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NASA NTRS2005

Constraining Oceanic Sources and Sinks of CO2 by Inverse Modeling: A First Step Toward Assimilation of In Situ and Remote Data

The primary goal of this collaborative project, lead by Nicholas Gruber at UCLA, is to better quantify the air-sea fluxes of pre-industrial and anthropogenic carbon dioxide CO2 for the global ocean with a newly developed inverse method. The method involves using ocean general circulation models (OGCMs) to characterize the subsurface distribution of tracers injected for different regions of the ocean surface. By linearly combining a series of these basis (or Greens) functions, one should be able to recreate any arbitrary ocean tracer field. In our case, we are using the global database of ocean dissolved inorganic carbon (DIC) from the WOCE/JGOFS global CO2 survey from the 1990s. The observed DIC field has been partitioned into pre-industrial and anthropogenic components using the empirical C* technique. We then solve for an optimal combination of the regional basis functions that best match the two sets observations in a least-squares sense with singular value decomposition. This in turn provides solutions for the regional air-sea CO2 fluxes (pre-industrial and anthropogenic) that are independent of more traditional estimates derived from surface water disequilibria in the partial pressure of CO2 and wind speed gas exchange parameterizations. We are apply the method to a suite of nine different OGCMs to test the robustness of the derived surface fluxes fields to differences in ocean circulation. Together with a collegue Keith Lindsay at NCAR, S. Doney's role in this project was to create the appropriate steady-state and temporally evolving basis functions from the NCAR CSM-1 ocean circulation model. Because of a slow start across the project in defining a common set of experimental protocols, we requested and were given a one year, no cost extension. Both the steady-state and temporally evolving basis functions were delivered to the UCLA group for further analysis. Additional information is included in the original extended abstract.

Doney, Scott↗

NASA NTRS2018

An Evolutionary Computation System Design Concept for Developing Controlled Closed Ecosystems: An Intelligent Systems Approach to Foster Gravitational Ecosystem Research for Developing Sustainable Communities in Space and on Earth

An adjustably-autonomous intelligent systems approach for developing Closed Ecosystems (CESs) is presented, which includes a design concept and preliminary design details for the Controlled Closed-Ecosystem Development System (CCEDS) and the Orbiting Modular Artificial-Gravity Spacecraft (OMAGS). The paper is divided into three sections: CESs, the CCEDS Design Concept, and Orbiting Fractional-Gravity Closed Ecosystems OMAGS design concept. The first section briefly describes Closed EcoSystems (CESs), complex adaptive systems, biomes, microbial microbiomes, and their relevance for the study of astrobiology. This section also discusses initial efforts in the development of Closed Environment Life Support Systems (CELSSs) for sustainable communities in space and on Earth. This section concludes with a discussion of the bioregenerative life support system challenge of and the corresponding consequences due to the inverse relationship of the very small human biomass/non-human biomass ratio overall on the Earth with respect to the extremely large human biomass/non-human-biomass ratio found in cities and the International Space Station. The second section describes the CCEDS design concept, which consists of a population of controlled colonies of CES Modules (CESMs), each an integrated CES, continually generating data for an intelligent system that operates the CESs and their CESMs. A variety of CESM types and their use are briefly described. The CCEDS intelligent system uses an evolutionary computation algorithm described in this section to develop and optimize these CESs to increase their viability duration and the size of the animals they support with the ultimate goal to support populations of humans, both on Earth and in space. The CCEDS architecture, its five control subsystems, and its five evolutionary computation levels are also discussed. The section concludes with a discussion of several CCEDS design strategies. The third section summarizes the OMAGS design concept for a spacecraft with a payload consisting of CESs in an orbiting spacecraft centrifuge that operates for at least 5 years. The spacecraft concept is described including its 150cm-radius centrifuge with a 2 ton & 3,000 liter bioscience payload capacity for 24 CESMs. The centrifuge design has four physical levels for its CESMs, each level subject to a different fractional gravity level. This section presents the spacecraft benefits of being designed and operated such that the spacecraft and payload centrifuge wheel counter-rotate resulting in net zero angular momentum and zero gyroscopic forces. Artificial-gravity generation by centripetal acceleration is also discussed. This section concludes by showing the external specifications of the CESMs and their layout in the centrifuge, followed by discussing the multi-payload module rationale. In tandem, the CCEDS and OMAGS systems can be used to foster gravitational ecosystem research for developing sustainable communities in space and on Earth.

Dorais, Gregory A.↗

NASA NTRS2017

Characterization of a Method for Inverse Heat Conduction Using Real and Simulated Thermocouple Data

It is often impractical to instrument the external surface of high-speed vehicles due to the aerothermodynamic heating. Temperatures can instead be measured internal to the structure using embedded thermocouples, and direct and inverse methods can then be used to estimate temperature and heat flux on the external surface. Two thermocouples embedded at different depths are required to solve direct and inverse problems, and filtering schemes are used to reduce noise in the measured data. Accuracy in the estimated surface temperature and heat flux is dependent on several factors. Factors include the thermocouple location through the thickness of a material, the sensitivity of the surface solution to the error in the specified location of the embedded thermocouples, and the sensitivity to the error in thermocouple data. The effect of these factors on solution accuracy is studied using the methodology discussed in the work of Pizzo, et. al.1 A numerical study is performed to determine if there is an optimal depth at which to embed one thermocouple through the thickness of a material assuming that a second thermocouple is installed on the back face. Solution accuracy will be discussed for a range of embedded thermocouple depths. Moreover, the sensitivity of the surface solution to (a) the error in the specified location of the embedded thermocouple and to (b) the error in the thermocouple data are quantified using numerical simulation, and the results are discussed.

Pizzo, Michelle E.↗

DOE OSTI2024

Microstructural Underpinnings of Giant Intrinsic Exchange Bias in Epitaxial NiCo 2 O 4 Thin Films

Understanding intrinsic exchange bias in nominally single-component ferromagnetic or ferrimagnetic materials is crucial for simplifying related device architectures. However, the mechanisms behind this phenomenon and its tunability remain elusive, which hinders the efforts to achieve unidirectional magnetization for widespread applications. Inspired by the high tunability of ferrimagnetic inverse spinel NiCo 2 O 4 , the origin of intrinsic exchange bias in NiCo 2 O 4 (111) films deposited on Al 2 O 3 (0001) substrates are investigated. The comprehensive characterizations, including electron diffraction, X-ray reflectometry and spectroscopy, and polarized neutron reflectometry, reveal that intrinsic exchange bias in NiCo 2 O 4 (111)/Al 2 O 3 (0001) arises from a reconstructed antiferromagnetic rock-salt Ni x Co 1-x O layer at the interface between the film and the substrate due to a significant structural mismatch. Remarkably, by engineering the interfacial structure under optimal growth conditions, it can achieve exchange bias larger than coercivity, leading to unidirectional magnetization. Such giant intrinsic exchange bias can be utilized for realistic device applications. This work establishes a new material platform based on NiCo 2 O 4 , an emergent spintronics material, to study tunable interfacial magnetic and spintronic properties.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Mechanism-Resolved PFM of Ferroionic and Ferroelectric Responses in Thickness-Gradient Hf 0.5 Zr 0.5 O 2 Libraries

Resolving growth mechanisms and thickness evolution of functional properties is one of the key tasks in materials discovery and optimization involving thin-film materials, traditionally requiring significant experimental budgets. Here we introduce the combination of thickness-gradient libraries and automated scanning probe microscopy as a systematic pathway to elucidate growth modes and disentangle ferroelectric and electrochemical contributions in ferroelectric thin films. As a model system, we explore the Hf 0.5 Zr 0.5 O 2 (HZO) gradient thin films grown on La x Sr 1-x MnO 3 (LSMO) bottom electrode thin films. Automated piezoresponse force microscopy, spectroscopy, and lithography reveals that irreversible topographic deformation arises from electrochemical activity at the LSMO surface, whereas reversible phase inversion in HZO reflects ferroelectric switching. Automated topography height-map scans are used to further quantify nucleation density, particle-size evolution, and roughness correlations across the thickness-gradient, demonstrating that improved plume stabilization during growth suppresses interfacial reactions and promotes dense, fine-grained HZO conducive to ferroelectric phase formation. This combined materials-engineering and automated-SPM framework establishes a platform for high-throughput, mechanism-resolved characterization of ferroionic and ferroelectric responses in complex oxide films.

FOS: Physical sciences↗

DOE OSTI2025

Phase II Field Demonstration at Lansing Smith Generating Plant, Southport, Florida (Final Report)

The Final Technical Report: Field Test Design and Pressure Management Strategies for Phase II Field Demonstration of Optimal Pressure Control, Plume Management, and Produced Water Strategies presents the culmination of multi-year efforts under the U.S. Department of Energy’s Brine Extraction and Storage Test (BEST) program, led by EPRI in partnership with Gulf Power at the Plant Smith site near Panama City, Florida. The project was designed to evaluate and demonstrate the technical feasibility of managing subsurface pressures and fluid movement associated with large-scale CO₂ injection, using low-salinity water as a proxy fluid. Through a combination of field injection testing, reservoir modeling, and optimization studies, the research team developed and refined active and passive brine extraction strategies aimed at controlling injection-induced pressure buildup, mitigating risks of fault activation, and managing plume migration. The field demonstration incorporated a new injection well (TIW-2), a new monitoring/extraction well (TEMW-A), and an existing well (TIW-1) repurposed for passive pressure relief. Complementary geophysical monitoring was designed to track plume development and evaluate the effectiveness of pressure management strategies. The report details the integrated workflow encompassing wellfield development, regulatory permitting, model calibration, and survey design. It includes a comprehensive electromagnetic (EM) modeling and inversion study used to develop a cost-effective, time-lapse geophysical monitoring plan capable of imaging the evolving low-salinity plume within the highly saline Lower Tuscaloosa formation. Reservoir simulation results guided the design of the 17-month injection program and the timing of active extraction to maintain formation pressures below a hypothetical fault reactivation threshold. Supporting analyses evaluated potential injectivity risks related to clay fines migration and geochemical incompatibility, identifying practical mitigation measures such as salinity and pH control. The final design recommends a series of crosswell EM surveys—conducted before, during, and after injection—as the most effective and economical approach for plume imaging, supplemented by continuous downhole pressure and flow monitoring. Collectively, the study provides a field-ready framework for cost-effective pressure management and monitoring in support of future CO₂ storage projects.

01 COAL, LIGNITE, AND PEAT↗

DOE OSTI2025

Experimental study of energy-dependent angular broadening of MeV electron beams for high-resolution imaging in thick samples

In scanning transmission electron microscopy (STEM), spatial resolution is primarily influenced by the projected size of the electron probe within the specimen. In thin samples, a large semi-convergence angle enables a tightly focused beam and sub-nanometer resolution. However, in thick specimens, resolution is fundamentally limited by transverse beam broadening from multiple large-angle scattering events—for example, a probe with 10 mrad angular divergence can broaden by ∼100 nm over a 10 μm path. Since this broadening scales inversely with beam energy, MeV-STEM offers a promising route for high-resolution imaging in thick materials. To quantitatively assess this effect, we performed high-precision measurements at UCLA’s PEGASUS beamline, characterizing beam divergence and intensity profiles for 3–8 MeV electrons transmitted through a wedged-silicon sample of varying thickness. Our results reconcile discrepancies among analytical models and validate Monte Carlo simulations. Here, we find that increasing beam energy from 3.0 to 5.8 MeV reduces angular broadening by a factor of 2.6, with diminishing returns observed at 7.6 MeV. These findings provide a quantitative framework for optimizing MeV-STEM parameters in high-resolution imaging of thick biological and microelectronic specimens, and for guiding beam energy selection in other advanced imaging modes beyond STEM.

36 MATERIALS SCIENCE↗

NASA NTRS1996

The Structure and Stability of B(n)H(+) Clusters

The geometries of the B(n)H(+) clusters for n=1-13 have been optimized at the B3LYP level of theory. Excluding B2H(+) all clusters are low-spin coupled. The structures appear to be related to the bare B(+)(n) clusters in two ways: 1) an H atom is added to the B(+)(n) cluster or 2) a BH is added to the B(+)(n-1) cluster. The B(+)(n)-H binding energies are computed at the B3LYP level of theory, and calibrated n using the CCSD(T) level of theory. The computed results tend to fall between the experimental best estimates and the experimental lower bounds. The B(+)(n)-H binding energies show an inverse correspondence with the B+(n-1)-B values) while the B(+)(n-1)-B and B(+)(n-1)-BH results tend to parallel each other.

Ricca, Alessandra↗

NASA NTRS1999

Time of Arrival Retrievals on an Oblate Spheroidal Earth

The current National Lightning Detection Network (NLDN) retrieval algorithm is based on a chi sq minimization. This numerical approach finds the lightning ground strike location on an ellipsoidal Earth that optimally agrees with multiple station time of arrival (TOA) measurements and, if available, magnetic bearing data. In addition, an analytic solution for determining lightning ground strike locations on a spherical Earth using only TOA data has been recently introduced. In the current work, a quasi-analytic approach is suggested for determining lightning ground strike locations on an oblate spheroidal Earth by perturbing the spherical model results proposed by Koshak. Latitude, longitude, time, and the associated perturbed quantities are related through terms in a Taylor series. The correction terms may be considered collectively as a vector which may be calculated by an overconstrained inversion. Expressions for the derivatives contained in the Taylor series are given in terms of elliptic integrals.

Solakiewicz, Richard J.↗

NASA NTRS2013

Adaptive Sampling of Spatiotemporal Phenomena with Optimization Criteria

This work was designed to find a way to optimally (or near optimally) sample spatiotemporal phenomena based on limited sensing capability, and to create a model that can be run to estimate uncertainties, as well as to estimate covariances. The goal was to maximize (or minimize) some function of the overall uncertainty. The uncertainties and covariances were modeled presuming a parametric distribution, and then the model was used to approximate the overall information gain, and consequently, the objective function from each potential sense. These candidate sensings were then crosschecked against operation costs and feasibility. Consequently, an operations plan was derived that combined both operational constraints/costs and sensing gain. Probabilistic modeling was used to perform an approximate inversion of the model, which enabled calculation of sensing gains, and subsequent combination with operational costs. This incorporation of operations models to assess cost and feasibility for specific classes of vehicles is unique.

Chien, Steve A.↗

NASA NTRS2013

Downscaling Satellite Precipitation with Emphasis on Extremes: A Variational 1-Norm Regularization in the Derivative Domain

The increasing availability of precipitation observations from space, e.g., from the Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) Mission, has fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can handle large data sets in computationally efficient ways while optimally reproducing desired properties of the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism for downscaling satellite precipitation observations, which explicitly allows for the preservation of some key geometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to high-intensity regions embedded within lower-intensity areas), coherent spatial structures (due to regions of slowly varying rainfall),and thicker-than-Gaussian tails of precipitation gradients and intensities. Specifically, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational approach (variational downscaling) where the regularization term is selected to impose the desired smoothness in the solution while allowing for some steep gradients(called 1-norm or total variation regularization). We demonstrate the duality between this geometrically inspired solution and its Bayesian statistical interpretation, which is equivalent to assuming a Laplace prior distribution for the precipitation intensities in the derivative (wavelet) space. When the observation operator is not known, we discuss the effect of its misspecification and explore a previously proposed dictionary-based sparse inverse downscaling methodology to indirectly learn the observation operator from a database of coincidental high- and low-resolution observations. The proposed method and ideas are illustrated in case studies featuring the downscaling of a hurricane precipitation field.

Hurricanes↗

NASA NTRS2011

MoSST DAS: The First Working Geomagnetic Data Assimilation System

The Earth possesses an internal magnetic field (geomagnetic field) generated by convection in the outer core (geodynamo). Previous efforts have been focused along two distinct paths: (1) numerical geodynamo modeling to understand the origin of the geomagnetic field, and the mechanisms of geomagnetic secular variations (SV); and (2) geomagnetic field modeling to map the spatial/temporal variations of the field from geomagnetic data, and to derive core properties, e.g. inversion of core flow near the core-mantle boundary (CMB). Geomagnetic data assimilation is a new approach emerged over the past 5 years: surface observations are assimilated with geodynamo models for better understanding of the core dynamical state, and accurately prediction of SV. In collaboration with several geomagnetic research groups, we have developed the first working geomagnetic data assimilation system, Modular, Scalable, Self-consistent, and Three-dimensional (MoSST) DAS, that includes the MoSST numerical dynamo model; 7000 years of geomagnetic field maps from several field models utilizing satellite and ground observatory data, historical magnetic records and archeo/paleo magnetic data; and an ensemble based optimal interpolation (01) assimilation algorithm. With this system, we have demonstrated clearly that the assimilated core dynamical state is substantially different from those of pure geodynamo simulations. Ensemble assimilation runs also show the convergence of the assimilated solutions inside the core, suggesting that the simulation state is pulled closer to the truth via data assimilation. The forecasts from this system are also very accurate: the 5-year forecast of the geomagnetic field agrees very well with the observations; and the 5-year secular variation forecast is more accurate than the IGRF SV forecast models in the past. Using geomagnetic records up to 2009, we have made an SV forecast for the period from 2010-2015, and is a candidate SV model for IGRF-11.

Kuang, Weijia↗

DOE OSTI2025

Dynamics of inverse metal oxides on metal catalysts using spectro-kinetics: reversible Brønsted acid site formation and irreversible reduction

Brønsted acid sites (BASs) in inverse catalysts are vital for the selective hydrogenolysis of polyols, specifically cleaving secondary C–O bonds. These BASs form dynamically in situ in an H 2 environment. While H 2 enables rapid BAS generation on short timescales, it reduces the catalyst at prolonged exposures. The active center for BAS generation, the kinetics of BAS formation, its reverse decomposition, and the irreversible oxide reduction have lacked direct experimental evidence. Here, aided by advanced spectro-kinetic studies, we identify trimeric W 3 O x sites on Pt as the active centers for BAS generation, whereas isolated WO x species on SiO 2 act merely as spectator species, demonstrated using an inverse WO x /Pt catalyst as a representative system. A detailed kinetic profile capturing the dynamics of W 3 O x sites on Pt is also established. The rate constant for BAS formation is two orders of magnitude higher than for its decomposition, which is one order of magnitude faster than the irreversible site reduction. Co-fed H 2 O suppresses the site reduction by ∼50%. Furthermore, the H 2 partial pressure plays an important role. While lower gas-phase H 2 partial pressure does not influence the reversible BAS formation, it can significantly (∼3×) suppress catalyst reduction. Finally, these findings offer critical insights into optimizing reaction conditions through periodic H 2 pulsing, enhancing catalyst stability and performance in hydrogenolysis reactions.

Sourav, Sagar [Indian Inst. of Technology (IIT), M↗

NASA NTRS2008

Temporal Stability of Surface Roughness Effects on Radar Based Soil Moisture Retrieval During the Corn Growth Cycle

A representative soil surface roughness parameterization needed for the retrieval of soil moisture from active microwave satellite observation is difficult to obtain through either in-situ measurements or remote sensing-based inversion techniques. Typically, for the retrieval of soil moisture, temporal variations in surface roughness are assumed to be negligible. Although previous investigations have suggested that this assumption might be reasonable for natural vegetation covers (Moran et al. 2002, Thoma et al. 2006), insitu measurements over plowed agricultural fields (Callens et al. 2006) have shown that the soil surface roughness can change considerably over time. This paper reports on the temporal stability of surface roughness effects on radar observations and soil moisture retrieved from these radar observations collected once a week during a corn growth cycle (May 10th - October 2002). The data set employed was collected during the Optimizing Production Inputs for Economic and Environmental Enhancement (OPE3) field campaign covering this 2002 corn growth cycle and consists of dual-polarized (HH and VV) L-band (1.6 GHz) acquired at view angles of 15, 35, and 55 degrees. Cross-polarized L baud radar data were also collected as part of this experiment, but are not used in the analysis reported on here. After accounting for vegetation effects on radar observations, time-invariant optimum roughness parameters were determined using the Integral Equation Method (IEM) and radar observations acquired over bare soil and cropped conditions (the complete radar data set includes entire corn growth cycle). The optimum roughness parameters, soil moisture retrieval uncertainty, temporal distribution of retrieval errors and its relationship with the weather conditions (e.g. rainfall and wind speed) have been analyzed. It is shown that over the corn growth cycle, temporal roughness variations due to weathering by rain are responsible for almost 50% of soil moisture retrieval uncertainty depending on the sensing configuration. The effects of surface roughness variations are found to be smallest for observations acquired at a view angle of 55 degrees and HH polarization. A possible explanation for this result is that at 55 degrees and HH polarization the effect of vertical surface height changes on the observed radar response are limited because the microwaves travel parallel to the incident plane and as a result will not interact directly with vertically oriented soil structures.

Joseph, A.T.↗

NASA NTRS2011

Electromagnetic Scattering Analysis of Large Size Asteroids/Comets for Reflection/Transmission Tomography (RTT)

A precise knowledge of the interior structure of asteroids, comets, and Near Earth Objects (NEO) is important to assess the consequences of their impacts with the Earth and develop efficient mitigation strategies. Knowledge of their interior structure also provides opportunities for extraction of raw materials for future space activities. Low frequency radio sounding is often proposed for investigating interior structures of asteroids and NEOs. For designing and optimizing radio sounding instrument it is advantageous to have an accurate and efficient numerical simulation model of radio reflection and transmission through large size bodies of asteroid shapes. In this presentation we will present electromagnetic (EM) scattering analysis of electrically large size asteroids using (1) a weak form formulation and (2) also a more accurate hybrid finite element method/method of moments (FEM/MOM) to help estimate their internal structures. Assuming the internal structure with known electrical properties of a sample asteroid, we first develop its forward EM scattering model. From the knowledge of EM scattering as a function of frequency and look angle we will then present the inverse scattering procedure to extract its interior structure image. Validity of the inverse scattering procedure will be presented through few simulation examples.

Deshpande, Manohar↗

DOE OSTI2025

Demonstration of MgCr 2– x Mn x O 4 Spinel Oxide Cathodes in High-Voltage Mg Batteries

Solid-solution oxide spinels with high redox voltages and facile Mg 2+ mobility have been identified as promising candidates for practical, high-voltage cathodes in Mg batteries. In this work, we discuss the development of MgCr 2-x Mn x O 4 [x = 0.5, 1, 1.2] solid-solution spinel oxides as a cathode material and their electrochemical performance paired with an Mg anode in a full cell. This work presents the first demonstration of full cells with these materials. Mg-Cr-Mn spinel oxides with varying Cr and Mn contents were synthesized using alternative synthetic routes for optimal electrochemical performance. High-resolution synchrotron powder X-ray diffraction (PXRD), solid-state nuclear magnetic resonance (NMR) spectroscopy, and electron microscopy showed that these different synthetic routes resulted in changes in structures and particle morphologies, which in turn affect the electrochemical performance. Particularly, the urea coprecipitation synthetic route resulted in high-surface-area particles that enabled lower overpotentials and increased discharge capacity. The high surface area also resulted in expedited structural degradation caused by the irreversible migration of Mg 2+ into normally vacant 16c sites in the spinel lattice. This structural degradation was lessened by using a hydrosauna-urea synthesis method, which decreased the Mg/Mn inversion ratio while retaining high-surface-area particles with good cycling performance. Furthermore, our findings highlight the necessity for high surface area or nanostructured spinel oxide cathodes with minimized Mg-Mn inversion to enable spinel oxide cathodes in Mg full cells.

Mg anode↗

NASA NTRS1975

A random search algorithm for laboratory computers

The small laboratory computer is ideal for experimental control and data acquisition. Postexperimental data processing is often performed on large computers because of the availability of sophisticated programs, but costs and data compatibility are negative factors. Parameter optimization can be accomplished on the small computer, offering ease of programming, data compatibility, and low cost. A previously proposed random-search algorithm ('random creep') was found to be very slow in convergence. A method is proposed (the 'random leap' algorithm) which starts in a global search mode and automatically adjusts step size to speed convergence. A FORTRAN executive program for the random-leap algorithm is presented which calls a user-supplied function subroutine. An example of a function subroutine is given which calculates maximum-likelihood estimates of receiver operating-characteristic parameters from binary response data. Other applications in parameter estimation, generalized least squares, and matrix inversion are discussed.

Curry, R. E.↗

NASA NTRS1992

Inference in infinite-dimensional inverse problems - Discretization and duality

Many techniques for solving inverse problems involve approximating the unknown model, a function, by a finite-dimensional 'discretization' or parametric representation. The uncertainty in the computed solution is sometimes taken to be the uncertainty within the parametrization; this can result in unwarranted confidence. The theory of conjugate duality can overcome the limitations of discretization within the 'strict bounds' formalism, a technique for constructing confidence intervals for functionals of the unknown model incorporating certain types of prior information. The usual computational approach to strict bounds approximates the 'primal' problem in a way that the resulting confidence intervals are at most long enough to have the nominal coverage probability. There is another approach based on 'dual' optimization problems that gives confidence intervals with at least the nominal coverage probability. The pair of intervals derived by the two approaches bracket a correct confidence interval. The theory is illustrated with gravimetric, seismic, geomagnetic, and helioseismic problems and a numerical example in seismology.

Stark, Philip B.↗