Matrix probing: a randomized preconditioner for the wave-equation Hessian

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.Comment: 21 pages, 6 figure

Processing Structured Data Streams

We elaborate this study in order to choose the most suitable technology to develop our proposal. Second, we propose three methods to reduce the set of data to be processed by a query when working with large graphs, namely spatial, temporal and random approximations. These methods are based on Approximate Query Processing techniques and consist in discarding the information that is considered not relevant for the query. The reduction of the data is performed online with the processing and considers both spatial and temporal aspects of the data. Since discarding information in the source data may decrease the validity of the results, we also define the transformation error obtain with these methods in terms of accuracy, precision and recall. Finally, we present a preprocessing algorithm, called SDR algorithm, that is also used to reduce the set of data to be processed, but without compromising the accuracy of the results. It calculates a subgraph from the source graph that contains only the relevant information for a given query. Since this technique is a preprocessing algorithm it is run offline before the actual processing begins. In addition, an incremental version of the algorithm is developed in order to update the subgraph as new information arrives to the system.A large amount of data is daily generated from different sources such as social networks, recommendation systems or geolocation systems. Moreover, this information tends to grow exponentially every year. Companies have discovered that the processing of these data may be important in order to obtain useful conclusions that serve for decision-making or the detection and resolution of problems in a more efficient way, for instance, through the study of trends, habits or customs of the population. The information provided by these sources typically consists of a non-structured and continuous data flow, where the relations among data elements conform graph structures. Inevitably, the processing performance of this information progressively decreases as the size of the data increases. For this reason, non-structured information is usually handled taking into account only the most recent data and discarding the rest, since they are considered not relevant when drawing conclusions. However, this approach is not enough in the case of sources that provide graph-structured data, since it is necessary to consider spatial features as well as temporal features. These spatial features refer to the relationships among the data elements. For example, some cases where it is important to consider spatial aspects are marketing techniques, which require information on the location of users and their possible needs, or the detection of diseases, that use data about genetic relationships among subjects or the geographic scope. It is worth highlighting three main contributions from this dissertation. First, we provide a comparative study of seven of the most common processing platforms to work with huge graphs and the languages that are used to query them. This study measures the performance of the queries in terms of execution time, and the syntax complexity of the languages according to three parameters: number of characters, number of operators and number of internal variables

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

We present a finite-element solution method that is well suited for interactive simulations of cutting meshes in the regime of linear elastic models. Our approach features fast updates to the solution of the stiffness system of equations to account for real-time changes in mesh connectivity and boundary conditions. Updates are accomplished by augmenting the stiffness matrix to keep it consistent with changes to the underlying model, without refactoring the matrix at each step of cutting. The initial stiffness matrix and its Cholesky factors are used to implicitly form and solve a Schur complement system using an iterative solver. As changes accumulate over many simulation timesteps, the augmented solution method slows down due to the size of the augmented matrix. However, by periodically refactoring the stiffness matrix in a concurrent background process, fresh Cholesky factors that incorporate recent model changes can replace the initial factors. This controls the size of the augmented matrices and provides a way to maintain a fast solution rate as the number of changes to a model grows. We exploit sparsity in the stiffness matrix, the right-hand-side vectors and the solution vectors to compute the solutions fast, and show that the time complexity of the update steps is bounded linearly by the size of the Cholesky factor of the initial matrix. Our complexity analysis and experimental results demonstrate that this approach scales well with problem size. Results for cutting and deformation of 3D linear elastic models are reported for meshes representing the brain, eye, and model problems with element counts up to 167,000; these show the potential of this method for real-time interactivity. An application to limbal incisions for surgical correction of astigmatism, for which linear elastic models and small deformations are sufficient, is included

Random Sampling for Group-By Queries

Random sampling has been widely used in approximate query processing on large databases, due to its potential to significantly reduce resource usage and response times, at the cost of a small approximation error. We consider random sampling for answering the ubiquitous class of group-by queries, which first group data according to one or more attributes, and then aggregate within each group after filtering through a predicate. The challenge with group-by queries is that a sampling method cannot focus on optimizing the quality of a single answer (e.g. the mean of selected data), but must simultaneously optimize the quality of a set of answers (one per group).We present CVOPT, a query- and data-driven sampling framework for a set of group-by queries. To evaluate the quality of a sample, CVOPT defines a metric based on the norm (e.g. ℓ2 or ℓ∞) of the coefficients of variation (CVs) of different answers, and constructs a stratified sample that provably optimizes the metric. CVOPT can handle group-by queries on data where groups have vastly different statistical characteristics, such as frequencies, means, or variances. CVOPT jointly optimizes for multiple aggregations and multiple group-by clauses, and provides a way to prioritize specific groups or aggregates. It can be tuned to cases when partial information about a query workload is known, such as a data warehouse where queries are scheduled to run periodically.Our experimental results show that CVOPT outperforms current state-of-the-art on sample quality and estimation accuracy for group-by queries. On a set of queries on two real-world data sets, CVOPT yields relative errors that are 5x smaller than competing approaches, under the same space budget

Advanced techniques for atmospheric effects

Over the last few years, open world videogames have been gaining lots of interest in the gaming industry. Open world videogames not only allow the player to freely roam over a vast terrain but also aim to recreate a believable dynamic world. Thus, one of the basic elements that such a videogame should feature is a day and night cycle. In this thesis, all of the intricacies that are involved in developing a physically based day and night cycle solution in a real-time rendering context are discussed. The main topics that will be covered are atmosphere rendering, celestial bodies positioning, celestial bodies rendering and nighttime scenes rendering

Scheduling Issues in Partitioned Temporal Join

One of the major problems of temporal databases is to develop efficient algorithms for operations that involves the time attributes. An operation that has received much attention in recent years is the temporal join which matches records from two temporal relations whose time intervals overlap. Under a partition-based algorithm, temporal data are split into partitions. During the join process, a partition in one relation only needs to join with some, but not all, partitions of the other relation. In this paper, we address scheduling issues in such an algorithm. Depending on the orders in which partitions are read, the number of I/Os incurred varies. We propose a three-phase scheduling framework to minimize the number of I/Os incurred. From the framework, a large number of scheduling strategies can be derived. We also study several representative scheduling strategies and report our findings in this paper