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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,729 records · Page 96
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

D-Guaranteed Discrimination of Statistical Hypotheses: a Review of Results and Unsolved Problems

We compare two sequential d-guaranteed tests and an optimal d-guaranteed test based on a fixed number of observations with respect to the average number of observations within the most accepted practical applications of Bayesian probabilistic models. We consider the sequential “first skipping” test, the sequential locally efficient test based on the score statistic, and the test based on a fixed number of observations that minimizes the necessary sample size. We study various characteristics of these tests connected with the number of observations within three probabilistic models, namely, the normal (ϑ, σ{sup 2}) distribution of the observed random variable and the normal a priori distribution of ϑ with fixed σ{sup 2}; the exponential distribution with the intensity parameter ϑ and the a priori gamma distribution of ϑ; and the Bernoulli sampling with the success probability ϑ and the a priori beta distribution of ϑ. We discuss the connection of the d-posterior approach with the compound decision problem as applied to the analysis of data provided by microchips (when the false discovery rate, or FDR, for short, is treated as the d-risk of the first kind). We present the vast data on characteristics of the mentioned tests obtained by the method of mathematical modeling in several tables. We discuss unsolved problems of the d-guaranteed discrimination of hypotheses with the minimal number of observations and approaches to their solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Effects of the Electron Beam Welding Process on the Microstructure, Tensile, Fatigue and Fracture Properties of Nickel Alloy Nimonic 80A

The purpose of this study was to evaluate rotary bending high-cycle fatigue properties and crack growth of Nimonic 80A-based metal and electron beam-welded joints. All the tests were performed at room temperature. Fracture surfaces under high-cycle fatigue and fatigue crack growth were observed by scanning electron microscopy. Microstructure, hardness and tensile properties were also evaluated in order to understand the effects on the fatigue results obtained. It was found that the tensile properties, hardness and high-cycle fatigue properties of the welded joint are lower than the base metal. The fracture surface of the high-cycle fatigue shows that fatigue crack initiated from the surface under the high stress amplitude and from the subsurface under the low stress amplitude. The effect of the welding process on the statistical fatigue data was studied with a special focus on probabilistic life prediction and probabilistic lifetime limits. The fatigue crack growth rate versus stress intensity factor range data were obtained from the fatigue crack growth tests. From the results, it was evident that the fatigue crack growth rates of the welded are higher than the base metal. The mechanisms and fracture modes of fatigue crack growth of welded specimens were found to be related to the stress intensity factor range ΔK. In addition, the effective fatigue crack propagation thresholds and mismatch of welded joints were described and discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Enhancing teleportation fidelity by means of weak measurements or reversal

The enhancement of teleportation fidelity by weak measurement or quantum measurement reversal is investigated. One qubit of a maximally entangled state undergoes the amplitude damping, and the subsequent application of weak measurement or quantum measurement reversal could improve the teleportation fidelity beyond the classical region. The improvement could not be attributed to the increasing of entanglement, quantum discord, classical correlation or total correlation. We declare that it should be owed to the probabilistic nature of the method. - Highlights: • The method’s probabilistic nature should be responsible for the improvement. • Quantum or classical correlation cannot explain the improvement. • The receiver cannot apply weak measurements. • The sender’s quantum measurement reversal is only useful for |Ψ{sup ±}〉.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

A five-dimensional perspective on the Klein–Gordon equation

We discuss the Klein–Gordon (KG) equation using a path-integral approach in 5D space–time. We explicitly show that the KG equation in flat space–time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space–time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Investigation of Stochastic Unit Commitment to Enable Advanced Flexibility Measures for High Shares of Solar PV

As the share of solar photovoltaics (PV) in the power system increases, there is a growing need for flexibility from multiple, possibly interdependent sources to adjust to PV's variability, uncertainty, and diurnal dependence. This paper investigates how stochastic unit commitment leveraging probabilistic solar forecasts can support other flexibility measures under high solar shares. We consider two flexibility measures relevant to day-ahead scheduling: battery energy time-shifting and solar ancillary service provision. Unit commitment and economic dispatch simulations are conducted on a realistic test system based on Texas using day-ahead solar trajectories. The benefits of the two flexibility measures are pronounced when the instantaneous solar share is high, offering cost savings of 10%-20% in the spring. For a Texas-sized system, this translates to hundreds of millions of dollars in cost savings once the installed PV capacity enables instantaneous solar shares regularly exceeding 40%. Using probabilistic forecasts also greatly increases the reliability of upward reserve provision from solar PV, reducing unserved reserves by 50%-100%. Both day-ahead forecast resolution and errors can impact system reliability at high solar shares, but the stochastic formulation has significant value, mitigating reliability impacts on over-forecast days.

ancillary services↗
DOE OSTI2022

Insights into Methodologies and Operational Details of Resource Adequacy Assessment: A Case Study with Application to a Broader Flexibility Framework

Assessing and maintaining resource adequacy (RA) is a core pillar of power systems. However, recent changes in the physical makeup of these systems and the conditions under which these systems must operate have yielded a renewed interest in the methods, metrics, and assumptions that underpin RA assessments. In this paper, we systematically explore a wide range of RA modeling dimensions, including: the objective function and level of operational detail in the underlying model formulation; the quantity (look-ahead) and quality (accuracy) of data that is available for making operational decisions within those models; and the physical configuration of solar photovoltaics (PV) with battery storage hybrid resources. We apply a set of probabilistic RA tools and production cost modeling tools to a realistic test system based loosely on a future Electric Reliability Council of Texas power system dominated by solar PV resources. Under the assumptions of our system and models, we find that multi-stage probabilistic assessments may provide a more robust evaluation of RA by capturing a wider range of operational and system interactions, but this comes at a computational cost of 1-2 orders of magnitude longer run time depending on the specific configuration. In addition, the information on thermal generator availability impacts RA performance by an order of magnitude more than solar resource forecasts, which is driven by the comparatively larger magnitude of thermal outages than solar forecast errors within our test system. Lastly, the flexibility provided by hybrid and other resources can help reduce system load-shedding event frequencies and enable the system to be more robust to inaccurate forecast information, and alternative hybrid inverter sizes can impact RA levels by 1-2 orders of magnitude. Our results point to the importance of a broader flexibility framework to describe the interaction between (1) flexibility "supply" from both physical resource capabilities and operational constraints considered in the modeling, and (2) flexibility "demand" from forecast errors, thermal generator outages, and other sources of uncertainty, as well as their RA impacts. Results are likely sensitive to the system buildout explored; future work could consider additional system configurations and conditions.

ENERGY PLANNING, POLICY, AND ECONOMY,POWER TRANSMI↗
DOE OSTI2021

Seawater sea-sand engineered/strain-hardening cementitious composites (ECC/SHCC): Assessment and modeling of crack characteristics

Highlights: • A probabilistic model was proposed to model the crack width evolution of SS-ECC at different strain levels. • The Weibull distribution fit the crack width distribution better than the log-normal distribution. • A 5-D representation was proposed to assess the SS-ECC by considering both cracking and mechanical performance. • Larger sea-sand size and lower fiber dosage led to larger crack widths in SS-ECC under the same tensile strain. • 18-mm PE fiber led to larger crack widths in SS-ECC at strain >2%, due to a large fraction of fiber rupture. Seawater sea-sand Engineered Cementitious Composites (SS-ECC) is a new version of ECC for marine constructions facing the scarcity of freshwater and river/manufactured sand. This study aims to assess and model the crack characteristics of SS-ECC, which are critical for its applications with non-corrosive reinforcements. The influence of sea-sand size, fiber length and fiber dosage on the crack characteristics of SS-ECC was explored. A five-dimensional representation was proposed to assess the overall performance of SS-ECC, by comprehensively considering both the crack characteristics (i.e., crack width and its variation) and the mechanical properties (i.e., compressive and tensile properties). A probabilistic model was also proposed to describe the stochastic nature and evolution of crack width, and it can be used to estimate the critical tensile strain on SS-ECC for a given crack-width limit and cumulative probability. The findings and proposed methods can facilitate the design of SS-ECC in marine and coastal structures.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Cost-Effectiveness Analysis of Intensity Modulated Radiation Therapy Versus 3-Dimensional Conformal Radiation Therapy for Anal Cancer

Purpose: To compare the cost-effectiveness of intensity modulated radiation therapy (IMRT) and 3-dimensional conformal radiation therapy (3D-CRT) for anal cancer and determine disease, patient, and treatment parameters that influence the result. Methods and Materials: A Markov decision model was designed with the various disease states for the base case of a 65-year-old patient with anal cancer treated with either IMRT or 3D-CRT and concurrent chemotherapy. Health states accounting for rates of local failure, colostomy failure, treatment breaks, patient prognosis, acute and late toxicities, and the utility of toxicities were informed by existing literature and analyzed with deterministic and probabilistic sensitivity analysis. Results: In the base case, mean costs and quality-adjusted life expectancy in years (QALY) for IMRT and 3D-CRT were $32,291 (4.81) and $28,444 (4.78), respectively, resulting in an incremental cost-effectiveness ratio of $128,233/QALY for IMRT compared with 3D-CRT. Probabilistic sensitivity analysis found that IMRT was cost-effective in 22%, 47%, and 65% of iterations at willingness-to-pay thresholds of $50,000, $100,000, and $150,000 per QALY, respectively. Conclusions: In our base model, IMRT was a cost-ineffective strategy despite the reduced acute treatment toxicities and their associated costs of management. The model outcome was sensitive to variations in local and colostomy failure rates, as well as patient-reported utilities relating to acute toxicities.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2015

Individualized Positron Emission Tomography–Based Isotoxic Accelerated Radiation Therapy Is Cost-Effective Compared With Conventional Radiation Therapy: A Model-Based Evaluation

Purpose: To evaluate long-term health effects, costs, and cost-effectiveness of positron emission tomography (PET)-based isotoxic accelerated radiation therapy treatment (PET-ART) compared with conventional fixed-dose CT-based radiation therapy treatment (CRT) in non-small cell lung cancer (NSCLC). Methods and Materials: Our analysis uses a validated decision model, based on data of 200 NSCLC patients with inoperable stage I-IIIB. Clinical outcomes, resource use, costs, and utilities were obtained from the Maastro Clinic and the literature. Primary model outcomes were the difference in life-years (LYs), quality-adjusted life-years (QALYs), costs, and the incremental cost-effectiveness and cost/utility ratio (ICER and ICUR) of PET-ART versus CRT. Model outcomes were obtained from averaging the predictions for 50,000 simulated patients. A probabilistic sensitivity analysis and scenario analyses were carried out. Results: The average incremental costs per patient of PET-ART were €569 (95% confidence interval [CI] €−5327-€6936) for 0.42 incremental LYs (95% CI 0.19-0.61) and 0.33 QALYs gained (95% CI 0.13-0.49). The base-case scenario resulted in an ICER of €1360 per LY gained and an ICUR of €1744 per QALY gained. The probabilistic analysis gave a 36% probability that PET-ART improves health outcomes at reduced costs and a 64% probability that PET-ART is more effective at slightly higher costs. Conclusion: On the basis of the available data, individualized PET-ART for NSCLC seems to be cost-effective compared with CRT.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

Cost-Effectiveness Analysis of Chemoradiation Therapy Versus Transoral Robotic Surgery for Human Papillomavirus–Associated, Clinical N2 Oropharyngeal Cancer

Purpose: To perform a cost-effectiveness analysis of primary chemoradiation therapy (CRT) versus transoral robotic surgery (TORS) for clinical N2, human papillomavirus (HPV)-positive oropharyngeal carcinoma. Methods and Materials: We developed a Markov model to describe the health states after treatment with CRT or TORS, followed by adjuvant radiation therapy or CRT in the presence of high-risk pathology (positive margins or extracapsular extension). Outcomes, toxicities, and costs were extracted from the literature. One-way sensitivity analyses (SA) were performed over a wide range of parameters, as were 2-way SA between the key variables. Probabilistic SA and value of information studies were performed over key parameters. Results: The expected quality-adjusted life years (QALYs)/total costs for CRT and TORS were 7.31/$50,100 and 7.29/$62,200, respectively, so that CRT dominated TORS. In SA, primary CRT was almost always cost-effective up to a societal willingness-to-pay of $200,000/QALY, unless the locoregional recurrence risk after TORS was 30% to 50% lower, at which point it became cost effective at a willingness-to-pay of $50-100,000/QALY. Probabilistic SA confirmed the importance of locoregional recurrence risk, and the value of information in precisely knowing this parameter was more than $7M per year. If the long-term utility after TORS was 0.03 lower than CRT, CRT was cost-effective over nearly any assumption. Conclusions: Under nearly all assumptions, primary CRT was the cost-effective therapy for HPV-associated, clinical N2 OPC. However, in the hypothetical event of a large relative improvement in LRR with surgery and equivalent long-term utilities, primary TORS would become the higher-value treatment, arguing for prospective, comparative study of the 2 paradigms.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

Establishing Cost-Effective Allocation of Proton Therapy for Breast Irradiation

Purpose: Cardiac toxicity due to conventional breast radiation therapy (RT) has been extensively reported, and it affects both the life expectancy and quality of life of affected women. Given the favorable oncologic outcomes in most women irradiated for breast cancer, it is increasingly paramount to minimize treatment side effects and improve survivorship for these patients. Proton RT offers promise in limiting heart dose, but the modality is costly and access is limited. Using cost-effectiveness analysis, we provide a decision-making tool to help determine which breast cancer patients may benefit from proton RT referral. Methods and Materials: A Markov cohort model was constructed to compare the cost-effectiveness of proton versus photon RT for breast cancer management. The model was analyzed for different strata of women based on age (40 years, 50 years, and 60 years) and the presence or lack of cardiac risk factors (CRFs). Model entrants could have 1 of 3 health states: healthy, alive with coronary heart disease (CHD), or dead. Base-case analysis assumed CHD was managed medically. No difference in tumor control was assumed between arms. Probabilistic sensitivity analysis was performed to test model robustness and the influence of including catheterization as a downstream possibility within the health state of CHD. Results: Proton RT was not cost-effective in women without CRFs or a mean heart dose (MHD) <5 Gy. Base-case analysis noted cost-effectiveness for proton RT in women with ≥1 CRF at an approximate minimum MHD of 6 Gy with a willingness-to-pay threshold of $100,000/quality-adjusted life-year. For women with ≥1 CRF, probabilistic sensitivity analysis noted the preference of proton RT for an MHD ≥5 Gy with a similar willingness-to-pay threshold. Conclusions: Despite the cost of treatment, scenarios do exist whereby proton therapy is cost-effective. Referral for proton therapy may be cost-effective for patients with ≥1 CRF in cases for which photon plans are unable to achieve an MHD <5 Gy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

Stochastic Predictions of Cell Kill During Stereotactic Ablative Radiation Therapy: Do Hypoxia and Reoxygenation Really Matter?

Purpose: To simulate stereotactic ablative radiation therapy on hypoxic and well-oxygenated in silico tumors, incorporating probabilistic parameter distributions and linear-quadratic versus linear-quadratic-cubic methodology and the evaluation of optimal fractionation schemes using biological effective dose (BED{sub α/β=10} {sub or} {sub 3}) comparisons. Methods and Materials: A temporal tumor growth and radiation therapy algorithm simulated high-dose external beam radiation therapy using stochastic methods. Realistic biological proliferative cellular hierarchy and pO{sub 2} histograms were incorporated into the 10{sup 8}-cell tumor model, with randomized radiation therapy applied during continual cell proliferation and volume-based gradual tumor reoxygenation. Dose fractions ranged from 6-35 Gy, with predictive outcomes presented in terms of the total doses (converted to BED) required to eliminate all cells that could potentially regenerate the tumor. Results: Well-oxygenated tumor control BED{sub 10} outcomes were not significantly different for high-dose versus conventional radiation therapy (BED{sub 10}: 79-84 Gy; Equivalent Dose in 2 Gy fractions with α/β of 10: 66-70 Gy); however, total treatment times decreased from 7 down to 1-3 weeks. For hypoxic tumors, an additional 28 Gy (51 Gy BED{sub 10}) was required, with BED{sub 10} increasing with dose per fraction due to wasted dose in the final fraction. Fractions of 9 Gy compromised well for total treatment time and BED, with BED{sub 10}:BED{sub 3} of 84:176 Gy for oxic and 132:278 Gy for non-reoxygenating hypoxic tumors. Initial doses of 12 Gy followed by 6 Gy further increased the therapeutic ratio. When delivering ≥9 Gy per fraction, applying reoxygenation and/or linear-quadratic-cubic cell survival both affected tumor control doses by a significant 1-2 fractions. Conclusions: The complex temporal dynamics of tumor oxygenation combined with probabilistic cell kinetics in the modeling of radiation therapy requires sophisticated stochastic modeling to predict tumor cell kill. For stereotactic ablative radiation therapy, high doses in the first week followed by doses that are more moderate may be beneficial because a high percentage of hypoxic cells could be eradicated early while keeping the required BED{sub 10} relatively low and BED{sub 3} toxicity to tolerable levels.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2017

Cost-Effectiveness Analysis of Preoperative Versus Postoperative Radiation Therapy in Extremity Soft Tissue Sarcoma

Purpose: Surgery combined with radiation therapy (RT) is the cornerstone of multidisciplinary management of extremity soft tissue sarcoma (STS). Although RT can be given in either the preoperative or the postoperative setting with similar local recurrence and survival outcomes, the side effect profiles, costs, and long-term functional outcomes are different. The aim of this study was to use decision analysis to determine optimal sequencing of RT with surgery in patients with extremity STS. Methods and Materials: A cost-effectiveness analysis was conducted using a state transition Markov model, with quality-adjusted life years (QALYs) as the primary outcome. A time horizon of 5 years, a cycle length of 3 months, and a willingness-to-pay threshold of $50,000/QALY was used. One-way deterministic sensitivity analyses were performed to determine the thresholds at which each strategy would be preferred. The robustness of the model was assessed by probabilistic sensitivity analysis. Results: Preoperative RT is a more cost-effective strategy ($26,633/3.00 QALYs) than postoperative RT ($28,028/2.86 QALYs) in our base case scenario. Preoperative RT is the superior strategy with either 3-dimensional conformal RT or intensity-modulated RT. One-way sensitivity analyses identified the relative risk of chronic adverse events as having the greatest influence on the preferred timing of RT. The likelihood of preoperative RT being the preferred strategy was 82% on probabilistic sensitivity analysis. Conclusions: Preoperative RT is more cost effective than postoperative RT in the management of resectable extremity STS, primarily because of the higher incidence of chronic adverse events with RT in the postoperative setting.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2013

A flexible numerical approach for quantification of epistemic uncertainty

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty. And there have been few studies and methods to deal with epistemic uncertainty. A recent work can be found in [J. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty, J. Comput. Phys. 229 (2010) 4648–4663], where a framework for numerical treatment of epistemic uncertainty was proposed. The method is based on solving an encapsulation problem, without using any probability information, in a hypercube that encapsulates the unknown epistemic probability space. If more probabilistic information about the epistemic variables is known a posteriori, the solution statistics can then be evaluated at post-process steps. In this paper, we present a new method, similar to that of Jakeman et al. but significantly extending its capabilities. Most notably, the new method (1) does not require the encapsulation problem to be in a bounded domain such as a hypercube; (2) does not require the solution of the encapsulation problem to converge point-wise. In the current formulation, the encapsulation problem could reside in an unbounded domain, and more importantly, its numerical approximation could be sought in L{sup p} norm. These features thus make the new approach more flexible and amicable to practical implementation. Both the mathematical framework and numerical analysis are presented to demonstrate the effectiveness of the new approach.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

A partially reflecting random walk on spheres algorithm for electrical impedance tomography

In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute the voltage-to-current map in an embarrassingly parallel manner, which is an important issue with regard to the corresponding inverse problem. Our method uses the well-known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions. We analyze the global bias and the variance of the new estimator both theoretically and experimentally. Subsequently, the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Bayesian analysis of rare events

In many areas of engineering and science there is an interest in predicting the probability of rare events, in particular in applications related to safety and security. Increasingly, such predictions are made through computer models of physical systems in an uncertainty quantification framework. Additionally, with advances in IT, monitoring and sensor technology, an increasing amount of data on the performance of the systems is collected. This data can be used to reduce uncertainty, improve the probability estimates and consequently enhance the management of rare events and associated risks. Bayesian analysis is the ideal method to include the data into the probabilistic model. It ensures a consistent probabilistic treatment of uncertainty, which is central in the prediction of rare events, where extrapolation from the domain of observation is common. We present a framework for performing Bayesian updating of rare event probabilities, termed BUS. It is based on a reinterpretation of the classical rejection-sampling approach to Bayesian analysis, which enables the use of established methods for estimating probabilities of rare events. By drawing upon these methods, the framework makes use of their computational efficiency. These methods include the First-Order Reliability Method (FORM), tailored importance sampling (IS) methods and Subset Simulation (SuS). In this contribution, we briefly review these methods in the context of the BUS framework and investigate their applicability to Bayesian analysis of rare events in different settings. We find that, for some applications, FORM can be highly efficient and is surprisingly accurate, enabling Bayesian analysis of rare events with just a few model evaluations. In a general setting, BUS implemented through IS and SuS is more robust and flexible.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗