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The Bayesian Reconstruction (BR) Method

Alexander Rothkopf


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    <dcat:keyword>Bayesian inference, spectral reconstruction, spectral function, inverse problem, BR Method, Maximum Entropy Method,</dcat:keyword>
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    <dct:description>&lt;p&gt;//*********************************************************************************&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;Bayesian Spectral Reconstruction and Extended MEM&lt;br&gt; //&lt;br&gt; // &amp;nbsp; &amp;nbsp;Code:&amp;nbsp;Dr. Alexander Karl Rothkopf&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; DATE: July 14th 2021&lt;br&gt; //&lt;br&gt; // &amp;nbsp; &amp;nbsp;Method: Dr. Yannis Burnier, Dr. Alexander Karl Rothkopf DATE: 2013&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;========================================================================&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;THIS PROGRAM IS AN IMPLEMENTATION OF TWO BAYESIAN APPROACHES&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;FOR THE NUMERICAL INVERSION OF THE TYPICALLY ILL DEFINED TASK&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;DATA[t] = Int_-WMIN^WMAX [ K[w,t] * SPEC[w] ]dw&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;WHERE THE GOAL IS TO EXTRACT THE SPECTRAL FUNCTION SPEC[w] FROM&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;THE PROVIDED LATTICE DATA DATA[t] USING A PREVIOUSLY SPECIFIED&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;KERNEL FUNCTION K[w,t].&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;References:&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;Y.&amp;nbsp;Burnier and A.&amp;nbsp;Rothkopf,&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;quot;Bayesian Approach to Spectral Function Reconstruction for&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp; Euclidean Quantum Field Theories&amp;quot;&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;arXiv:1307.6106 [hep-lat]&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;Phys. Rev. Lett. 111, 182003 (2013)&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;A.&amp;nbsp;Rothkopf&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;quot;Improved Maximum Entropy Analysis with an Extended Search Space&amp;quot;&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;arXiv:1110.6285 [physics.comp-ph]&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;J.Comput.Phys. 238 (2013) 106-114&lt;br&gt; //&lt;br&gt; //*********************************************************************************&lt;/p&gt; &lt;p&gt;//*********************************************************************************&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;This program uses adapted code from two libraries:&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;1. The GNU Scientific Library:&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;SINGULAR VALUE DECOMPOSITION&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;EIGENSYSTEMS OF SYMMETRIC MATRICES&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;MATRIX INVERSION BASED ON QR DECOMPOSITION&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;SOLVER FOR Ax=b BASED ON QR DECOMPOSITION&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;2. The ALGLIB Library (MPFR part 2.6.):&lt;br&gt; //&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;LBFGS MINIMIZER&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;LEVENBERG MARQUARD + LBFGS PRECONDITIONER MINIMIZER&lt;br&gt; //&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;EIGENVALUES FOR NON_SYMMETRIC MATRICES&lt;br&gt; //&lt;br&gt; //*********************************************************************************&lt;/p&gt; &lt;p&gt;&amp;nbsp;&lt;/p&gt;</dct:description>
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