Statistics And Numerical Methods By G Balaji Pdf

[Kemal Allan]

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In recent years, alignment-free methods have been widely applied in comparing genome sequences, as these methods compute efficiently and provide desirable phylogenetic analysis results. These methods have been successfully combined with hierarchical clustering methods for finding phylogenetic trees. However, it may not be suitable to apply these alignment-free methods directly to existing statistical classification methods, because an appropriate statistical classification theory for integrating with the alignment-free representation methods is still lacking. In this article, we propose a discriminant analysis method which uses the discrete wavelet packet transform to classify whole genome sequences. The proposed alignment-free representation statistics of features follow a joint normal distribution asymptotically. The data analysis results indicate that the proposed method provides satisfactory classification results in real time.

Ali Mani is an associate professor of Mechanical Engineering at Stanford University. He is a faculty affiliate of the Institute for Computational and Mathematical Engineering at Stanford. He received his PhD in Mechanical Engineering from Stanford in 2009. Prior to joining the faculty in 2011, he was an engineering research associate at Stanford and a senior postdoctoral associate at Massachusetts Institute of Technology in the Department of Chemical Engineering. His research group builds and utilizes large-scale high-fidelity numerical simulations, as well as methods of applied mathematics, to develop quantitative understanding of transport processes that involve strong coupling with fluid flow and commonly involve turbulence or chaos. His teaching includes the undergraduate engineering math classes and graduate courses on fluid mechanics and numerical analysis.

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The understanding of the dynamics of the velocity gradients in turbulent flows is critical to understanding various nonlinear turbulent processes. Several simplified dynamical equations have been proposed earlier that model the Lagrangian velocity gradient evolution equation. A robust model for the velocity gradient evolution equation can ultimately lead to the closure of the system of equations in the Lagrangian probability distribution function method. The pressure Hessian and the viscous Laplacian are the two important processes that govern the Lagrangian evolution of the velocity gradients. These processes are nonlocal in nature and unclosed from a mathematical point of view. The recent fluid deformation closure model (RFDM) has been shown to retrieve excellent statistics of the viscous process. However, the pressure Hessian modeled by the RFDM has various physical limitations. In this work, we first demonstrate such limitations of the RFDM. Subsequently, we employ a tensor basis neural network (TBNN) to model the pressure Hessian using the information about the velocity gradient tensor itself. Our neural network is trained on high-resolution data obtained from direct numerical simulation (DNS) of isotropic turbulence at the Reynolds number of 433. The predictions made by the TBNN are evaluated against several other DNS datasets. Evaluation is made in terms of the alignment statistics of the pressure-Hessian eigenvectors with the strain-rate eigenvectors. Our analysis of the predicted solution leads to the finding of ten unique coefficients of the tensor basis of the strain-rate and the rotation-rate tensors, the linear combination of which is used to accurately capture key alignment statistics of the pressure-Hessian tensor.

Schematic of the feedforward densely connected neural network. W(i) and b(i) are the learnable parameters called weight matrix and the bias vector of the ith layer, respectively. ϕ is a nonlinear activation function.

Schematic of the TBNN network. W(i) and b(i) are the weight matrix and the bias vector of the ith layer. Rectified linear unit (RELU) nonlinear activation function is used. Both W(i) and b(i) are the learnable parameters of the neural network, which are optimized using the RMSprop optimizer [54].

Alignment of PTBNN eigenvectors (pi) obtained from modified TBNN with S eigenvectors (si). Here, i (=α, β, or γ) denotes the three eigenvectors corresponding to the three eigenvalues α>β>γ. (Decaying isotropic turbulence testing dataset B, Table 1, initial Reynolds number 250, and turbulent Mach number of 0.4.) Note that the field is extracted at t=2.7t0, where t0 is the eddy turnover time. At this instant, the Reynolds number of the flow has decayed to the value of 67.4 and the turbulent Mach number is 0.3.

Also remember that indexes can contain multiple columns. Don't think of each column being a candidate for an index, think of each SELECT or WHERE clause as potentially having a "most helpful" index, perhaps a compound (multicolumn) index. For instance, in indexing a long phone book you wouldn't have two indexes, one on givenname, one on familyname; you'd be better off with a single index on (familyname, givenname).

Creating indexes you don't use is, as you wrote, wasteful. Not only does the work of maintaining the index take time, but the extra pages of space that the index uses in the database file make it bigger, more annoying to handle, and need more resources each time you back it up. One does try to avoid it.

All the columns in the partition by clause, in order, followed by all the columns in the order by clause, in order. If you wish the index to be covering then you need to append any columns used in filter expressions, followed by any columns actually used in calculations or in the output projection.

Thank you, Simon and Keith. Your answers make perfect sense. I had done some other research on the internet and the consensus seems to be that it is best to index on the partition by columns and then on the sort by columns.

But one part of Simon's answer makes me think either this whole exercise needs to be rethought from scratch or I need to give up on making my query more efficient using just indexes. And that part is the sentence "a query can use zero or one index".

Right now, in my query, I have 14 different windows because I am calculating these stats for 14 different measures. Obviously each window partitions and sorts by a different measure. I could create an index for each of them, but if the query can use only one of those indexes, it probably won't make a huge difference. Either I have to live with the time it takes my query right now (which is not bad at all), or I have to come up with a different query structure where I have one window in each query, and then join up the results of all those queries to get the final output I want. I have a feeling the latter will be slower than what I currently have even if each of the individual queries are sped up greatly by the creation of better indexes.

I don't think it's quite true that "a query can use zero or one index". That would be true for a simple projection query, selecting from a single table. But in general subqueries and joins will involve multiple tables and possibly as many indexes as table usages.

uses two tables and each one will use a different index (if available). The most efficient index for the outer query is t(b) (to be able to traverse the rows of t in order and avoid having to sort) and that the most efficient index for the correlated subquery will be a covering index t(a,c) to permit the appropriate traversal of the table in order and generating the sum without accessing the underlying table).

then again for t1 it will use one index or the full table, and for t2 it will use one index or the full table. It can't use an index on t2.id2 to speed up the ON clause and use an index on t2.b to speed up the WHERE and do some cross comparing. It can only pick one.

Taking the indexes and tables available it then enumerates all the possible solution methods to the query. It may also consider generating indexes just for the duration of the single query. It then decides which of the possible millions of alternate solution methods is likely to have the "lowest cost" given all the available information (including distribution statistics for the tables and indexes, if available).

The whole purpose of SQL is that it is a declarative language. You describe 'what' you want, not 'how' to get it. And the computer using all the information available to it computes the most efficient 'how' to to get your 'what' each time it is requested. After all, the purpose of computers is to compute.

The creation of indexes is not really an optimization. The Query Planner will obtain your 'what' according to its computation of the most efficient 'how' giving appropriate consideration to the extant circumstances. Adding indexes and statistics gives the Query Planner more 'possibilities' to consider when generating the 'how' to get 'what' you asked for but does not actually change the process or the result, merely perhaps the ability to generate a better 'how' for your 'what'.

Thank you, everyone. Very educational for me. Since my entire query uses just one table with windows on multiple fields in that table, there doesn't seem to be a way to create an index that will encompass the needs of each of those windows.

Symplectic Time Integration Methods for the Material Point Method, Experiments, Analysis and Order Reduction

M. Berzins.In WCCM-ECCOMAS2020 virtual Conference, Note: Minor typographical correction in March 2024, January, 2021.

The provision of appropriate time integration methods for the Material Point Method (MPM) involves considering stability, accuracy and energy conservation. A class of methods that addresses many of these issues are the widely-used symplectic time integration methods. Such methods have good conservation properties and have the potential to achieve high accuracy. In this work we build on the work in [5] and consider high order methods for the time integration of the Material Point Method. The results of practical experiments show that while high order methods in both space and time have good accuracy initially, unless the problem has relatively little particle movement then the accuracy of the methods for later time is closer to that of low order methods. A theoretical analysis explains these results as being similar to the stage error found in Runge Kutta methods, though in this case the stage error arises from the MPM differentiations and interpolations from particles to grid and back again, particularly in cases in which there are many grid crossings.

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