Maximum A posteriori interval estimation
Maximum a posteriori estimate of random process state vector in interval
RANDOM PROCESS↗
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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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On unifying randomized methods for inverse problems
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. Here, we present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.
42 ENGINEERING↗
Karhunen–Loève deep learning method for surrogate modeling and approximate Bayesian parameter estimation
We evaluate the performance of the Karhunen-Loève Deep Neural Network (KL-DNN) framework for surrogate modeling and approximate Bayesian parameter estimation in partial differential equation models. In the surrogate model, the Karhunen-Loève (KL) expansions are used for the dimensionality reduction of the number of unknown parameters and variables, and a deep neural network is employed to relate the reduced space of parameters to that of the state variables. The KL-DNN surrogate model is used to formulate a maximum-a-posteriori-like least-squares problem, which is randomized to draw samples of the posterior distribution of the parameters. We test the proposed framework for a hypothetical unconfined aquifer via comparison with the forward MODFLOW and inverse PEST++ iterative ensemble smoother (IES) solutions as well as the state-of-the-art Fourier neural operator (FNO) and deep operator networks (DeepONets) operator learning surrogate models. Our results show that the KL-DNN surrogate model outperforms FNO and DeepONet for forward predictions. For solving inverse problems, the randomized algorithm provides the same or more accurate Bayesian predictions of the parameters than IES as evidenced by the higher log-predictive probability of both the estimated parameter field and the forecast hydraulic head. The posterior mean obtained from the randomized algorithm is closer to the reference parameter field than that obtained with FNO as the maximum a posteriori estimate.
Approximate Bayesian inference↗
Maximum a posteriori decoder for digital communications
A system and method for decoding by identification of the most likely phase coded signal corresponding to received data. The present invention has particular application to communication with signals that experience spurious random phase perturbations. The generalized estimator-correlator uses a maximum a posteriori (MAP) estimator to generate phase estimates for correlation with incoming data samples and for correlation with mean phases indicative of unique hypothesized signals. The result is a MAP likelihood statistic for each hypothesized transmission, wherein the highest value statistic identifies the transmitted signal.
Altes, Richard A.↗
Maximum a posteriori interval estimation.
Maximum a posteriori /MAP/ interval estimation for state vector of random process, solving fixed delay filtering and deriving error matrix
RANDOM PROCESS↗
Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation
The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.
Gauss-Newton↗
Desmearing Bonse–Hart USANS data using Bayesian Gaussian process regression
Ultra-small-angle neutron scattering (USANS) enables access to micrometer-scale structures but is intrinsically affected by strong, anisotropic resolution smearing arising from slit-geometry optics. As a result, recovery of the intrinsic scattering intensity constitutes an ill-posed inverse problem, and commonly used iterative desmearing methods lack rigorous uncertainty quantification. We present a Bayesian desmearing framework for slit-geometry USANS based on Gaussian process regression. In this approach, the scattering intensity is modeled as a smooth random function, and the instrumental point spread function is incorporated explicitly as a forward operator. The resulting formulation yields a closed-form maximum a posteriori solution with well-defined credibility intervals. Computational benchmarks and experimental validation using combined USANS and small-angle neutron scattering (SANS) measurements demonstrate that the framework enables stable desmearing, suppresses experimental noise, and preserves physically meaningful structural features under realistic conditions.
Tung, Chi-Huan [Oak Ridge National Laboratory (ORN↗
A generalization of the near-optimum binary coherent state receiver concept
The structure of a maximum a posteriori decoder is derived for binary optical fields processed according to Kennedy's near-optimum scheme, but operating in the presence of random optical phase-error and interfering background fields. An exact expression is found for the error probability, and examples to illustrate decoder performance under various operating conditions of interest are provided.
Vilnrotter, V. A.↗
Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration
This paper proposes novel noise-free Bayesian optimization strategies that rely on a random exploration step to enhance the accuracy of Gaussian process surrogate models. The new algorithms retain the ease of implementation of the classical GP-UCB algorithm, but the additional random exploration step accelerates their convergence, nearly achieving the optimal convergence rate. Furthermore, to facilitate Bayesian inference with intractable likelihoods, we propose to utilize optimization iterates for maximum a posteriori estimation to build a Gaussian process surrogate model for the unnormalized log-posterior density. We provide bounds for the Hellinger distance between the true and the approximate posterior distributions in terms of the number of design points. We demonstrate the effectiveness of our Bayesian optimization algorithms in nonconvex benchmark objective functions, in a machine learning hyperparameter tuning problem, and in a black-box engineering design problem. The effectiveness of our posterior approximation approach is demonstrated in two Bayesian inference problems for parameters of dynamical systems.
Bayesian inference↗
On the design of turbo codes
In this article, we design new turbo codes that can achieve near-Shannon-limit performance. The design criterion for random interleavers is based on maximizing the effective free distance of the turbo code, i.e., the minimum output weight of codewords due to weight-2 input sequences. An upper bound on the effective free distance of a turbo code is derived. This upper bound can be achieved if the feedback connection of convolutional codes uses primitive polynomials. We review multiple turbo codes (parallel concatenation of q convolutional codes), which increase the so-called 'interleaving gain' as q and the interleaver size increase, and a suitable decoder structure derived from an approximation to the maximum a posteriori probability decision rule. We develop new rate 1/3, 2/3, 3/4, and 4/5 constituent codes to be used in the turbo encoder structure. These codes, for from 2 to 32 states, are designed by using primitive polynomials. The resulting turbo codes have rates b/n (b = 1, 2, 3, 4 and n = 2, 3, 4, 5, 6), and include random interleavers for better asymptotic performance. These codes are suitable for deep-space communications with low throughput and for near-Earth communications where high throughput is desirable. The performance of these codes is within 1 dB of the Shannon limit at a bit-error rate of 10(exp -6) for throughputs from 1/15 up to 4 bits/s/Hz.
Divsalar, D.↗
Maximum a posteriori classification of multifrequency, multilook, synthetic aperture radar intensity data
We present a maximum a posteriori (MAP) classifier for classifying multifrequency, multilook, single polarization SAR intensity data into regions or ensembles of pixels of homogeneous and similar radar backscatter characteristics. A model for the prior joint distribution of the multifrequency SAR intensity data is combined with a Markov random field for representing the interactions between region labels to obtain an expression for the posterior distribution of the region labels given the multifrequency SAR observations. The maximization of the posterior distribution yields Bayes's optimum region labeling or classification of the SAR data or its MAP estimate. The performance of the MAP classifier is evaluated by using computer-simulated multilook SAR intensity data as a function of the parameters in the classification process. Multilook SAR intensity data are shown to yield higher classification accuracies than one-look SAR complex amplitude data. The MAP classifier is extended to the case in which the radar backscatter from the remotely sensed surface varies within the SAR image because of incidence angle effects. The results obtained illustrate the practicality of the method for combining SAR intensity observations acquired at two different frequencies and for improving classification accuracy of SAR data.
Rignot, E.↗
Combined Uncertainty and A-Posteriori Error Bound Estimates for General CFD Calculations: Theory and Software Implementation
This workshop presentation discusses the design and implementation of numerical methods for the quantification of statistical uncertainty, including a-posteriori error bounds, for output quantities computed using CFD methods. Hydrodynamic realizations often contain numerical error arising from finite-dimensional approximation (e.g. numerical methods using grids, basis functions, particles) and statistical uncertainty arising from incomplete information and/or statistical characterization of model parameters and random fields. The first task at hand is to derive formal error bounds for statistics given realizations containing finite-dimensional numerical error [1]. The error in computed output statistics contains contributions from both realization error and the error resulting from the calculation of statistics integrals using a numerical method. A second task is to devise computable a-posteriori error bounds by numerically approximating all terms arising in the error bound estimates. For the same reason that CFD calculations including error bounds but omitting uncertainty modeling are only of limited value, CFD calculations including uncertainty modeling but omitting error bounds are only of limited value. To gain maximum value from CFD calculations, a general software package for uncertainty quantification with quantified error bounds has been developed at NASA. The package provides implementations for a suite of numerical methods used in uncertainty quantification: Dense tensorization basis methods [3] and a subscale recovery variant [1] for non-smooth data, Sparse tensorization methods[2] utilizing node-nested hierarchies, Sampling methods[4] for high-dimensional random variable spaces.
CFD↗
Maximum a posteriori Ly α estimator (MAPLE): band power and covariance estimation of the 3D Ly α forest power spectrum
We present a novel maximum a posteriori estimator to jointly estimate band powers and the covariance of the three-dimensional power spectrum (P3D) of Ly $\alpha$ forest flux fluctuations, called MAPLE. Our Wiener-filter based algorithm reconstructs a window-deconvolved P3D in the presence of complex survey geometries typical for Ly $\alpha$ surveys that are sparsely sampled transverse to and densely sampled along the line of sight. We demonstrate our method on idealized Gaussian random fields with two selection functions: (i) a sparse sampling of 30 background sources per square degree designed to emulate the current Dark Energy Spectroscopic Instrument; (ii) a dense sampling of 900 background sources per square degree emulating the upcoming Prime Focus Spectrograph Galaxy Evolution Survey. Our proof-of-principle shows promise, especially since the algorithm can be extended to marginalize jointly over nuisance parameters and contaminants, i.e. offsets introduced by continuum fitting. Our code is implemented in JAX and is publicly available on GitHub.
79 ASTRONOMY AND ASTROPHYSICS↗
On estimating the phase of a periodic waveform in additive Gaussian noise, part 1
Motivated by recent advances in technology, a new look is taken at the problem of estimating the phase of a periodic waveform in additive Gaussian noise. The maximum a posteriori probability criterion with signal space interpretation is used to obtain the structures of optimum and some suboptimum phase estimators for the following cases: (1) known constant frequency and unknown constant phase with an a priori distribution; (2) unknown constant frequency and phase with a joint a priori distribution; (3) frequency a parameterized function of time with a joint a priori distribution on parameters and phase and (4) frequency a Gaussian random process.
Rauch, L. L.↗