An adaptive POD approximation method for the control of advection-diffusion equations

We present an algorithm for the approximation of a finite horizon optimal control problem for advection-diffusion equations. The method is based on the coupling between an adaptive POD representation of the solution and a Dynamic Programming approximation scheme for the corresponding evolutive Hamilton-Jacobi equation. We discuss several features regarding the adaptivity of the method, the role of error estimate indicators to choose a time subdivision of the problem and the computation of the basis functions. Some test problems are presented to illustrate the method.Comment: 17 pages, 18 figure

Parameter estimation for the Euler-Bernoulli-beam

An approximation involving cubic spline functions for parameter estimation problems in the Euler-Bernoulli-beam equation (phrased as an optimization problem with respect to the parameters) is described and convergence is proved. The resulting algorithm was implemented and several of the test examples are documented. It is observed that the use of penalty terms in the cost functional can improve the rate of convergence

Northern fur seal reproductive rates and early maternal care

The majority of the world's breeding population of northern fur seals (Callorhinus ursinus) is found on the Pribilof Islands (St. Paul and St. George) in the Bering Sea, Alaska. Pup production on these islands experienced an irregular but overall decline since the early 1970's. Between 1998 and 2010, pup production declined precipitously at an annual rate of 4.9% on the Pribilof Islands, and 5.5% on St. Paul Island. Specific reasons for this decline remain unknown, and contemporary estimates for many vital rate parameters including reproductive rates are unavailable. This study determined a contemporary estimate of natality and fertility rates, as well as reproductive timing on the Polovina Cliffs rookery of St. Paul Island during the 2008 (30 June-31 August) and 2009 (1 July-25 August) breeding seasons. Natality rate (defined as the number of pups born divided by the number of reproductively mature females) was determined from visual observations of parturition or associated maternal behavior in 208 and 217 individually marked females (via flipper tags) in 2008 and 2009, respectively. Data yielded observed natality estimates of 0.79 in 2008 and 0.88 in 2009. The fertility rate (defined as the number of pups born divided by the total number of females present, irrespective of reproductive maturity/age) was determined for the 2008 breeding season only. This ratio of total pup to female counts was derived from adjusted daily cross-sectional counts conducted through the breeding season. Maximum pup and female counts were derived as asymptotes of sigmoid growth models fitted to corrected daily counts. Live pup counts were corrected for mortalities by estimates of cumulative pup mortalities. Daily counts of females present in the rookery were corrected for reduced detection probabilities resulting from increased maternal foraging trip durations through the season, typical of attendance patterns associated with colonial, income breeders. Daily detection probabilities for individually marked females were generated from Cormack-Jolly-Seber (CJS) open population models using maximum likelihood estimators (MLE) in Program MARK. Multiple a priori models accounting for the effects of possible covariates on detection probabilities were evaluated in an information-theoretic approach using Akaike's Information Criterion (AIC) and AICc model weights. Data yielded a minimum fertility rate estimate of 0.60 in 2008. Detection probabilities derived from the top CJS model for dual flipper-tagged females only were used to adjust the daily cross-section counts of all (marked and unmarked) females. Therefore, the actual fertility rate is probably higher than the estimate presented here, which should be regarded as the lowest likely value for 2008. However, AICc model weights also demonstrated the absence of density effects on detection probability estimates. This supports the applicability of marked female-based detection probabilities for correcting cross-sectional counts of all females and further suggests that the actual fertility estimate likely does not differ much from the presented estimate. Median dates of birth were calculated as the date closest to 50% of modeled corrected pup count asymptotes, yielding median dates of 17 July in 2008 and 15 July in 2009. Pregnant females are highly consistent in their arrival dates, with parturition occurring approximately 1 day after arrival. Median observed dates of arrival from individually marked females resulted in dates of 16 July in 2008 and 15 July in 2009. These dates occurred 5 to 13 days later than historic reports from 1951 through 1995. With median arrival dates 1 day prior to parturition, the observed match between birth dates derived from pup counts and from observed arrival dates of marked females supports the finding of a contemporary delay in the timing of parturition. Median arrival derived as the date closest to 50% of the asymptote from corrected and modeled female counts yielded 13 July in 2008. This earlier data is likely an effect of the inclusion of immature and nulli-parous females. In a subset of 62 females with pregnancy confirmed through a trans-rectal ultrasonography procedure in November 2007 and 29 females in 2008, the return rate for the following reproductive season was 0.92 and 0.76, respectively. In 2008, the return and natality rate was measured by radiotelemetry data, detected from females outfitted with VHF-radio transmitter. In 2009 both rates were determined by observational data. Observed natality rates for returned females of a known pregnancy status were 0.95 in 2008 and 0.96 in 2009. Radiotelemetry data from 76 females was analyzed for early maternal attendance patterns (duration and ratio of presences and absences) in 2008. The mean date of detected return was 18 July. The mean duration of the perinatal period was 7.5 days (+/- 1.3 SD). Excluding the perinatal period, the mean duration of presence on shore for the first five visits was 1.47 days (+/- 0.21 SD). The mean duration of absence at sea for the first five trips was 7.07 days (+/- 0.42 SD). Results presented from this study do not provide any direct evidence of a contemporary reduction in natality or fertility rates in northern fur seals. Since observed rates were comparably high and consistent between 2008 and 2009, it is unlikely that reduced natality rates are contributing to the current population trajectory. Attendance patterns do not provide any evidence of increased maternal foraging effort or secondarily, reduced prey availability. Interestingly, median pupping dates were found to occur significantly later than historical estimates. Potential reasons for this shift could be an increase in younger females within the reproductive female population at this rookery, or a shift in the timing of ocean climate conditions and peak prey availability during the breeding season.Keywords: fur seal, Callorhinus ursinus, reproductive timing, reproduction, natality, reproductive rate

Stabilization by sparse controls for a class of semilinear parabolic equations

Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P. The second author was supported by the Austrian Science Fund (FWF) under grant SFB F32 (SFB “Mathematical Optimization and Applications in Biomedical Sciences”) and by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program

Finite element approximation of sparse parabolic control problems

We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

A review on sparse solutions in optimal control of partial differential equations

In this paper a review of the results on sparse controls for partial differential equations is presented. There are two different approaches to the sparsity study of control problems. One approach consists of taking functions to control the system, putting in the cost functional a convenient term that promotes the sparsity of the optimal control. A second approach deals with controls that are Borel measures and the norm of the measure is involved in the cost functional. The use of measures as controls allows to obtain optimal controls supported on a zero Lebesgue measure set, which is very interesting for practical implementation. If the state equation is linear, then we can carry out a complete analysis of the control problem with measures. However, if the equation is nonlinear the use of measures to control the system is still an open problem, in general, and the use of functions to control the system seems to be more appropriate.This work was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P