included the evaluation of price signals from the Pennsylvania-New Jersey-Maryland (PJM) market, and Electric Reliability Council of Texas (ERCOT), using an autoregressive moving average (ARMA) model. Risk Analysis Virtual Environment (RAVEN) tool and its dispatch optimization plugin, the Holistic Energy Resource Optimization Network (HERON), were used to perform dispatch and capacity optimization, using the price data provided by the ARMA models. The results from the Nuclear-TES use cases will be used to design and characterize dynamic integrated system behavior and feedback.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Optimal Control for Stochastic Delay Evolution Equations

In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

97 MATHEMATICS AND COMPUTING↗

The Aeronautical Data Link: Decision Framework for Architecture Analysis

A decision analytic approach that develops optimal data link architecture configuration and behavior to meet multiple conflicting objectives of concurrent and different airspace operations functions has previously been developed. The approach, premised on a formal taxonomic classification that correlates data link performance with operations requirements, information requirements, and implementing technologies, provides a coherent methodology for data link architectural analysis from top-down and bottom-up perspectives. This paper follows the previous research by providing more specific approaches for mapping and transitioning between the lower levels of the decision framework. The goal of the architectural analysis methodology is to assess the impact of specific architecture configurations and behaviors on the efficiency, capacity, and safety of operations. This necessarily involves understanding the various capabilities, system level performance issues and performance and interface concepts related to the conceptual purpose of the architecture and to the underlying data link technologies. Efficient and goal-directed data link architectural network configuration is conditioned on quantifying the risks and uncertainties associated with complex structural interface decisions. Deterministic and stochastic optimal design approaches will be discussed that maximize the effectiveness of architectural designs.

Morris, A. Terry↗

Optimizing Microgrid Energy Delivery Under High Uncertainty

One of the largest transitions in the power system today is the shift to a more sustainable and resilient power system. This is being driven by public opinion, changes in regulatory policies, and advancements in smart grid technologies. The most noticeable changes taking place is the integration of distributed energy sources (DERs); this study uses the term DER in the most general way as a resource that can be manipulated to alter energy delivery and flow in the transmission and distribution networks. Also, here it is preferred to focus on energy as the true need while power is a function of the equipment rating. As such, wind and solar, demand that can be manipulated, electric vehicles, electric energy storage, thermal storage, and storage in water system are all considered DERs. These additions to the distribution system are evolving the operation of distribution feeders into microgrids- communication, computing, and control-enabled resources that produce, transport, and utilize energy in a manner that provides cost, reliability, and resilience benefits. As this evolution progresses, the planning and operational management (scheduling and control) must explicitly include the consideration of risk. The management of system risk is currently in the purview of the utility and will likely remain so in the future. However, as each microgrid, as well as federation of microgrids, sees autonomy in order to provide maximum benefits to their constituents, they must assume responsibility to manage their internal risk. The primary scope of this study is the scheduling of resources in a distribution feeder(s) operating as microgrids. The study explores a distribution algorithm to develop the transactive schedule for the DERs, to minimize cost and risk over a time horizon, and an initial laboratory-scale to conduct implementation on distributed hardware. Results from case studies are presented that show that solutions derived by the distributed algorithm are valid. This study also discusses the continuing work on the expansion of: 1) the distributed algorithm from a deterministic to stochastic optimization formulation, and 2) implementation of the distributed algorithm into real-time simulation within the Power System laboratory at New Mexico State University (NMSU) and expanding to the Southwest Technology Development Institute located at NMSU where actual solar, energy storage, and demand response resources are installed.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Flight Test of an Intelligent Flight-Control System

The F-15 Advanced Controls Technology for Integrated Vehicles (ACTIVE) airplane (see figure) was the test bed for a flight test of an intelligent flight control system (IFCS). This IFCS utilizes a neural network to determine critical stability and control derivatives for a control law, the real-time gains of which are computed by an algorithm that solves the Riccati equation. These derivatives are also used to identify the parameters of a dynamic model of the airplane. The model is used in a model-following portion of the control law, in order to provide specific vehicle handling characteristics. The flight test of the IFCS marks the initiation of the Intelligent Flight Control System Advanced Concept Program (IFCS ACP), which is a collaboration between NASA and Boeing Phantom Works. The goals of the IFCS ACP are to (1) develop the concept of a flight-control system that uses neural-network technology to identify aircraft characteristics to provide optimal aircraft performance, (2) develop a self-training neural network to update estimates of aircraft properties in flight, and (3) demonstrate the aforementioned concepts on the F-15 ACTIVE airplane in flight. The activities of the initial IFCS ACP were divided into three Phases, each devoted to the attainment of a different objective. The objective of Phase I was to develop a pre-trained neural network to store and recall the wind-tunnel-based stability and control derivatives of the vehicle. The objective of Phase II was to develop a neural network that can learn how to adjust the stability and control derivatives to account for failures or modeling deficiencies. The objective of Phase III was to develop a flight control system that uses the neural network outputs as a basis for controlling the aircraft. The flight test of the IFCS was performed in stages. In the first stage, the Phase I version of the pre-trained neural network was flown in a passive mode. The neural network software was running using flight data inputs with the outputs provided to instrumentation only. The IFCS was not used to control the airplane. In another stage of the flight test, the Phase I pre-trained neural network was integrated into a Phase III version of the flight control system. The Phase I pretrained neural network provided realtime stability and control derivatives to a Phase III controller that was based on a stochastic optimal feedforward and feedback technique (SOFFT). This combined Phase I/III system was operated together with the research flight-control system (RFCS) of the F-15 ACTIVE during the flight test. The RFCS enables the pilot to switch quickly from the experimental- research flight mode back to the safe conventional mode. These initial IFCS ACP flight tests were completed in April 1999. The Phase I/III flight test milestone was to demonstrate, across a range of subsonic and supersonic flight conditions, that the pre-trained neural network could be used to supply real-time aerodynamic stability and control derivatives to the closed-loop optimal SOFFT flight controller. Additional objectives attained in the flight test included (1) flight qualification of a neural-network-based control system; (2) the use of a combined neural-network/closed-loop optimal flight-control system to obtain level-one handling qualities; and (3) demonstration, through variation of control gains, that different handling qualities can be achieved by setting new target parameters. In addition, data for the Phase-II (on-line-learning) neural network were collected, during the use of stacked-frequency- sweep excitation, for post-flight analysis. Initial analysis of these data showed the potential for future flight tests that will incorporate the real-time identification and on-line learning aspects of the IFCS.

Davidson, Ron↗

A scalable planning framework of energy storage systems under frequency dynamics constraints

As the penetration of renewables increases in power systems, the declining system inertia can cause frequency stability issues. Battery energy storage systems (BESSs) respond fast and therefore can relieve the low inertia difficulty but need to be appropriately sized considering the associated cost. This paper presents a novel stochastic optimization model for economically planning BESS capacity while considering the spatial–temporal correlation of wind generation and generator outages under frequency stability constraints, which include the rate-of-change of frequency (RoCoF), frequency nadir (FN), and quasi-steady-state (QSS) frequency. A set of new FN constraints that can be easily linearized is developed. To account for renewable uncertainties, a realistic uncertainty modeling approach, Random Field, is adopted to generate wind generation scenarios by considering both spatial and temporal evolutions of wind speed profiles. The ESS sizing is formulated as a mixed-integer linear programming problem and solved by using a scalable decomposition-and-coordination approach, Surrogate Absolute Value Lagrangian Relaxation (SAVLR). To further improve the scalability and reduce computational burdens, a rolling-horizon-based update is developed and incorporated into SAVLR for providing a practical solution to the long-term planning of very large-scale power systems. Finally, a modified IEEE 118-bus system and the Polish system are used to validate the effectiveness and scalability of the model and solution methodology.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Evaluation of a Machine-Learning Algorithm for Treatment Planning in Prostate Low-Dose-Rate Brachytherapy

Purpose: This work presents the application of a machine learning (ML) algorithm to automatically generate high-quality, prostate low-dose-rate (LDR) brachytherapy treatment plans. The ML algorithm can mimic characteristics of preoperative treatment plans deemed clinically acceptable by brachytherapists. The planning efficiency, dosimetry, and quality (as assessed by experts) of preoperative plans generated with an ML planning approach was retrospectively evaluated in this study. Methods and Materials: Preimplantation and postimplantation treatment plans were extracted from 100 high-quality LDR treatments and stored within a training database. The ML training algorithm matches similar features from a new LDR case to those within the training database to rapidly obtain an initial seed distribution; plans were then further fine-tuned using stochastic optimization. Preimplantation treatment plans generated by the ML algorithm were compared with brachytherapist (BT) treatment plans in terms of planning time (Wilcoxon rank sum, α = 0.05) and dosimetry (1-way analysis of variance, α = 0.05). Qualitative preimplantation plan quality was evaluated by expert LDR radiation oncologists using a Likert scale questionnaire. Results: The average planning time for the ML approach was 0.84 ± 0.57 minutes, compared with 17.88 ± 8.76 minutes for the expert planner (P=.020). Preimplantation plans were dosimetrically equivalent to the BT plans; the average prostate V150% was 4% lower for ML plans (P=.002), although the difference was not clinically significant. Respondents ranked the ML-generated plans as equivalent to expert BT treatment plans in terms of target coverage, normal tissue avoidance, implant confidence, and the need for plan modifications. Respondents had difficulty differentiating between plans generated by a human or those generated by the ML algorithm. Conclusions: Prostate LDR preimplantation treatment plans that have equivalent quality to plans created by brachytherapists can be rapidly generated using ML. The adoption of ML in the brachytherapy workflow is expected to improve LDR treatment plan uniformity while reducing planning time and resources.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

Near-Optimal Performance of Stochastic Model Predictive Control

Here, this article presents a regret analysis for stochastic model predictive control (SMPC) in linear systems with quadratic performance index and additive and multiplicative uncertainties. Under a finite support assumption, the problem can be cast as a finite-dimensional quadratic program, but the problem becomes quickly intractable as the problem size grows exponentially in the horizon length. SMPC aims to compute approximate solutions by solving a sequence of problems with truncated prediction horizons and committing the solution in a receding-horizon fashion. Although this approach is widely used in practice, its performance relative to the optimal solution is not well understood. This article reports for the first time a rigorous near-optimal performance guarantee of SMPC: under stabilizability and detectability conditions, the regret of SMPC is exponentially small in the prediction horizon length, allowing SMPC to achieve near-optimal performance at a substantially reduced computational expense.

93E20, 93B45↗

A primal–dual algorithm for risk minimization

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. Here, this lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method.

97 MATHEMATICS AND COMPUTING↗

Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients

We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with space-dependent and state-dependent parameters. We demonstrate that these methods provide accurate and cost-effective alternatives to Markov Chain Monte Carlo simulation. We assume a parameterized Gaussian prior on the unknown functions, and approximate the posterior density by a parameterized multivariate Gaussian density. The parameters of the prior and posterior are estimated from sparse observations of the PDE model's states and the unknown functions themselves by maximizing the evidence lower bound (ELBO), a lower bound on the log marginal likelihood of the observations. The first method, Laplace-EM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the E-step, and minimization of a Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradient-based stochastic optimization, with nosiy gradients computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply these methods to identifying diffusion coefficients in linear and nonlinear diffusion equations, and we find that both methods provide accurate estimates of posterior densities and the hyperparameters of Gaussian priors. While the Laplace-EM method is more accurate, it requires computing Hessians of the physics model. The DSVI-EB method is found to be less accurate but only requires gradients of the physics model.

approximate Bayesian inference, model inversion, v↗

Solving geosteering inverse problems by stochastic Hybrid Monte Carlo method

The inverse problems arise in almost all fields of science where the real-world parameters are extracted from a set of measured data. The geosteering inversion plays an essential role in the accurate prediction of oncoming strata as well as a reliable guidance to adjust the borehole position on the fly to reach one or more geological targets. This mathematical treatment is not easy to solve, which requires finding an optimum solution among a large solution space, especially when the problem is non-linear and non-convex. Nowadays, a new generation of logging-while-drilling (LWD) tools has emerged on the market. The so-called azimuthal resistivity LWD tools have azimuthal sensitivity and a large depth of investigation. Hence, the associated inverse problems become much more difficult since the earth model to be inverted will have more detailed structures. The conventional deterministic methods are incapable to solve such a complicated inverse problem, where they suffer from the local minimum trap. Alternatively, stochastic optimizations are in general better at finding global optimal solutions and handling uncertainty quantification. In this article, we investigate the Hybrid Monte Carlo (HMC) based statistical inversion approach and suggest that HMC based inference is more efficient in dealing with the increased complexity and uncertainty faced by the geosteering problems.

97 MATHEMATICS AND COMPUTING↗