dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS
We provide a numerical package for the computation of a <i>d</i>-variate near G-optimal polynomial regression design of degree <i>m</i> on a finite design space <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1122 |
| _version_ | 1797562954799906816 |
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| author | Monica Dessole Fabio Marcuzzi Marco Vianello |
| author_facet | Monica Dessole Fabio Marcuzzi Marco Vianello |
| author_sort | Monica Dessole |
| collection | DOAJ |
| description | We provide a numerical package for the computation of a <i>d</i>-variate near G-optimal polynomial regression design of degree <i>m</i> on a finite design space <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> </mrow> </semantics> </math> </inline-formula>, by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of the discrete measure keeping the reached G-efficiency, via an accelerated version of the Lawson-Hanson algorithm for Non-Negative Least Squares (NNLS) problems. This package can solve on a personal computer large-scale problems where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mi>a</mi> <mi>r</mi> <mi>d</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>×</mo> <mo form="prefix">dim</mo> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>d</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is up to <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>8</mn> </msup> </semantics> </math> </inline-formula>–<inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>9</mn> </msup> </semantics> </math> </inline-formula>, being <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">dim</mo> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>d</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mi>d</mi> </mrow> <mi>d</mi> </mfrac> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mi>d</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mfenced> </mrow> </semantics> </math> </inline-formula>. Several numerical tests are presented on complex shapes in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and on hypercubes in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>></mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-03-10T18:35:56Z |
| format | Article |
| id | doaj.art-c4602b7ac3b943e0b0cad8555599d1a2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:35:56Z |
| publishDate | 2020-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c4602b7ac3b943e0b0cad8555599d1a22023-11-20T06:17:53ZengMDPI AGMathematics2227-73902020-07-0187112210.3390/math8071122dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLSMonica Dessole0Fabio Marcuzzi1Marco Vianello2Department of Mathematics “Tullio Levi Civita”, University of Padova, Via Trieste 63, 35131 Padova, ItalyDepartment of Mathematics “Tullio Levi Civita”, University of Padova, Via Trieste 63, 35131 Padova, ItalyDepartment of Mathematics “Tullio Levi Civita”, University of Padova, Via Trieste 63, 35131 Padova, ItalyWe provide a numerical package for the computation of a <i>d</i>-variate near G-optimal polynomial regression design of degree <i>m</i> on a finite design space <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> </mrow> </semantics> </math> </inline-formula>, by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of the discrete measure keeping the reached G-efficiency, via an accelerated version of the Lawson-Hanson algorithm for Non-Negative Least Squares (NNLS) problems. This package can solve on a personal computer large-scale problems where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mi>a</mi> <mi>r</mi> <mi>d</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>×</mo> <mo form="prefix">dim</mo> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>d</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is up to <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>8</mn> </msup> </semantics> </math> </inline-formula>–<inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>9</mn> </msup> </semantics> </math> </inline-formula>, being <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">dim</mo> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>d</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mi>d</mi> </mrow> <mi>d</mi> </mfrac> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mfrac linethickness="0pt"> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mi>d</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mfenced> </mrow> </semantics> </math> </inline-formula>. Several numerical tests are presented on complex shapes in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and on hypercubes in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>></mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/7/1122multivariate polynomial regression designsG-optimalityD-optimalitymultiplicative algorithmsG-efficiencyCaratheodory-Tchakaloff discrete measure compression |
| spellingShingle | Monica Dessole Fabio Marcuzzi Marco Vianello dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS Mathematics multivariate polynomial regression designs G-optimality D-optimality multiplicative algorithms G-efficiency Caratheodory-Tchakaloff discrete measure compression |
| title | dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS |
| title_full | dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS |
| title_fullStr | dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS |
| title_full_unstemmed | dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS |
| title_short | dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS |
| title_sort | dcatch a numerical package for d variate near g optimal tchakaloff regression via fast nnls |
| topic | multivariate polynomial regression designs G-optimality D-optimality multiplicative algorithms G-efficiency Caratheodory-Tchakaloff discrete measure compression |
| url | https://www.mdpi.com/2227-7390/8/7/1122 |
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