Reliable Error Estimates for Optimal Control of Linear Elliptic PDEs with Random Inputs

We discretize a risk-neutral optimal control problem governed by a linear elliptic partial differential equation with random inputs using a Monte Carlo sample-based approximation and a finite element discretization, yielding finite dimensional control problems. We establish an exponential tail bound for the distance between the finite dimensional problems' solutions and the risk-neutral problem's solution. The tail bound implies that solutions to the risk-neutral optimal control problem can be reliably estimated with the solutions to the finite dimensional control problems. Numerical simulations illustrate our theoretical findings.Comment: 26 pages, 11 figure

Consistency of sample-based stationary points for infinite-dimensional stochastic optimization

We consider stochastic optimization problems with possibly nonsmooth integrands posed in Banach spaces and approximate these stochastic programs via a sample-based approaches. We establish the consistency of approximate Clarke stationary points of the sample-based approximations. Our framework is applied to risk-averse semilinear PDE-constrained optimization using the average value-at-risk and to risk-neutral bilinear PDE-constrained optimization.Comment: 20 page

Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program's solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.Comment: 20 pages, 4 figure

Sample Size Estimates for Risk-Neutral Semilinear PDE-Constrained Optimization

The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented.Comment: 26 pages, 10 figure

Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the literature.Comment: 24 page

The BDNFVal66Met SNP modulates the association between beta-amyloid and hippocampal disconnection in Alzheimer’s disease

In Alzheimer’s disease (AD), a single-nucleotide polymorphism in the gene encoding brain-derived neurotrophic factor (BDNFVal66Met) is associated with worse impact of primary AD pathology (beta-amyloid, Aβ) on neurodegeneration and cognitive decline, rendering BDNFVal66Met an important modulating factor of cognitive impairment in AD. However, the effect of BDNFVal66Met on functional networks that may underlie cognitive impairment in AD is poorly understood. Using a cross-validation approach, we first explored in subjects with autosomal dominant AD (ADAD) from the Dominantly Inherited Alzheimer Network (DIAN) the effect of BDNFVal66Met on resting-state fMRI assessed functional networks. In seed-based connectivity analysis of six major large-scale networks, we found a stronger decrease of hippocampus (seed) to medial-frontal connectivity in the BDNFVal66Met carriers compared to BDNFVal homozogytes. BDNFVal66Met was not associated with connectivity in any other networks. Next, we tested whether the finding of more pronounced decrease in hippocampal-medial-frontal connectivity in BDNFVal66Met could be also found in elderly subjects with sporadically occurring Aβ, including a group with subjective cognitive decline (N = 149, FACEHBI study) and a group ranging from preclinical to AD dementia (N = 114, DELCODE study). In both of these independently recruited groups, BDNFVal66Met was associated with a stronger effect of more abnormal Aβ-levels (assessed by biofluid-assay or amyloid-PET) on hippocampal-medial-frontal connectivity decreases, controlled for hippocampus volume and other confounds. Lower hippocampal-medial-frontal connectivity was associated with lower global cognitive performance in the DIAN and DELCODE studies. Together these results suggest that BDNFVal66Met is selectively associated with a higher vulnerability of hippocampus-frontal connectivity to primary AD pathology, resulting in greater AD-related cognitive impairment

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA solutions. Our analysis exploits problem structure in PDE-constrained optimization problems, allowing us to construct deterministic, compact subsets of the feasible set that contain the solutions to the risk-neutral problem and eventually those to the SAA problems. The construction is used to study the consistency using results established in the literature on stochastic programming. The assumptions of our framework are verified on three nonlinear optimization problems under uncertainty.Comment: Revised introduction and appendix; 25 page

Osseous microarchitecture in frequent fracture zones of the distal clavicle.

BACKGROUND Fracture classifications of the distal clavicle are based on ligamentous integrity. The influence of osseous microarchitecture on fracture occurrence, morphology, and the lesion's stability has not yet been investigated. We aimed to characterize osseous microarchitecture according to common fracture classification systems based on ligamentous integrity and investigated the possible effects of age, gender, and osteoporosis in distal clavicle fractures. METHODS N = 20 human cadaveric distal clavicles were scanned using XtremeCT with an isometric voxel size of 82 μm. In the sagittal plane, each data set was evaluated in 11 sections of approximately 7 mm thickness. Three topographic regions were defined: the bone lateral to the trapezoid (LTR), intertubercular (ITR), and medial to the conoid (MCR) ligament. Cortical bone mineral density (BMD) [mgHA/cm3] and cortical porosity (1- (BV/TV) [%]) were determined and evaluated relative to age and gender. RESULTS Along the mediolateral axis, there was an >20-fold increase in median cortical porosity (P ≤ .001). There were significant differences in cortical porosity between LTR and ITR (P ≤ .001) but not between ITR and MCR (P = .09). In ITR, cortical porosity was significantly greater in >60-year-old compared to younger donors (P = .01). For BMD, there was an >2-fold decrease toward the distal apex (P ≤ .001). BMD was significantly greater in ITR compared to LTR (P ≤ .001) and in MCR compared to ITR (P = .02). In ITR and MCR, clavicles of >60-year-old donors had significantly lower BMD values compared to younger donors (P 60-year-olds (P > .6). CONCLUSION The distal clavicle features a characteristic bony microarchitecture. The present study revealed a significant difference in bone quality of lateral, intertubercular, and medial zones of the distal clavicle and could specify target areas and strategies for surgical treatment of unstable fractures. Age, gender, and osteoporosis have a limited effect on bone quality and fracture genesis. In contrast, ligamentous quality is supposed to exert a substantial influence on fracture characteristics, especially in ITR. Fracture morphology of the distal clavicle is determined by a bony-ligamentous conjunction, which remains to be characterized