Luther W. White

Coefficient estimation in a first-order nonlinear hyberbolic Cauchy problem

This paper deals with the estimation and approximation of coefficient function in a first-order, nonlinear, hyperbolic Cauchy problem. The estimation is accomplished by minimizing a functional which measures the error between a finite set of given observations and the corresponding values of the solution generated by the coefficient function. A class of admissible coefficient functions is defined, and it is proved that minimizing coefficient function always exists within this class. We also develop an approximation by a sequence of solutions of associated finite-dimensional minimization problems.Based on a review of existing algorithms, a general branch-and-bound concept in global optimization is presented. A sufficient and necessary convergence condition is established, and a broad class of realizations is derived that include existing and several new approaches for concave minimization problems.

ELEVATED CO2 DIFFERENTIATES ECOSYSTEM CARBON PROCESSES: DECONVOLUTION ANALYSIS OF DUKE FOREST FACE DATA

Quantification of the flux of carbon (C) through different pathways is critical to predict the impact of global change on terrestrial ecosystems. Past research has en- countered considerable difficulty in separating root exudation, root turnover rate, and other belowground C fluxes as affected by elevated CO2. In this study we adopted a deconvolution analysis to differentiate C flux pathways in forest soils and to quantify the flux through those pathways. We first conducted forward analysis using a terrestrial-C sequestration (TCS) model to generate four alternative patterns of convolved responses of soil surface respiration to a step increase in atmospheric CC)2. The model was then validated against measured soil respiration at ambient CO2 before it was used to deconvolve the CO2 stim- ulation of soil respiration. Deconvolved data from the Duke Forest free-air CO2 enrichment (FACE) experiment suggest that fast C transfer processes, e.g., root exudation, are of minor importance in the ecosystem C cycling in the Duke Forest and were not affected by elevated CO2. The analysis indicates that the fine-root turnover is a major process adding C to the rhizosphere. This C has a residence time of several months to -2 yr and increases signif- icantly with increased CO2. In addition, the observed phase shift in soil respiration caused by elevated CO2 can be only reproduced by incorporation of a partial time delay function in C fluxes into the model. This paper also provides a detailed explanation of deconvolution analysis, since it is a relatively new research technique in ecology.

Estimation of optimal parameters for a surface hydrology model

The estimation of infiltration, roughness, and rainfall intensity parameters in a surface flow model is studied. The problem is formulated as an optimal control problem in which the parameters to be estimated are viewed as controls in the surface flow model and a fit-to-data functional is viewed as a control functional. The control system is obtained as the approximation of the distributed model by means of finite elements in the spatial domain and finite differences in the time domain. The leap-frog and implicit differences are used for the time differencing approximations. The resulting minimization problem is solved by a descent method, The adjoint equations are introduced in the context of a general method to calculate efficiently the derivatives of the functional. The adjoint equations are developed for the continuous as well as the finite difference equations. A numerical example is presented. This example illustrates the effect of the accuracy of prior rainfall intensity information on the estimates of roughness and infiltration parameters.

Identifiablity under approximation for an elliptic boundary value problem

Necessary and sufficient conditions for identifiability of the diffusion coefficient in Galerkin approximations to a two point boundary value problem are derived for various choices of Galerkin subspaces. The results are further used to investigate output least squares identifiability and output least squares stability of the diffusion coefficient.

The Parameter Estimation Problem for Parabolic Equations and Discontinuous Observation Operators

Parameter estimation problems are studied for a class of linear autonomous parabolic partial differential equations with various fit-to-data criteria, which may be discontinuous with respect to the state variable. We analyze the convergence of Galerkin schemes approximating the optimization problems and generalize results to higher dimensional problems. An example is then presented for the case of point observation fit-to-data criteria in higher dimensions. Finally, discretization of coefficients is discussed for identification problems with variable coefficients.

The Adjoint Newton Algorithm for Large-Scale Unconstrained Optimization in Meteorology Applications

A new algorithm is presented for carrying out large-scale unconstrained optimization required in variational data assimilation using the Newton method. The algorithm is referred to as the adjoint Newton algorithm. The adjoint Newton algorithm is based on the first- and second-order adjoint techniques allowing us to obtain the Newton line search direction by integrating a tangent linear equations model backwards in time (starting from a final condition with negative time steps). The error present in approximating the Hessian (the matrix of second-order derivatives) of the cost function with respect to the control variables in the quasi-Newton type algorithm is thus completely eliminated, while the storage problem related to the Hessian no longer exists since the explicit Hessian is not required in this algorithm. The adjoint Newton algorithm is applied to three one-dimensional models and to a two-dimensional limited-area shallow water equations model with both model generated and First Global Geophysical Experiment data. We compare the performance of the adjoint Newton algorithm with that of truncated Newton, adjoint truncated Newton, and LBFGS methods. Our numerical tests indicate that the adjoint Newton algorithm is very efficient and could find the minima within three or four iterations for problems tested here. In the case of the two-dimensional shallow water equations model, the adjoint Newton algorithm improves upon the efficiencies of the truncated Newton and LBFGS methods by a factor of at least 14 in terms of the CPU time required to satisfy the same convergence criterion.The Newton, truncated Newton and LBFGS methods are general purpose unconstrained minimization methods. The adjoint Newton algorithm is only useful for optimal control problems where the model equations serve as strong constraints and their corresponding tangent linear model may be integrated backwards in time. When the backwards integration of the tangent linear model is ill-posed in the sense of Hadamard, the adjoint Newton algorithm may not work. Thus, the adjoint Newton algorithm must be used with some caution. A possible solution to avoid the current weakness of the adjoint Newton algorithm is proposed.

Application of a New Adjoint Newton Algorithm to the 3D ARPS Storm-Scale Model Using Simulated Data

The adjoint Newton algorithm (ANA) is based on the first- and second-order adjoint techniques allowing one to obtain the ‘‘Newton line search direction’’ by integrating a ‘‘tangent linear model’’ backward in time (with negative time steps). Moreover, the ANA provides a new technique to find Newton line search direction without using gradient information. The error present in approximating the Hessian (the matrix of second-order derivatives) of the cost function with respect to the control variables in the quasi-Newton-type algorithm is thus completely eliminated, while the storage problem related to storing the Hessian no longer exists since the explicit Hessian is not required in this algorithm. The ANA is applied here, for the first time, in the framework of 4D variational data assimilation to the adiabatic version of the Advanced Regional Prediction System, a threedimensional, compressible, nonhydrostatic storm-scale model. The purpose is to assess the feasibility and efficiency of the ANA as a large-scale minimization algorithm in the setting of 4D variational data assimilation. Numerical results using simulated observations indicate that the ANA can efficiently retrieve high quality model initial conditions. It improves upon the efficiency of the usual adjoint method employing the LBFGS algorithm by more than an order of magnitude in terms of both CPU time and number of iterations for test problems presented here. Numerical results also show that the ANA obtains a fast linear convergence rate.

Control and optimal design of distributed parameter systems

A shape optimization problem in inverse acoustics.- On a variational equation for thin shells.- Oriented distance functions in shape analysis and optimization.- On the density of the range of the semigroup for semilinear heat equations.- Relaxation in semilinear infinite dimensional systems modeling fluid flow control problems.- Multidimensional inverse scattering problems in deterministic and random media.- Decay estimates for the wave equation.- Asymptotic behavior and attractors for nonlinear Von Karman plate equations with boundary dissipation.- Stabilization of the Korteweg-De Vries equation on a periodic domain.- A note concerning boundary effects and long time vibrations of layered media.- On the linearised dynamics of linked mechanical structures.

Estimation of parameters in carbon sequestration models from net ecosystem exchange data

The use of net ecosystem exchange (NEE) data to estimate carbon transfer coefficients is investigated in the context of a deterministic compartmental carbon sequestration system. Sensitivity and approximation properties are investigated for the underlying model initial value problems. Joint probability distributions are obtained by including NEE data along with corresponding synthetic NEE values generated from the model and are compared with a priori distributions. These distributions are used to estimate individual transfer parameters and to predict future carbon states. Results are compared with those obtained using only a priori information without the benefit of data. Shannon information content is introduced to measure the dependence of results on the lengths of observational intervals and provide an additional indicator of the value added by the inclusion of NEE data.

Surface flow model: inverse problems and predictions

The inverse problem to estimate the scaling coefficients for infiltration, rainfall, intensity, and roughness in a surface flow model is considered. A probability density function (pdf) is constructed from an observed hydrograph associated with a given rainfall event, the deterministic surface flow model, prior information on bounds on the scale coefficients, and prior information on the coefficients. From this pdf a basin pdf is obtained describing the likelihood of basin infiltration and roughness properties within a sample space of basin properties. With the occurrence of a new rainfall event, the basin pdf is utilized to describe the behavior of the basin in the presence of this event with a basin-event pdf. From this information predictions of flooding behavior may be obtained as well as the expected water levels along the stream. Numerical examples are presented that include different observational errors.