NONLINEAR EQUATIONS AND ITERATIVE METHODS. APPLICATIONS TO OPTIMISATION PROBLEMS AND MATRIX EQUATIONS. (Q3162744): Difference between revisions
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(Created claim: summary (P836): THIS PROJECT PROPOSES 11 LINES OF RESEARCH RELATED TO THE NUMERICAL RESOLUTION OF NONLINEAR EQUATIONS. AMONG THE OBJECTIVES OF THESE LINES, WE CAN HIGHLIGHT THE DESIGN OF EFFICIENT ITERATIVE METHODS FOR THE NUMERICAL RESOLUTION OF EQUATIONS AND NONLINEAR SYSTEMS, ANALYSING THEIR CONVERGENCE AND COMPUTATIONAL EFFICIENCY. SO, FOR EXAMPLE, WE LOOK FOR FREE DERIVATIVE AND/OR REVERSE-FREE METHODS FOR THEIR APPLICABILITY IN OPTIMISATION PROBLEMS AND I...) |
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NONLINEAR EQUATIONS AND ITERATIVE METHODS. APPLICATIONS TO OPTIMISATION PROBLEMS AND MATRIX EQUATIONS. | |||
Revision as of 16:45, 12 October 2021
Project Q3162744 in Spain
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | NONLINEAR EQUATIONS AND ITERATIVE METHODS. APPLICATIONS TO OPTIMISATION PROBLEMS AND MATRIX EQUATIONS. |
Project Q3162744 in Spain |
Statements
25,410.0 Euro
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50,820.0 Euro
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50.0 percent
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1 January 2015
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31 December 2018
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UNIVERSIDAD DE LA RIOJA
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42°27'58.03"N, 2°26'22.81"W
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26089
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EL PRESENTE PROYECTO PLANTEA 11 LINEAS DE INVESTIGACION RELACIONADAS CON LA RESOLUCION NUMERICA DE ECUACIONES NO LINEALES. ENTRE LOS OBJETIVOS DE ESTAS LINEAS, PODEMOS DESTACAR EL DE DISEÑAR METODOS ITERATIVOS EFICIENTES PARA LA RESOLUCION NUMERICA DE ECUACIONES Y SISTEMAS NO LINEALES, ANALIZANDO SU CONVERGENCIA Y EFICIENCIA COMPUTACIONAL. ASI POR EJEMPLO, BUSCAMOS METODOS LIBRES DE DERIVADAS Y/O LIBRES DE INVERSOS POR SU APLICABILIDAD EN PROBLEMAS DE OPTIMIZACION Y EN PROBLEMAS QUE PRESENTEN SINGULARIDADES EN LA SOLUCION BUSCADA. EL USO DE DIFERENCIAS DIVIDIDAS PARA APROXIMAR LAS DERIVADAS EN ESTE TIPO DE PROBLEMAS ES UNA TECNICA QUE EMPLEAMOS CON FRECUENCIA. DISTINGUIMOS EN ESTE ASPECTO A SU VEZ DOS FAMILIAS DE PROCESOS: CON Y SIN MEMORIA. EN ESTE CAMPO, TRABAJAMOS INDISTINTAMENTE CON ECUACIONES ESCALARES Y CON SISTEMAS DE ECUACIONES. OTRA LINEA DE TRABAJO SE DEDICA A LOS METODOS DIRECCIONALES, ESPECIFICAMENTE DISEÑADOS PARA ECUACIONES NO LINEALES, Y CON APLICACIONES EN PROBLEMAS DE OPTIMIZACION Y GEOMETRICOS (INTERSECCION DE SUPERFICIES). DEDICAMOS TAMBIEN UNA PARTE DE ESTE PROYECTO AL ESTUDIO DE METODOS HIBRIDOS, CONSISTENTES EN COMBINAR DOS O MAS PROCESOS ITERATIVOS PARA INTENTAR EXPLOTAR AL MAXIMO LAS VENTAJAS DE CADA UNO (VELOCIDAD DE CONVERGENCIA DEL UNO, REGION DE ACCESIBILIDAD DEL OTRO, ETC.). UNA DE LAS TECNICAS NOVEDOSAS QUE INTRODUCIMOS EN ESTE PROYECTO PARA CONSTRUIR PROCESOS ITERATIVOS EN LAS LINEAS ANTERIORES ES EL USO DE FUNCIONES PESO. DICHA TECNICA PERMITE AUMENTAR EL ORDEN DE CONVERGENCIA DE UN PROCESO ITERATIVO DADO SIN AUMENTAR DE MANERA EXCESIVA SU COSTE OPERACIONAL. DENTRO DEL ESTUDIO GENERAL DE LOS PROCESOS ITERATIVOS CONSTRUIDOS DE LAS DIFERENTES FORMAS CITADAS ANTERIORMENTE, HAREMOS ESPECIAL HINCAPIE EN EL ESTUDIO DE LO QUE OCURRE CUANDO SE APROXIMAN RAICES MULTIPLES. COMO NORMA GENERAL, SABEMOS QUE EL ORDEN DE CONVERGENCIA SE REDUCE DRASTICAMENTE EN ESTOS CASOS. PERO ADEMAS, EN EL CASO MULTIDIMENSIONAL, APARECEN OTROS PROBLEMAS ASOCIADOS CON LA SINGULARIDAD DE LA MATRIZ JACOBIANA O EL MAL CONDICIONAMIENTO CERCA DE LA SOLUCION._x000D_ OTRA IMPORTANTE LINEA DE TRABAJO ES EL ANALISIS DE LA DINAMICA DE LOS METODOS ITERATIVOS. ESTUDIAREMOS DESDE UN PUNTO DE VISTA ANALITICO, NUMERICO Y GRAFICO CUESTIONES TALES COMO LAS CUENCAS DE ATRACCION DE LAS RAICES, LOS PLANOS DE PARAMETROS, LOS CONJUNTOS DE JULIA, LA PRESENCIA DE CICLOS ATRACTORES Y DE ATRACTORES EXTRAÑOS (PUNTOS FIJOS QUE NO SON SOLUCION DE LA ECUACION BUSCADA), LA CONVERGENCIA GENERAL Y LA PRESENCIA DE COMPORTAMIENTOS CAOTICOS. DESDOBLAMOS TODOS LOS ESTUDIOS REALIZADOS EN EL ESTUDIO DE LA DINAMICA REAL Y COMPLEJA, PUES AMBAS NO SON EQUIVALENTES Y UTILIZAN DIFERENTES HERRAMIENTAS Y RESULTADOS. EN OTRA LINEA DE INVESTIGACION ANALIZAMOS LA CONEXION ENTRE EL COMPORTAMIENTO DE LOS METODOS ITERATIVOS PARA RESOLVER ECUACIONES NO LINEALES CON LOS METODOS NUMERICOS PARA RESOLVER ECUACIONES DIFERENCIALES (PROBLEMAS DE VALOR INICIAL)._x000D_ SIN CERRAR LA PUERTA A OTRAS APLICACIONES, PRESTAREMOS ESPECIAL ATENCION A DOS APLICACIONES CONCRETAS QUE NECESITAN DE LA RESOLUCION DE ECUACIONES NO LINEALES PARA SU RESOLUCION: LOS PROBLEMAS DE OPTIMIZACION NO LINEAL Y LAS ECUACIONES MATRICIALES NO LINEALES. (Spanish)
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THIS PROJECT PROPOSES 11 LINES OF RESEARCH RELATED TO THE NUMERICAL RESOLUTION OF NONLINEAR EQUATIONS. AMONG THE OBJECTIVES OF THESE LINES, WE CAN HIGHLIGHT THE DESIGN OF EFFICIENT ITERATIVE METHODS FOR THE NUMERICAL RESOLUTION OF EQUATIONS AND NONLINEAR SYSTEMS, ANALYSING THEIR CONVERGENCE AND COMPUTATIONAL EFFICIENCY. SO, FOR EXAMPLE, WE LOOK FOR FREE DERIVATIVE AND/OR REVERSE-FREE METHODS FOR THEIR APPLICABILITY IN OPTIMISATION PROBLEMS AND IN PROBLEMS THAT PRESENT SINGULARITIES IN THE SOUGHT SOLUTION. THE USE OF DIVIDED DIFFERENCES TO APPROXIMATE THOSE DERIVED IN THIS TYPE OF PROBLEM IS A TECHNIQUE THAT WE USE FREQUENTLY. IN THIS RESPECT WE DISTINGUISH TWO FAMILIES OF PROCESSES: WITH AND WITHOUT MEMORY. IN THIS FIELD, WE WORK INDISTINCTLY WITH SCALAR EQUATIONS AND SYSTEMS OF EQUATIONS. ANOTHER LINE OF WORK IS DEDICATED TO DIRECTIONAL METHODS, SPECIFICALLY DESIGNED FOR NON-LINEAR EQUATIONS, AND WITH APPLICATIONS IN OPTIMISATION AND GEOMETRICAL PROBLEMS (SURFACE INTERSECTION). WE ALSO DEDICATE PART OF THIS PROJECT TO THE STUDY OF HYBRID METHODS, CONSISTING OF COMBINING TWO OR MORE ITERATIVE PROCESSES TO TRY TO MAXIMISE THE ADVANTAGES OF EACH ONE (SPEED OF CONVERGENCE OF THE ONE, REGION ACCESSIBILITY OF THE OTHER, ETC.). ONE OF THE NOVEL TECHNIQUES THAT WE INTRODUCE IN THIS PROJECT TO BUILD ITERATIVE PROCESSES IN THE PREVIOUS LINES IS THE USE OF WEIGHT FUNCTIONS. THIS TECHNIQUE MAKES IT POSSIBLE TO INCREASE THE ORDER OF CONVERGENCE OF A GIVEN ITERATIVE PROCESS WITHOUT EXCESSIVELY INCREASING ITS OPERATIONAL COST. WITHIN THE GENERAL STUDY OF THE ITERATIVE PROCESSES CONSTRUCTED OF THE DIFFERENT FORMS MENTIONED ABOVE, WE WILL MAKE SPECIAL EMPHASIS IN THE STUDY OF WHAT HAPPENS WHEN MULTIPLE ROOTS APPROACH. AS A GENERAL RULE, WE KNOW THAT THE ORDER OF CONVERGENCE IS DRASTICALLY REDUCED IN THESE CASES. But in addition, in the case of a multiple, there are other problems associated with the synergistic nature of the Jacobean matrix or the bad condition of the solution._x000D_ Another important working line is the analysis of the DINAMIC of the iterative methods. WE WILL STUDY FROM AN ANALYTIC, NUMERIC AND GRAPHICAL POINT OF VIEW ISSUES SUCH AS THE BASINS OF ATTRACTION OF THE ROOTS, THE PLANES OF PARAMETERS, THE SETS OF JULIA, THE PRESENCE OF ATTRACTOR CYCLES AND STRANGE ATTRACTORS (FIXED POINTS THAT ARE NOT THE SOLUTION OF THE EQUATION SOUGHT), THE GENERAL CONVERGENCE AND THE PRESENCE OF CHAOTIC BEHAVIORS. WE UNFOLD ALL THE STUDIES CARRIED OUT IN THE STUDY OF REAL AND COMPLEX DYNAMICS, AS BOTH ARE NOT EQUIVALENT AND USE DIFFERENT TOOLS AND RESULTS. In another research line we analyse the connection between the transport of the iterative methods for the resolution of non-linear ECUs with the NUMERICAL methods for the resolution of different situations (initial VALOR PROBLEMS)._x000D_ Without closing the door to other applications, we will pay special attention to two concret applications that you need from the non-linear ECU resolution for your resolution: NONLINEAR OPTIMISATION PROBLEMS AND NONLINEAR MATRIX EQUATIONS. (English)
12 October 2021
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Logroño
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Identifiers
MTM2014-52016-C2-1-P
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