We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

As companies strive to develop artefacts intended for services instead of traditional sell-off, new challenges in the product development process arise to promote continuous improvement and increasing market profits. This creates a focus on product life-cycle components as companies then make life-cycle commitments, where they are responsible for the function availability during the extent of the life-cycle, i.e. functional products. One of these life-cycle components is manufacturing; therefore, companies search for new approaches of success during manufacturability evaluation already in engineering design. Efforts have been done to support early engineering design, as this phase sets constraints and opportunities for manufacturing. These efforts have turned into design for manufacturing methods and guidelines. A further step to improve the life-cycle focus during early engineering design is to reuse results and use experience from earlier projects. However, because results and experiences created during project work are often not documented for reuse, only remembered by some people, there is a need for design support. Knowledge engineering (KE) is a methodology for creating knowledge-based systems, e.g. systems that enable reuse of earlier results and make available both explicit and tacit corporate knowledge, enabling the automated generation and evaluation of new engineering design solutions during early product development. There are a variety of KE-approaches, such as knowledge-based engineering, case-based reasoning and programming, which have been used in research to develop design for manufacturing methods and applications. There are, however, opportunities for research where several approaches and their interdependencies, to create a transparent picture of how KE can be used to support engineering design, are investigated. The aim of the research presented in this thesis is to create new methods for design for manufacturing, by using several approaches of KE, and find the beneficial and less beneficial aspects of these methods in comparison to each other and earlier research. This thesis presents methods and applications for design for manufacturing using KE. KE has been employed in several ways, namely rule-based, rule-, programmingand finite element analysis (FEA)-based, and ruleand plan-based, which are tested and compared with each other. Results show that KE can be used to generate information about manufacturing in several ways. The rule-based way is suitable for supporting life-cycle commitments, as engineering design and manufacturing can be integrated with maintenance and performance predictions during early engineering design, though limited to the firing of production rules. The rule-, programmingand FEA-based way can be used to integrate computer-aided design tools and virtual manufacturing for non-linear stress and displacement analysis. This way may also bridge the gap between engineering designers and computational experts, even though this way requires a larger effort to program than the rule-based. The ruleand planbased way can enable design for manufacturing in two fashions – based on earlier manufacturing plans and based on rules. Because earlier manufacturing plans, together with programming algorithms, can handle knowledge that may be more intricate to capture as rules, as opposed to the time demanding routine work that is often automated by means of rules, several opportunities for designing for manufacturing exist.

Methods and Applications

We construct a stream-cipher S whose implementation is secure even if a bounded amount of arbitrary (adversarially chosen) information on the internal state ofS is leaked during computation. This captures all possible side-channel attacks on S where the amount of information leaked in a given period is bounded, but overall can be arbitrary large. The only other assumption we make on the implementation of S is that only data that is accessed during computation leaks information. The stream-cipher S generates its output in chunks K1, K2, . . . and arbitrary but bounded information leakage is modeled by allowing the adversary to adaptively chose a function fl : {0,1}* rarr {0, 1}lambda before Kl is computed, she then gets fl(taul) where taul is the internal state ofS that is accessed during the computation of Kg. One notion of security we prove for S is that Kg is indistinguishable from random when given K1,..., K1-1,f1(tau1 ),..., fl-1(taul-1) and also the complete internal state of S after Kg has been computed (i.e. S is forward-secure). The construction is based on alternating extraction (used in the intrusion-resilient secret-sharing scheme from FOCS'07). We move this concept to the computational setting by proving a lemma that states that the output of any PRG has high HILLpseudoentropy (i.e. is indistinguishable from some distribution with high min-entropy) even if arbitrary information about the seed is leaked. The amount of leakage lambda that we can tolerate in each step depends on the strength of the underlying PRG, it is at least logarithmic, but can be as large as a constant fraction of the internal state of S if the PRG is exponentially hard.

/pdf/leakage-resilient-cryptography-15i0byrphr.pdf

Leakage-Resilient Cryptography

Network management requires accurate estimates of metrics for traffic engineering (e.g., heavy hitters), anomaly detection (e.g., entropy of source addresses), and security (e.g., DDoS detection). Obtaining accurate estimates given router CPU and memory constraints is a challenging problem. Existing approaches fall in one of two undesirable extremes: (1) low fidelity general-purpose approaches such as sampling, or (2) high fidelity but complex algorithms customized to specific application-level metrics. Ideally, a solution should be both general (i.e., supports many applications) and provide accuracy comparable to custom algorithms. This paper presents UnivMon, a framework for flow monitoring which leverages recent theoretical advances and demonstrates that it is possible to achieve both generality and high accuracy. UnivMon uses an application-agnostic data plane monitoring primitive; different (and possibly unforeseen) estimation algorithms run in the control plane, and use the statistics from the data plane to compute application-level metrics. We present a proof-of-concept implementation of UnivMon using P4 and develop simple coordination techniques to provide a ``one-big-switch'' abstraction for network-wide monitoring. We evaluate the effectiveness of UnivMon using a range of trace-driven evaluations and show that it offers comparable (and sometimes better) accuracy relative to custom sketching solutions.

https://dl.acm.org/doi/pdf/10.1145/2934872.2934906

One Sketch to Rule Them All: Rethinking Network Flow Monitoring with UnivMon

Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was two-fold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the state-of-the-art results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas.

/pdf/graph-stream-algorithms-a-survey-1m1oq4kh5y.pdf

Graph stream algorithms: a survey

Weighted sampling without replacement has proved to be a very important tool in designing new algorithms. Efraimidis and Spirakis (IPL 2006) presented an algorithm for weighted sampling without replacement from data streams. Their algorithm works under the assumption of precise computations over the interval [0,1]. Cohen and Kaplan (VLDB 2008) used similar methods for their bottom-k sketches. 
Efraimidis and Spirakis ask as an open question whether using finite precision arithmetic impacts the accuracy of their algorithm. In this paper we show a method to avoid this problem by providing a precise reduction from k-sampling without replacement to k-sampling with replacement. We call the resulting method Cascade Sampling.

Weighted Sampling Without Replacement from Data Streams

The task of finding heavy hitters is one of the best known and well studied problems in the area of data streams. One is given a list $i_1,i_2,\ldots,i_m\in[n]$ and the goal is to identify the items among $[n]$ that appear frequently in the list. In sub-polynomial space, the strongest guarantee available is the $\ell_2$ guarantee, which requires finding all items that occur at least $\epsilon\|f\|_2$ times in the stream, where the vector $f\in\mathbb{R}^n$ is the count histogram of the stream with $i$th coordinate equal to the number of times~$i$ appears $f_i:=\#\{j\in[m]:i_j=i\}$. The first algorithm to achieve the $\ell_2$ guarantee was the CountSketch of [CCF04], which requires $O(\epsilon^{-2}\log n)$ words of memory and $O(\log n)$ update time and is known to be space-optimal if the stream allows for deletions. The recent work of [BCIW16] gave an improved algorithm for insertion-only streams, using only $O(\epsilon^{-2}\log\epsilon^{-1}\log\log n)$ words of memory. In this work, we give an algorithm \bptree for $\ell_2$ heavy hitters in insertion-only streams that achieves $O(\epsilon^{-2}\log\epsilon^{-1})$ words of memory and $O(\log\epsilon^{-1})$ update time, which is the optimal dependence on $n$ and $m$. In addition, we describe an algorithm for tracking $\|f\|_2$ at all times with $O(\epsilon^{-2})$ memory and update time. Our analyses rely on bounding the expected supremum of a Bernoulli process involving Rademachers with limited independence, which we accomplish via a Dudley-like chaining argument that may have applications elsewhere.

BPTree: an $\ell_2$ heavy hitters algorithm using constant memory

Given a stream with frequency vector f in n dimensions, we characterize the space necessary for approximating the frequency negative moments Fp, where p<0, in terms of n, the accuracy, and the L_1 length of the vector f. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function g, we characterize the space necessary for any streaming algorithm that outputs a (1 +/- eps)-approximation to the sum of the coordinates of the vector f transformed by g. The storage required is expressed in the form of the solution to a relatively simple nonlinear optimization problem, and the algorithm is universal for (1 +/- eps)-approximations to any such sum where the applied function is nonnegative, nonincreasing, and has the same or smaller space complexity as g. This partially answers an open question of Nelson (IITK Workshop Kanpur, 2009).

/pdf/universal-sketches-for-the-frequency-negative-moments-and-2wcugsbdg5.pdf

Universal Sketches for the Frequency Negative Moments and Other Decreasing Streaming Sums

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set Although many existing algorithms in the sliding window model use or borrow elements from the smooth histogram framework (Braverman and Ostrovsky, FOCS 2007), we show that many interesting linear-algebraic problems, including spectral and vector induced matrix norms, generalized regression, and low-rank approximation, are not amenable to this approach in the row-arrival model To overcome this challenge, we first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_{1}$ -subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime Our algorithms are based on “reverse online” versions of offline sampling distributions such as (ridge) leverage scores, $\ell_{1}$ sensitivities, and Lewis weights to quantify both the importance and the recency of a row; our structural results on these distributions may be of independent interest for future algorithmic design Although our techniques initially address numerical linear algebra in the sliding window model, our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et al (FOCS 2019) We also give the first online algorithm for $\ell_{1}$ -subspace embeddings We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets, which we define as a weighted subset of rows of the input matrix that can be used to compute a good approximation to some given function on all of its prefixes Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_{1}$ - subspace embeddings in the sliding window model that use nearly optimal space

Near Optimal Linear Algebra in the Online and Sliding Window Models

Network performance problems are notoriously difficult to diagnose. Prior profiling systems collect performance statistics by keeping information about each network flow, but maintaining per-flow state is not scalable on resource-constrained NIC and switch hardware. Instead, we propose sketch-based performance monitoring using memory that is sublinear in the number of flows. Existing sketches estimate flow monitoring metrics based on flow sizes. In contrast, performance monitoring typically requires combining information across pairs of packets, such as matching a data packet with its acknowledgment to compute a round-trip time. We define a new class of \emph{lean} algorithms that use memory sublinear in both the size of input data and the number of flows. We then introduce lean algorithms for a set of important statistics, such as identifying flows with high latency, loss, out-of-order, or retransmitted packets. We implement prototypes of our lean algorithms on a commodity programmable switch using the P4 language. Our experiments show that lean algorithms detect $\sim$82\% of top 100 problematic flows among real-world packet traces using just 40KB memory.

Vladimir Braverman

Papers

Weighted Sampling Without Replacement from Data Streams

BPTree: an $\ell_2$ heavy hitters algorithm using constant memory

Universal Sketches for the Frequency Negative Moments and Other Decreasing Streaming Sums

Near Optimal Linear Algebra in the Online and Sliding Window Models

Memory-Efficient Performance Monitoring on Programmable Switches with Lean Algorithms.