I'm working for quite some time on finding a numerical instance for solution of a 8 variables system of 7 very complicated inequalities plus region specification. Unfortunately I cannot produce a MWE or nothing of the sort since the inputs are really long.
My current method is Mathematica's NMinimize routine, minimizing one of the 7 inequalities subject to every other condition as constraint -- The FindInstance command simply quits the kernel without being able to finish running.
The NMinimize is able to produce output, but besides being slower than would be optimal, produce results that do not obey every constraint.
The thing is that I need to be certain, for each benchmark I run, that if the output doesn't satisfy every constraint it is because such a set of real numbers doesn't exist -- which with my current method I can't be, by experience.
So: is there a foolproof, as efficient as possible, computational method for me to find a single instance of numerical solution to 7 complicated inequalities (involving trigonometric functions) of 8 variables or be sure that such a set doesn't exist?
It could be a Mathematica/python/fortran package, genetic algorithm or anything -- as long as there is clear enough documentation.
You need to give importance multiplier to constraints and the optimization method should not be greedy.
A genetic algorithm combined with multiple-starting points (or simulated annealing for diminishing mutations) tends to converge to global minima (hence not greedy) with more time given to it but there is not guarantee that the heuristic will complete X function in Y time. The more time given to it, the better it converges to global minima.
In genetic algorithm, you can add big constraint penalties like this:
fitness_minima = some_function_output_between_1_and_10 +
constraints_breached?1000.0f:0;
so that the DNAs with no contraint-violations will be favored for the crossover part of GA.
"As efficient as possible" depends on your algorithm. If you can parallelize the algorithm and run it on multiple GPUs, it should give substantial speedup over CPU. Compared to some hours of Mona-Lisa painting by CPU, a parallelized version running on 3 low-end GPUs complete within 10 minutes (https://www.youtube.com/watch?v=QRZqBLJ6brQ). At least some OpenCL/CUDA supporting libraries/frameworks (like Tensorflow) should be able to accelerate your algorithm if you don't want to do the work distribution yourself.
I'm working on an algorithm that has to do a small number
of operations on a large numbers of small arrays, somewhat independently.
To give an idea:
1k sorting of arrays of length typically of 0.5k-1k elements.
1k of LU-solve of matrices that have rank 10-20.
everything is in floats.
Then, there is some horizontality to this problem: the above
operations have to be carried independently on 10k arrays.
Also, the intermediate results need not be stored: for example, i don't
need to keep the sorted arrays, only the sum of the smallest $m$ elements.
The whole thing has been programmed in c++ and runs. My question is:
would you expect a problem like this to enjoy significant speed ups
(factor 2 or more) with CUDA?
You can run this in 5 lines of ArrayFire code. I'm getting speedups of ~6X with this over the CPU. I'm getting speedups of ~4X with this over Thrust (which was designed for vectors, not matrices). Since you're only using a single GPU, you can run ArrayFire Free version.
array x = randu(512,1000,f32);
array y = sort(x); // sort each 512-element column independently
array x = randu(15,15,1000,f32), y;
gfor (array i, x.dim(2))
y(span,span,i) = lu(x(span,span,i)); // LU-decomposition of each 15x15 matrix
Keep in mind that GPUs perform best when memory accesses are aligned to multiples of 32, so a bunch of 32x32 matrices will perform better than a bunch of 31x31.
If you "only" need a factor of 2 speed up I would suggest looking at more straightforward optimisation possibilities first, before considering GPGPU/CUDA. E.g. assuming x86 take a look at using SSE for a potential 4x speed up by re-writing performance critical parts of your code to use 4 way floating point SIMD. Although this would tie you to x86 it would be more portable in that it would not require the presence of an nVidia GPU.
Having said that, there may even be simpler optimisation opportunities in your code base, such as eliminating redundant operations (useless copies and initialisations are a favourite) or making your memory access pattern more cache-friendly. Try profiling your code with a decent profiler to see where the bottlenecks are.
Note however that in general sorting is not a particularly good fit for either SIMD or CUDA, but other operations such as LU decomposition may well benefit.
Just a few pointers, you maybe already incorporated:
1) If you just need the m smallest elements, you are probably better of to just search the smallest element, remove it and repeat m - times.
2) Did you already parallelize the code on the cpu? OpenMP or so ...
3) Did you think about buying better hardware? (I know it´s not the nice think to do, but if you want to reach performance goals for a specific application it´s sometimes the cheapest possibility ...)
If you want to do it on CUDA, it should work conceptually, so no big problems should occur. However, there are always the little things, which depend on experience and so on.
Consider the thrust-library for the sorting thing, hopefully someone else can suggest some good LU-decomposition algorithm.
I got to solve a pretty standard problem on the GPU, but I'm quite new to practical GPGPU, so I'm looking for ideas to approach this problem.
I have many points in 3-space which are assigned to a very small number of groups (each point belongs to one group), specifically 15 in this case (doesn't ever change). Now I want to compute the mean and covariance matrix of all the groups. So on the CPU it's roughly the same as:
for each point p
{
mean[p.group] += p.pos;
covariance[p.group] += p.pos * p.pos;
++count[p.group];
}
for each group g
{
mean[g] /= count[g];
covariance[g] = covariance[g]/count[g] - mean[g]*mean[g];
}
Since the number of groups is extremely small, the last step can be done on the CPU (I need those values on the CPU, anyway). The first step is actually just a segmented reduction, but with the segments scattered around.
So the first idea I came up with, was to first sort the points by their groups. I thought about a simple bucket sort using atomic_inc to compute bucket sizes and per-point relocation indices (got a better idea for sorting?, atomics may not be the best idea). After that they're sorted by groups and I could possibly come up with an adaption of the segmented scan algorithms presented here.
But in this special case, I got a very large amount of data per point (9-10 floats, maybe even doubles if the need arises), so the standard algorithms using a shared memory element per thread and a thread per point might make problems regarding per-multiprocessor resources as shared memory or registers (Ok, much more on compute capability 1.x than 2.x, but still).
Due to the very small and constant number of groups I thought there might be better approaches. Maybe there are already existing ideas suited for these specific properties of such a standard problem. Or maybe my general approach isn't that bad and you got ideas for improving the individual steps, like a good sorting algorithm suited for a very small number of keys or some segmented reduction algorithm minimizing shared memory/register usage.
I'm looking for general approaches and don't want to use external libraries. FWIW I'm using OpenCL, but it shouldn't really matter as the general concepts of GPU computing don't really differ over the major frameworks.
Even though there are few groups, I don't think you will be able to avoid the initial sorting into groups while still keeping the reduction step efficient. You will probably also want to perform the full sort, not just sorting indexes, because that will help keep memory access efficient in the reduction step.
For sorting, read about general strategies here:
http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter46.html
For reduction (old but still good):
http://developer.download.nvidia.com/compute/cuda/1.1-Beta/x86_website/projects/reduction/doc/reduction.pdf
For an example implementation of parallel reduction:
http://developer.nvidia.com/cuda-cc-sdk-code-samples#reduction
In User table i have more than 1 million records so how can i manage using MySQL, Symfony 1.4. Make performance better.
So that it can give quick output.
To significantly improve performance of well designed system all you can do is increase the resources. Typically, these days, the cheapest way to do this is to distribute the task.
For example a slow thing in RDBM system is reading and writing to an from the storage (typically RDBMs systems start as I/O bound, that is, they mostly wait for data to get read or written to storage).
So, to offset, very commonly the RDBMS will allow you to split the table across multiple HDDs, effectively multiplying the I/O performance (approach similar to RAID0).
Adding more hard disks increases the performance. This goes on up to maximum I/O that your system could support (either simply because the system can not push more data through circuits or because it does need to crunch the numbers a bit when it fetches them so it becomes CPU bound; optimally you would be utilising both)
After that you have to start multiplying the systems distributing the data across database nodes. For this to work either RDBMS must support it or there should be application layer that will coordinate distributing the tasks and merging the results, but normally things would still scale.
I would say that with 512 systems you could have all trillion records effectively cached (10^12) and achieve relatively nice performance. But really you should specify what kind of performance you are looking for - there is a difference between full text searches on terra-records and running mostly simple fetches and updates. Also, for certain work 500ms (or even more) is considered good performance and then for other work it would be horrible.
at first: theres a big difference between 1 trillion and 1 million.
to your performance problems: show us the query thats running slow, without seeing it, it's hard to tell whats wrong with it. what you could try:
use EXPLAIN to get more information about your slow querys, see if they're using your indexes or if not (and if not, why not?)
use correct and reasonable indexes
At what point does a MySQL database start to lose performance?
Does physical database size matter?
Do number of records matter?
Is any performance degradation linear or exponential?
I have what I believe to be a large database, with roughly 15M records which take up almost 2GB. Based on these numbers, is there any incentive for me to clean the data out, or am I safe to allow it to continue scaling for a few more years?
The physical database size doesn't matter. The number of records don't matter.
In my experience the biggest problem that you are going to run in to is not size, but the number of queries you can handle at a time. Most likely you are going to have to move to a master/slave configuration so that the read queries can run against the slaves and the write queries run against the master. However if you are not ready for this yet, you can always tweak your indexes for the queries you are running to speed up the response times. Also there is a lot of tweaking you can do to the network stack and kernel in Linux that will help.
I have had mine get up to 10GB, with only a moderate number of connections and it handled the requests just fine.
I would focus first on your indexes, then have a server admin look at your OS, and if all that doesn't help it might be time to implement a master/slave configuration.
In general this is a very subtle issue and not trivial whatsoever. I encourage you to read mysqlperformanceblog.com and High Performance MySQL. I really think there is no general answer for this.
I'm working on a project which has a MySQL database with almost 1TB of data. The most important scalability factor is RAM. If the indexes of your tables fit into memory and your queries are highly optimized, you can serve a reasonable amount of requests with a average machine.
The number of records do matter, depending of how your tables look like. It's a difference to have a lot of varchar fields or only a couple of ints or longs.
The physical size of the database matters as well: think of backups, for instance. Depending on your engine, your physical db files on grow, but don't shrink, for instance with innodb. So deleting a lot of rows, doesn't help to shrink your physical files.
There's a lot to this issues and as in a lot of cases the devil is in the details.
The database size does matter. If you have more than one table with more than a million records, then performance starts indeed to degrade. The number of records does of course affect the performance: MySQL can be slow with large tables. If you hit one million records you will get performance problems if the indices are not set right (for example no indices for fields in "WHERE statements" or "ON conditions" in joins). If you hit 10 million records, you will start to get performance problems even if you have all your indices right. Hardware upgrades - adding more memory and more processor power, especially memory - often help to reduce the most severe problems by increasing the performance again, at least to a certain degree. For example 37 signals went from 32 GB RAM to 128GB of RAM for the Basecamp database server.
I'm currently managing a MySQL database on Amazon's cloud infrastructure that has grown to 160 GB. Query performance is fine. What has become a nightmare is backups, restores, adding slaves, or anything else that deals with the whole dataset, or even DDL on large tables. Getting a clean import of a dump file has become problematic. In order to make the process stable enough to automate, various choices needed to be made to prioritize stability over performance. If we ever had to recover from a disaster using a SQL backup, we'd be down for days.
Horizontally scaling SQL is also pretty painful, and in most cases leads to using it in ways you probably did not intend when you chose to put your data in SQL in the first place. Shards, read slaves, multi-master, et al, they are all really shitty solutions that add complexity to everything you ever do with the DB, and not one of them solves the problem; only mitigates it in some ways. I would strongly suggest looking at moving some of your data out of MySQL (or really any SQL) when you start approaching a dataset of a size where these types of things become an issue.
Update: a few years later, and our dataset has grown to about 800 GiB. In addition, we have a single table which is 200+ GiB and a few others in the 50-100 GiB range. Everything I said before holds. It still performs just fine, but the problems of running full dataset operations have become worse.
I would focus first on your indexes, than have a server admin look at your OS, and if all that doesn't help it might be time for a master/slave configuration.
That's true. Another thing that usually works is to just reduce the quantity of data that's repeatedly worked with. If you have "old data" and "new data" and 99% of your queries work with new data, just move all the old data to another table - and don't look at it ;)
-> Have a look at partitioning.
2GB and about 15M records is a very small database - I've run much bigger ones on a pentium III(!) and everything has still run pretty fast.. If yours is slow it is a database/application design problem, not a mysql one.
It's kind of pointless to talk about "database performance", "query performance" is a better term here. And the answer is: it depends on the query, data that it operates on, indexes, hardware, etc. You can get an idea of how many rows are going to be scanned and what indexes are going to be used with EXPLAIN syntax.
2GB does not really count as a "large" database - it's more of a medium size.
I once was called upon to look at a mysql that had "stopped working". I discovered that the DB files were residing on a Network Appliance filer mounted with NFS2 and with a maximum file size of 2GB. And sure enough, the table that had stopped accepting transactions was exactly 2GB on disk. But with regards to the performance curve I'm told that it was working like a champ right up until it didn't work at all! This experience always serves for me as a nice reminder that there're always dimensions above and below the one you naturally suspect.
Also watch out for complex joins. Transaction complexity can be a big factor in addition to transaction volume.
Refactoring heavy queries sometimes offers a big performance boost.
A point to consider is also the purpose of the system and the data in the day to day.
For example, for a system with GPS monitoring of cars is not relevant query data from the positions of the car in previous months.
Therefore the data can be passed to other historical tables for possible consultation and reduce the execution times of the day to day queries.
Performance can degrade in a matter of few thousand rows if database is not designed properly.
If you have proper indexes, use proper engines (don't use MyISAM where multiple DMLs are expected), use partitioning, allocate correct memory depending on the use and of course have good server configuration, MySQL can handle data even in terabytes!
There are always ways to improve the database performance.
It depends on your query and validation.
For example, i worked with a table of 100 000 drugs which has a column generic name where it has more than 15 characters for each drug in that table .I put a query to compare the generic name of drugs between two tables.The query takes more minutes to run.The Same,if you compare the drugs using the drug index,using an id column (as said above), it takes only few seconds.
Database size DOES matter in terms of bytes and table's rows number. You will notice a huge performance difference between a light database and a blob filled one. Once my application got stuck because I put binary images inside fields instead of keeping images in files on the disk and putting only file names in database. Iterating a large number of rows on the other hand is not for free.
No it doesnt really matter. The MySQL speed is about 7 Million rows per second. So you can scale it quite a bit
Query performance mainly depends on the number of records it needs to scan, indexes plays a high role in it and index data size is proportional to number of rows and number of indexes.
Queries with indexed field conditions along with full value would be returned in 1ms generally, but starts_with, IN, Between, obviously contains conditions might take more time with more records to scan.
Also you will face lot of maintenance issues with DDL, like ALTER, DROP will be slow and difficult with more live traffic even for adding a index or new columns.
Generally its advisable to cluster the Database into as many clusters as required (500GB would be a general benchmark, as said by others it depends on many factors and can vary based on use cases) that way it gives better isolation and gives independence to scale specific clusters (more suited in case of B2B)