{"id":389,"date":"2021-08-08T16:15:23","date_gmt":"2021-08-08T14:15:23","guid":{"rendered":"https:\/\/alma-consulting.eu\/?p=389"},"modified":"2022-05-28T13:49:17","modified_gmt":"2022-05-28T11:49:17","slug":"working-wonders-with-adpl-math-illustrated-ep03-data-reduction","status":"publish","type":"post","link":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/","title":{"rendered":"Working Wonders with ADPL Math\u00a0– Ep 03: Data Reduction Fundamentals"},"content":{"rendered":"\n
What are we doing here?<\/h2>\n\n\n\n
<\/p>\n\n\n\n
This post may come as a surprise to many, as ANSYS APDL is at its core a tool aiming at simulating physical phenomena so that \u00abdata reduction\u00bb, being more oriented to the data analysis community, might sound a bit out-of-place.<\/p>\n\n\n\n
As a matter of fact, being a numerical tool, it does have ubiquitous applications and -as we shall shortly see- it can be also beneficial to down-to-earth, goal-oriented folks like say -engineers.<\/p>\n\n\n\n
Browsing the APDL Math commands, I started being curious about the *COMPRESS command, which I had so far ignored: as it was, I had assumed that it merely was a functionality aimed at compressing sparse matrices, i.e. a lossless procedure, detecting and eliminating near-zero entries. And yes, that\u2019s exactly what it can do, but there is more to it: it will also compress data using Singular Values Decomposition (SVD), which is probably one of the most important numerical tool there is. This is not the place to provide too much background on the topic, and for those interested there is a wealth of books and articles on the subject, one prominent contribution being the online videos by Steven Brunton and Nathan Kuntz, see [1] for an introduction.<\/p>\n\n\n\n
Before discussing applications, I will briefly introduce the topic of SVD, how it relates to data compression, and which APDL Math capabilities we need to use.<\/p>\n\n\n\n\n\n\n\n
Compression using SVD: a minimal working example<\/h2>\n\n\n\n
Say we have a nx3 matrix \u00abM\u00bb, for which the first two columns are linearly independent (geometrically speaking, those vectors are \u00abnon parrallel\u00bb), and the third one is a linear combination of the first two.<\/p>\n\n\n\n
From linear algebra, we know that this matrix has rank 2, and hence is non-invertible. So far so good. But that\u2019s not our point, since we don\u2019t want to solve for the matrix. Quite the contrary, we assume that there are redundancies (or repetitions) in its vectors, so that it\u2019s non invertible, hence amenable to compression. Then, from this perspective, we know that it would be more economical to describe this matrix as \u00abits first vector is like this, the second vector like that, and the third is a linear combination of the first two, with the following coefficients\u00bb.<\/p>\n\n\n\n
Well, SVD does exactly that. Let\u2019s see how.<\/p>\n\n\n\n
Mathematically, SVD will factor the original matrix M (nxm) into a product of 3 matrices, namely<\/p>\n\n\n\n
In this factorization, the number of non-zero entries in \uf053\uf020is equal to the rank of M. Obviously, if M is rectangular, M has rank at most equal to min(n,m). But here, we are specifically looking at situations where redundancies exist, so we expect the rank r to be much lower than that.<\/p>\n\n\n\n
Right, let\u2019s see how it works within ANSYS. Assume for simplicity a 3×3 matrix.<\/p>\n\n\n\n
First column vector <\/p>\n\n\n\n
Second column vector <\/p>\n\n\n\n
Third column vector <\/p>\n\n\n\n
The corresponding input commands would be:<\/p>\n\n\n\n
n=3\t! nb of \"DOFs\"\nm=3\t! nb of \"snapshots\"\n\n! Generate dummy \"snapshots\" matrix\n*DMAT,dummyMat,D,ALLOC,n,m\n\n! First and second vectors (should be linearly independent, hence rank=2)\n! first column \ndummyMat(1,1)=1\t\ndummyMat(2,1)=1\t \ndummyMat(3,1)=1\t \n! second column\ndummyMat(1,2)=1\t\t\ndummyMat(2,2)=-1\ndummyMat(3,2)=1\n! Add a third column, a linear combination of the first two\nalpha=1\nbeta=1\ndummyMat(1,3)=alpha*dummyMat(1,1)+beta*dummyMat(1,2)\ndummyMat(2,3)=alpha*dummyMat(2,1)+beta*dummyMat(2,2)\ndummyMat(3,3)=alpha*dummyMat(3,1)+beta*dummyMat(3,2)<\/code><\/pre>\n\n\n\n
From there, we can simply invoke the matrix compression by first allocating for the Matrix U, which will contain the left singular vectors, corresponding to singular values above a user defined threshold (more on this later). To do that, we simply copy the original matrix (it will be read and replaced by a skinnier version, same number of rows, but lower number of columns). The sigma matrix is diagonal and will be stored in a vector. Note that both vector sigmaVec and matrix Vtran will be created on the fly, and don\u2019t require manual allocation.<\/p>\n\n\n\n
! Perform Singular Value Decomposition \n*DMAT,U,D,COPY,dummyMat\t\n\n! M=U SIGMA V*\nthreshold=1e-2\n*COMPRESS,U,SVD,threshold,SigmaVec,Vconj<\/code><\/pre>\n\n\n\n
A number of important features are already visible:<\/p>\n\n\n\n
– the U matrix has been compressed to size 3×2, consistent with the fact that only 2 singular values are above the threshold (the last one being a numerical zero)<\/p>\n\n\n\n
– U and V are unitary matrices (vectors have unit norm, and are orthogonal), hence U does not <\/em>contain our \u00abbasis\u00bb vectors, but rather an orthogonal basis for the r-dimensional space in which the matrix M projects the n-dimensional space.<\/p>\n\n\n\n
– Vconj has 3 columns, but also 3 rows: however, the last row corresponds to the amplitudes of the discarded 3rd<\/sup> right singular vector, and has no usage.<\/p>\n\n\n\n
This decomposition and the \u00abpadding\u00bb is explained in an article by Gregory Undersen, see [2]. In order to check that everything ran as expected, we might want to re-generate the original matrix from its truncated factors, as follows:<\/p>\n\n\n\n
! Number of Singular values above threshold\nnbSingularValues=U_coldim\t\n\n! CHECK: REGENERATE ORIGINAL MATRIX FROM COMPRESSED \n! Step1: Generate SigmaMat (diagonal matrix) from vector SigmaVec\n*SMAT,SigmaMat,D,ALLOC,DIAG,nbSingularValues\n*do,ind,1,nbSingularValues\n\tSigmaMat(ind,ind)=SigmaVec(ind)\n*enddo\n\n! STEP2: GENERATE TRUNCATED RIGHT VECTORS MATRICE \n*DMAT,resizedVconj,D,COPY,Vconj\n*DMAT,resizedVconj,D,RESIZE,nbSingularValues,nbSnapshots\n\n!*MULT, M1, T1, M2, T2, M3 (M3=M1(T1)*M2(T2))\n*MULT,SigmaMat,,Vconj,,SigmaVconj\n*PRINT,SigmaVconj\n*MULT,U,,SigmaVconj,,shouldBeDummyMat\n*PRINT,shouldBeDummyMat<\/code><\/pre>\n\n\n\n
The output is consistent with our expectations:<\/p>\n\n\n\n
This is all well, but one could (and in fact, should) argue that this particular example was mathematically designed to showcase repeated vectors identification, and obviously works to perfection. In a real situation, one should expect a gradual decrease of singular values or, stated differently, a fuzzier boundary between \u00abneeded\u00bb and \u00absuperfluous\u00bb components, which renders the singular values selection much more empiric.<\/p>\n\n\n\n
A bulletproof -though na\u00efve- approach would be to iteratively generate data using an increasingly large number of singular vectors, until a satisfactory result is obtained, but this may not even be required. As it appears, since U and V matrices are unitary, so that only the singular values contain information about amplitude (i.e. they are scaling factors). What\u2019s more, those values are provided in decreasing order, and the total energy of the original signals (the square of the Frobenius norm of the original matrix) is equal to the sum of the squared singular values.<\/p>\n\n\n\n
Luckily for us ANSYS dutifully follows this rule. By construction, the Froebenius (energy) norm of our dummy matrix M is the sum of the norm of its vectors, namely:<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
and <\/p>\n\n\n\n
<\/p>\n\n\n\n
So that its total energy norm is :<\/p>\n\n\n\n
<\/p>\n\n\n\n
This value compares favorably with the sum of the squared singular values calculated by ANSYS:<\/p>\n\n\n\n
<\/p>\n\n\n\n
Hence, by selecting a number of singular values, the amount of the total energy is known beforehand, and the energy loss (sum of squared unused<\/em> singular values) can be controlled. To reemphasize, compression will only be efficient when the singular values decrease sufficiently rapidly, which may seem difficult to detect with the naked eye. A very clear explanation has been given by Pr.Alex Townsend in [3], which sheds light on the fact that propagating waves phenomena are ill-suited for SVD compression.<\/p>\n\n\n\n\n\n\n\n
Compression using SVD: sample usage<\/h2>\n\n\n\n
Remember the aluminum bracket-shaped support we used in Episode 1 to illustrate Thermal Modal Analysis? It wouldn\u2019t hurt to use it again, since it can be handy to demonstrate the similarities and differences between the POD basis (i.e. the left singular vectors) and classical Normal modes<\/p>\n\n\n\n
\n
\n
<\/figure>
\n
Let’s assume our bracket is submitted to a random temperature at its base, and see how its thermal response can be compressed using SVD. <\/p>\n<\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n
For ease of understanding, this temperature signal is filtered white noise, using a order-1 low pass filter with time constant equal to 30s. Its amplitude is adjusted so that its standard deviation is unity (i.e. 1K RMS). We apply this temperature signal as a boundary conditions at the bracket support, and estimate temperature fields every second, for a total duration of 1000s, hence 1000 snapshots. In terms of space-wise discretisation, let\u2019s first use the most simple element we have i.e. the good old PLANE55 (linear, i.e. 4 nodes element).<\/p>\n\n\n\n
Now, let\u2019s apply POD magic to these values and see what happens. To test the method efficiency, we will assume decreasing values for the threshold parameter, and extract the maximum RMS error in terms of nodal temperature over the entire model.<\/p>\n\n\n\n
Config#<\/td>
Threshold<\/td>
Model Order<\/td>
Max. Nodal Error [K RMS]<\/td><\/tr>
1<\/td>
1e-2<\/td>
2<\/td>
1.76<\/td><\/tr>
2<\/td>
1e-3<\/td>
4<\/td>
0.60<\/td><\/tr>
3<\/td>
1e-4<\/td>
6<\/td>
0.057<\/td><\/tr>
4<\/td>
1e-5<\/td>
8<\/td>
0.015<\/td><\/tr>
5<\/td>
1e-6<\/td>
10<\/td>
0.0016<\/td><\/tr>
6<\/td>
1e-7<\/td>
12<\/td>
0.00014<\/td><\/tr><\/tbody><\/table>Convergence indicators as a function of selected threshold<\/figcaption><\/figure>\n\n\n\n
From this, it is quite obvious that the procedure converges extremely rapidly, so that using 8 vectors, one readily reduces the compression error to about 1.5% of the original values, and below 0.01% for 12 vectors, well below the most daring engineering accuracy. It is also interesting to look at the corresponding spatial error distribution, see below. For the first two cases (model order 2 and 4), the maximum error is located about halfway along the heat propagation direction, while for higher order cases, the error is mostly localized at the bottom of the bracket.<\/p>\n\n\n\nThermal Reduction error distribution using 2 POD vectors<\/figcaption><\/figure>\n\n\n\nThermal Reduction error distribution using 4 POD vectors<\/figcaption><\/figure>\n\n\n\n<\/figure>\n\n\n\n
Quite impressive, if you ask me. Now, let’s turn on to visually inspecting those POD vectors, which are so efficient at capturing the physics at hand:<\/p>\n\n\n\n
\n\n\n\n
POD Vectors unveiled<\/h2>\n\n\n\n
POD (aka left singular vectors) order 1 to 9 are as follows: <\/p>\n\n\n\n
\n
\n
\n
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li>
<\/figure><\/li><\/ul>POD vectors 1 to 9<\/figcaption><\/figure>\n\n\n\n
<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n
Obviously we see that POD vectors are only loosely connected to normal modes. In this particular case, the mass (thermal inertia) is homogeneous, hence the very first POD and normal modes are somehow similar, but this is not the case for the other modes: POD vectors corresponds to temperature distribution that are actually <\/em>existing in the solution (based on the specific loading applied, both space and time-wise), ranging them in terms of contribution, while normal modes capture all<\/em> possible<\/em> temperature distributions, ranging them from lowest to highest \u00abstrain energy\u00bb (or, in thermal terms, from the longest to the shortest time constant).<\/p>\n\n\n\n
To reiterate, normal modes are orthogonal with respect to the \u00abmass\u00bb and \u00abstiffness\u00bb matrices (here, thermal capacity and conductivity) while POD vectors are orthogonal with each other. So far, so good. Now, we have a means to compress data, in our case from 228 nodal values per time point down to a mere dozen.<\/p>\n\n\n\n
A word of caution about threshold : in ANSYS, this is interpreted as the ratio of the current eigenvalue to the first (largest one). To clarify this, one can inspect the sigma matrix (which is actually a diagonal matrix, and stored as a vector). Dumping this vector to a text file one can readily check that actually, by requiring a threshold of 1e-2, only the first two POD vectors were required. Also, the decreasing rate is really both very<\/em> fast and very <\/em>steady, as illustrated by the singular values vector entries (see below). The amplitude decreases by a factor of 10 every other POD vector, down to 1e-12. This is in total contrast with normal modes, especially with thermal problems, for which the convergence rate is generally extremely slow.<\/p>\n\n\n\n
Other tests conducted with thermal conduction\/diffusion models in the range of up to 100k DOF showed the exact same trends. It was never necessary to include more than a dozen POD vectors to achieve excellent convergence (relative error below 10^-4). The POD reduction process, by itself, never required more computation time than about one transient time step.<\/p>\n\n\n\n
\n\n\n\n
Summary and Conclusion<\/h2>\n\n\n\n
As we know, engineering simulations generate significant quantities of output data, which are certainly not spatially random -even if the forcing function is- but always organized by the physics of the system. Hence, a compression tool like POD will find repeating patterns (correlation) between DOF responses, and find an optimal base for reduced order representation of system response, capture both system dynamics and forcing function characteristics<\/em>.POD reduction will always function, on one condition: the data used for identification must be rich enough, which in turns requires the loading to be \u00abharsh\u00bb enough. (And yes, engineering judgement will still be around for a while, very much so).<\/p>\n\n\n\n
This nice property can be used to our advantage for situations where large data volumes must be stored. But even more important, for all situations where linear<\/em> downstream analysis are to be subsequently performed, it opens the possibility to directly estimate responses by linearly combining the unitary responses of a handful of base vectors. This can be very handy for say-thermal elastic problems, for which the mechanical analysis can be obtained \u00abfor free\u00bb, once the thermal transient analysis has been performed. The additional cost of projection of the thermal solution onto the POD basis, and the estimation of the mechanical response are numerically inexpensive, and this can be invaluable when long, complex (a-periodic) transients must be analyzed.<\/p>\n\n\n\n
Beyond that, for truly non-linear problems, POD method allows to find the lowest dimensional space in which to search for a solution, leading to efficient model reduction techniques.<\/p>\n\n\n\n
Both aspects will be treated in upcoming articles so stay tuned.<\/p>\n\n\n\n
Happy reduction!<\/p>\n<\/div>\n<\/div>\n\n\n\n
References<\/h2>\n\n\n\n
[1] Brunton and Kuntz – Data Driven Science & Engineering : Machine Learning, Dynamical Systems, and Control<\/li>
What are we doing here? This post may come as a surprise to many, as ANSYS APDL is at its core a tool aiming at simulating physical phenomena so that \u00abdata reduction\u00bb, being more oriented to the data analysis community, might sound a bit out-of-place. As a matter of fact, being a numerical tool, it […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[4,5],"class_list":["post-389","post","type-post","status-publish","format-standard","hentry","category-technical-literature","tag-ansys-apdl-math","tag-model-reduction"],"yoast_head":"\n
Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\" \/>\n<link rel=\"next\" href=\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting\" \/>\n<meta property=\"og:description\" content=\"What are we doing here? This post may come as a surprise to many, as ANSYS APDL is at its core a tool aiming at simulating physical phenomena so that \u00abdata reduction\u00bb, being more oriented to the data analysis community, might sound a bit out-of-place. As a matter of fact, being a numerical tool, it […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\" \/>\n<meta property=\"og:site_name\" content=\"Alma Consulting\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-08T14:15:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-05-28T11:49:17+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\" \/>\n<meta name=\"author\" content=\"admin9661\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin9661\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\"},\"author\":{\"name\":\"admin9661\",\"@id\":\"https:\/\/alma-consulting.eu\/#\/schema\/person\/67349dff7b3613d00b9310b853e1544c\"},\"headline\":\"Working Wonders with ADPL Math\u00a0– Ep 03: Data Reduction Fundamentals\",\"datePublished\":\"2021-08-08T14:15:23+00:00\",\"dateModified\":\"2022-05-28T11:49:17+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\"},\"wordCount\":2106,\"publisher\":{\"@id\":\"https:\/\/alma-consulting.eu\/#organization\"},\"image\":{\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\",\"keywords\":[\"ANSYS APDL Math\",\"Model Reduction\"],\"articleSection\":[\"Technical Literature\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\",\"url\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\",\"name\":\"Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting\",\"isPartOf\":{\"@id\":\"https:\/\/alma-consulting.eu\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\",\"datePublished\":\"2021-08-08T14:15:23+00:00\",\"dateModified\":\"2022-05-28T11:49:17+00:00\",\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage\",\"url\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\",\"contentUrl\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\",\"width\":200,\"height\":273},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/alma-consulting.eu\/#website\",\"url\":\"https:\/\/alma-consulting.eu\/\",\"name\":\"Alma Consulting\",\"description\":\"No limits, except for physics\",\"publisher\":{\"@id\":\"https:\/\/alma-consulting.eu\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/alma-consulting.eu\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/alma-consulting.eu\/#organization\",\"name\":\"Alma Consulting\",\"url\":\"https:\/\/alma-consulting.eu\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/alma-consulting.eu\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2020\/06\/LOGO_V1_bis.png\",\"contentUrl\":\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2020\/06\/LOGO_V1_bis.png\",\"width\":1579,\"height\":447,\"caption\":\"Alma Consulting\"},\"image\":{\"@id\":\"https:\/\/alma-consulting.eu\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.linkedin.com\/in\/nicolas-jobert\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/alma-consulting.eu\/#\/schema\/person\/67349dff7b3613d00b9310b853e1544c\",\"name\":\"admin9661\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/alma-consulting.eu\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/4a042c685169086f29596c15b956a482?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/4a042c685169086f29596c15b956a482?s=96&d=mm&r=g\",\"caption\":\"admin9661\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/","next":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/2\/","og_locale":"en_US","og_type":"article","og_title":"Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting","og_description":"What are we doing here? This post may come as a surprise to many, as ANSYS APDL is at its core a tool aiming at simulating physical phenomena so that \u00abdata reduction\u00bb, being more oriented to the data analysis community, might sound a bit out-of-place. As a matter of fact, being a numerical tool, it […]","og_url":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/","og_site_name":"Alma Consulting","article_published_time":"2021-08-08T14:15:23+00:00","article_modified_time":"2022-05-28T11:49:17+00:00","og_image":[{"url":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png","type":"","width":"","height":""}],"author":"admin9661","twitter_card":"summary_large_image","twitter_misc":{"Written by":"admin9661","Est. reading time":"12 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#article","isPartOf":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/"},"author":{"name":"admin9661","@id":"https:\/\/alma-consulting.eu\/#\/schema\/person\/67349dff7b3613d00b9310b853e1544c"},"headline":"Working Wonders with ADPL Math\u00a0– Ep 03: Data Reduction Fundamentals","datePublished":"2021-08-08T14:15:23+00:00","dateModified":"2022-05-28T11:49:17+00:00","mainEntityOfPage":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/"},"wordCount":2106,"publisher":{"@id":"https:\/\/alma-consulting.eu\/#organization"},"image":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage"},"thumbnailUrl":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png","keywords":["ANSYS APDL Math","Model Reduction"],"articleSection":["Technical Literature"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/","url":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/","name":"Working Wonders with ADPL Math\u00a0- Ep 03: Data Reduction Fundamentals - Alma Consulting","isPartOf":{"@id":"https:\/\/alma-consulting.eu\/#website"},"primaryImageOfPage":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage"},"image":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage"},"thumbnailUrl":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png","datePublished":"2021-08-08T14:15:23+00:00","dateModified":"2022-05-28T11:49:17+00:00","inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/08\/08\/working-wonders-with-adpl-math-illustrated-ep03-data-reduction\/#primaryimage","url":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png","contentUrl":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png","width":200,"height":273},{"@type":"WebSite","@id":"https:\/\/alma-consulting.eu\/#website","url":"https:\/\/alma-consulting.eu\/","name":"Alma Consulting","description":"No limits, except for physics","publisher":{"@id":"https:\/\/alma-consulting.eu\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/alma-consulting.eu\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/alma-consulting.eu\/#organization","name":"Alma Consulting","url":"https:\/\/alma-consulting.eu\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/alma-consulting.eu\/#\/schema\/logo\/image\/","url":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2020\/06\/LOGO_V1_bis.png","contentUrl":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2020\/06\/LOGO_V1_bis.png","width":1579,"height":447,"caption":"Alma Consulting"},"image":{"@id":"https:\/\/alma-consulting.eu\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.linkedin.com\/in\/nicolas-jobert\/"]},{"@type":"Person","@id":"https:\/\/alma-consulting.eu\/#\/schema\/person\/67349dff7b3613d00b9310b853e1544c","name":"admin9661","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/alma-consulting.eu\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/4a042c685169086f29596c15b956a482?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/4a042c685169086f29596c15b956a482?s=96&d=mm&r=g","caption":"admin9661"}}]}},"_links":{"self":[{"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/posts\/389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/comments?post=389"}],"version-history":[{"count":49,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/posts\/389\/revisions"}],"predecessor-version":[{"id":562,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/posts\/389\/revisions\/562"}],"wp:attachment":[{"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/media?parent=389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/categories?post=389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/alma-consulting.eu\/index.php\/wp-json\/wp\/v2\/tags?post=389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"M=U \Sigma V\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-125881b785b1ee24c020c9a4e23ec452_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"\Sigma\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-14fb1e14301ad034b94e3db3ff52c0c9_l3.png\" "\\"Rendered")
= diagonal matrix, containing singular values (nxm)<\/li>![\"M_1= \begin{bmatrix} 1 \\1 \\ 1 \end{bmatrix}\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-ebf80310553a2497b346fc82b04872c2_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"M_2= \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-ed0f29b2cf88ef505ecde5303a224a69_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"M_3=M_1+M_2\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-bba07e3061c0a786b750bf45122bdcde_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"||M_1||^2=1^2+1^2+1^2=3\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-2e233df16c998a3ed79ddf48625bd867_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"||M_2||^2=1^2+(-1)^2+1^2=3\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-8f90058d3f8d3ab39b7a1eba29e36541_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"||M_3||^2=2^2+(0)^2+2^2=8\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-e52bf1bd1dfef966bbf94288c448f982_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"||M||=3+3+8=14\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-d5f0173c69f98b012a49fea286d27e05_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"Tr(\Sigma \Sigma ^t)=\sigma_1^2+\sigma_2^2=(3.464)^2+(1.414)^2=14\"](\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-bd83942bc022d161f2277389c601e318_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image.png\")
<\/figure>\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-1-1024x770.png\")
![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-2-1024x770.png\")
![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-3-1024x770.png\")
<\/figure>\n\n\n\n\n\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-5-1024x770.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-6-1024x770.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/image-7-1024x770.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_1_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_2_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_3_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_4_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_5_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_6_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_7_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_8_THERMAL-1024x770.png\")
<\/figure><\/li>![\"\"](\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/08\/POD_VECTOR_9_THERMAL-1024x770.png\")
<\/figure><\/li><\/ul>