Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to “boost” a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the “boosting” theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S_2^P ⊆ PP⊆(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice

Investigations of Material Response to Fatigue Phenomena in Contacting Bodies

Investigating the fatigue performance of machine components has been of significant interest to improve reliability and reduce the maintenance costs. In the current work, analytical as well as experimental approaches are used to investigate material response to contact fatigue damage. In particular, two fatigue phenomena namely; fretting fatigue and rolling contact fatigue (RCF) are studied. Fretting fatigue is a damage mechanism observed in machine components subjected to fretting in tandem with fluctuating bulk stresses. A fretting test fixture was developed to investigate fretting fatigue behavior of AISI 4140 vs. Ti-6Al-4V in a cylinder-on-flat contact configuration. The critical damage value for AISI 4140 was extracted using the method of variation of elasticity modulus. The fretting fatigue lives obtained from the proposed computational fatigue damage model were found to be in good agreement with the experimental results. The RCF investigation focuses on developing a modified 2D numerical model to simulate RCF damage in line contact configuration. First, a new computationally efficient approach is developed to investigate sub-surface initiated spalling in large bearings. Previously developed continuum damage mechanics based 2D fatigue model was modified to incorporate stress mapping procedure and dynamic remeshing tool to make the model computationally efficient. The new approach was validated against the previous numerical model for small rolling contacts. The scatter in the RCF lives and the progression of fatigue spalling for large bearings obtained from the model show good agreement with experimental results available in open literature. The ratio of L10 lives for different sized bearings computed from the model correlate well with the formula derived from the basic life rating for radial roller bearing as per ISO 281. Furthermore, the RCF model was extended to incorporate elastic-plastic material in order to investigate RCF of case carburized steels. A series of micro-indentation tests were conducted to obtain the hardness gradient in the case carburized 8620 steel. The hardness gradient in the material was modeled by changing the yield strength as a function of depth. The residual stress distribution due to carburization process was modeled by modifying the damage evolution law. The model was used to compare the rolling contact fatigue (RCF) lives of through hardened and case carburized bearing steel with different case depths. Based on the model results, the optimum case depths to maximize the RCF lives of the case carburized bearings at different loading conditions were obtained. This model was then modified to investigate RCF in refurbished case carburized bearings. Refurbishing process was simulated by removing a layer of material from the original surface after a set number of fatigue cycles. The original material properties, residual stresses and the fatigue damage accumulated prior to refurbishing in the remaining material were preserved. The refurbished geometry was then subjected to additional fatigue cycles until damage was detected. According to model results, more fatigue cycles prior to refurbishing enhance the total fatigue life of refurbished bearings. It was also found that beneficial impact of refurbishing on RCF lives of case carburized bearings depends on the relative values of case depth, contact half width, refurbishing depth

An Atypical Survey of Typical-Case Heuristic Algorithms

Heuristic approaches often do so well that they seem to pretty much always give the right answer. How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences? This article first discusses how a series of results by Berman, Buhrman, Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the early 1970s through the early 1990s, explicitly or implicitly limited how well heuristic algorithms can do on NP-hard problems. In particular, many desirable levels of heuristic success cannot be obtained unless severe, highly unlikely complexity class collapses occur. Second, we survey work initiated by Goldreich and Wigderson, who showed how under plausible assumptions deterministic heuristics for randomized computation can achieve a very high frequency of correctness. Finally, we consider formal ways in which theory can help explain the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012 issue of SIGACT New

Derandomization with Minimal Memory Footprint

Existing proofs that deduce BPL = ? from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ? DSPACE[c ? S] for c ? 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ? 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC?, that were not known before

Analytical and experimental investigation of microstructural alterations in bearing steel in rolling contact fatigue

Rolling Contact Fatigue (RCF) is one the most common failure modes in bearings. RCF is usually associated with particular microstructural alterations. Such alterations (i.e. white etching cracks, butterflies, etc.) which lead to RCF failure are known to be among the most concerning matters to bearing industry. In the current work, an analytical as well as experimental approaches are used to investigate “butterfly wing” formation, crack initiation and propagation from inclusions. A new damage evolution equation coupled with a FE model is employed to account for the effect of mean stresses and alternating stresses simultaneously to investigate butterfly formation. The proposed damage evolution law matches experimentally observed butterfly orientation, shape, and size successfully. The model is used to obtain S-N results for butterfly formation at different Hertzian load levels. The results corroborate well with the experimental data available in the open literature. The model is used to predict debonding at the inclusion/matrix interface and the most vulnerable regions for crack initiation on butterfly/matrix interface. A new variable called butterfly formation index (BFI) is introduced to manifest the dependence of wing formation on depth. The value of critical damage inside the butterfly wings was obtained experimentally and was then used to simulate damage evolution. Voronoi tessellation was used to develop the FEM domains to capture the effect of microstructural randomness on butterfly wing formation, crack initiation and propagation. Then, the effects of different inclusion characteristics such as size, depth, and stiffness on RCF life are studied. The results show that stiffness of an inclusion and its location has a significant effect on the RCF life: stiffer inclusions and inclusions located at the depth of maximum shear stress reversal are more detrimental to the RCF life. Stress concentrations are not significantly affected by inclusion size for the cases investigated; however, a stereology study showed that larger inclusions have a higher chance to be located at the critical depth and cause failure. Crack maps were recorded and compared to spall geometries observed experimentally. The results show that crack initiation locations and final spall shapes are similar to what has been observed in failed bearings

Pseudorandomness and Average-Case Complexity via Uniform Reductions

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP [not equal to] BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: 1. We obtain an optimal worst-case to average-case connection for EXP: if EXP is not a subset of BPTIME(t(n)), EXP has problems that cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{\Omega(1)}. 2. We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impaglaizzo and Wigderson, which used a a #P-complete problem with these properties. 3. We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.Engineering and Applied Science