Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Agent-based systems usually operate in real-time, stochastic and dynamic environments. Many theoretical and applied techniques have been applied to the investigation of agent architecture with respect to communication, cooperation, and learning, in order to provide a framework for implementing artificial intelligence and computing techniques. Intelligent agents are required to be able to adapt and learn in uncertain environments via communication and collaboration (in both competitive and cooperative situations). The ability of reasoning and learning is one fundamental feature for intelligent agents. Due to the inherent complexity, however, it is difficult to verify the properties of the complex and dynamic environments a priori. Since analytic techniques are inadequate for solving these problems, reinforcement learning (RL) has appeared as a popular approach by mapping states to actions, so as to maximise the long-term rewards. Computer simulation is needed to replicate an experiment for testing and verifying the efficiency of simulation-based optimisation techniques. In doing so, a simulation testbed called robot soccer is used to test the learning algorithms in the specified scenarios. This research involves the investigation of simulation-based optimisation techniques in agent-based systems. Firstly, a hybrid agent teaming framework is presented for investigating agent team architecture, learning abilities, and other specific behaviors. Secondly, the novel reinforcement learning algorithms to verify goal-oriented agents; competitive and cooperative learning abilities for decision-making are developed. In addition, the function approximation technique known as tile coding (TC), is used to avoid the state space growing exponentially with the curse of dimensionality. Thirdly, the underlying mechanism of eligibility traces is analysed in terms of on-policy algorithm and off-policy algorithm, accumulating traces and replacing traces. Fourthly, the "design of experiment" techniques, such as Simulated Annealing method and Response Surface methodology, are integrated with reinforcement learning techniques to enhance the performance. Fifthly, a methodology is proposed to find the optimal parameter values to improve convergence and efficiency of the learning algorithms. Finally, this thesis provides a serious full-fledged numerical analysis on the efficiency of various RL techniques.

The present work was developed in the Postgraduate Program in Mathematics Education and Physics Teaching (PPGEM & EF) of the Federal University of Santa Maria (UFSM) and inserted in the line of research Teaching and Learning of Mathematics and its Philosophical, Historical and Epistemological Foundations. Based on interviews with the pedagogical advisor of the 8ª Regional Education Coordination (8ª CRE), responsible for the Special Education sector, with teachers of Mathematics from state schools of Basic Education and the analysis of official documents of the Brazilian legislation on Special Education and Inclusive Education, curricula and pedagogical projects of undergraduate courses in Mathematics, this research aimed to investigate how these Mathematics teachers organize their classes for teaching in classes that have students included. For the data collection, semi-structured interviews were carried out with five Mathematics teachers from the state public school system who work or have already worked with students included in the regular education and with the pedagogical advisor responsible for the Special Education sector of the 8ª CRE. We used as a theoretical methodological reference for conducting these interviews Oral History, based on the ideas of José Carlos Sebe Bom Meihy (1996) and, mainly, the works developed by the Oral History and Mathematics Education Research Group, coordinated by Antonio Vicente Marafioti Garnica (2005, 2007, 2011, 2013). The analysis of the interviews was performed - according to the work carried out by Maria Edneia Martins-Salandim - in two moments: analysis of singularities and analysis of convergences. In the analysis of singularities we try to record what is characteristic of each deponent in his narrative, its peculiarities and its particularities. In the analysis of convergences we seek to confront the narratives of our deponents with what is written in the official documents of the legislation on Special Education and Inclusive Education and with the grades and pedagogical projects of the degree courses in Mathematics of the institutions where the teachers interviewed concluded the course Training. For this, we list three units of analysis: initial training, continuing training and work with included students. We emphasize, from the analysis carried out, that the Mathematics teachers interviewed consider that the work carried out with students included in regular education is hard work and difficult to be developed, since they did not receive orientation in the initial training course to work with these students. In addition, they have taken part in a few continuing education courses related to this topic, and the 8ª CRE does not provide guidelines for this work that be aimed at teachers in areas other than Special Education. / O presente trabalho foi desenvolvido junto ao Programa de Pós-Graduação em Educação Matemática e Ensino de Física (PPGEM&EF) da Universidade Federal de Santa Maria (UFSM) e inserido á linha de pesquisa Ensino e Aprendizagem da Matemática e seus Fundamentos Filosóficos, Históricos e Epistemológicos. Com base em entrevistas com o assessor pedagógico da 8ª Coordenadoria Regional de Educação (8ª CRE), responsável pelo setor de Educação Especial, com professores de Matemática de escolas estaduais da Educação Básica e na análise de documentos oficiais da legislação brasileira sobre Educação Especial e Educação Inclusiva, de currículos e projetos pedagógicos de cursos de Licenciatura em Matemática, essa pesquisa teve como objetivo investigar como esses professores de Matemática organizam suas aulas para a docência em turmas que possuem alunos incluídos. Para a coleta dos dados foram realizadas entrevistas semiestruturadas com cinco professoras de Matemática da rede pública estadual de ensino que trabalham ou já trabalharam com alunos incluídos no ensino regular e com a assessora pedagógica responsável pelo setor de Educação Especial da 8ª CRE. Utilizamos como referencial teórico metodológico para a realização dessas entrevistas a História Oral, nos fundamentando nas ideias de José Carlos Sebe Bom Meihy (1996) e, principalmente, nos trabalhos desenvolvidos pelo grupo de pesquisa Grupo História Oral e Educação Matemática, coordenado por Antonio Vicente Marafioti Garnica (2005; 2007; 2011; 2013). A análise das entrevistas foi realizada – conforme o trabalho realizado por Maria Edneia Martins-Salandim – em dois momentos: análise de singularidades e análise de convergências. Na análise de singularidades procuramos registrar o que é característico de cada depoente em sua narrativa, suas peculiaridades e suas particularidades. Na análise de convergências buscamos confrontar as narrativas dos nossos depoentes com o que está escrito nos documentos oficiais da legislação sobre Educação Especial e Educação Inclusiva e com as grades e os projetos pedagógicos dos cursos de Licenciatura em Matemática das instituições onde as professoras entrevistadas concluíram o curso de formação inicial. Para isso, elencamos três unidades de análise: formação inicial, formação continuada e trabalho com alunos incluídos. Destacamos, a partir da análise realizada, que as professoras de Matemática entrevistadas consideram que o trabalho realizado com alunos incluídos no ensino regular é um trabalho árduo e difícil de ser desenvolvido, pois elas não receberam, no curso de formação inicial, orientações para trabalhar com esses alunos. Além disso, participaram de poucos cursos de formação continuada relacionados a esse tema e a 8ª CRE também não fornece orientações para esse trabalho que sejam voltadas aos professores de áreas que não seja a Educação Especial.

nÃo hà / Este trabalho centra-se na avaliaÃÃo das modificaÃÃes ocorridas nos critÃrios de partilha da cota parte do ICMS distribuÃdo aos 184 municÃpios cearenses no perÃodo de 2009 a 2011. A lei estadual n 14.023/2007 alterou dispositivos da lei n 12.612/1996 que inclui e amplia indicadores sociais das Ãreas de educaÃÃo, saÃde e meio ambiente, no sentido de tornar os repasses do ICMS aos entes municipais mais distributivos e equitativos. Tem-se como objetivo verificar a existÃncia de convergÃncia da cota parte per capita recebida pelos municÃpios do CearÃ, o perfil socioeconÃmico dos municÃpios que foram beneficiados, bem como os efeitos da polÃtica implementada. Para isso, foram empregados alguns modelos economÃtricos com as finalidades metodolÃgicas de estimar beta convergÃncia; identificar as variÃveis que afetam a probabilidade de um municÃpio ter sido escolhido como beneficiado pela nova Lei; avaliar os efeitos da aÃÃo polÃtica adotada. Os resultados empÃricos indicam que no perÃodo estudado houve convergÃncia da cota parte transferida aos municÃpios. Identificou-se tambÃm que, entre as caracterÃsticas socioeconÃmicas, os municÃpios com baixo contingente populacional, mercado formal de trabalho menos desenvolvido, indicadores educacionais elevados e reduzida proporÃÃo de pobres tÃm maior probabilidade de serem beneficiados pela mudanÃa da Lei. Apesar da convergÃncia observada e das caracterÃsticas que indicam benefÃcios com o repasse da cota parte, o efeito da implantaÃÃo da polÃtica estabelecida pela Lei 14.023/2007 nÃo foi significativo quando se comparam os municÃpios cearenses com os municÃpios pernambucanos como grupo de controle. / This work focuses on the assessment of the changes that occurred in the quota share criteria of ICMS distributed to the 184 municipalities of Cearà in the period 2009-2011. State law 14.023/2007 amended provisions of Law 12.612/1996 by including and extending social indicators in the areas of education, health and environment, in order to make the ICMS transfers more distributive and promote equity among municipalities. Thus, it has been aimed at verifying the existence of convergence of per capita quota received by the municipalities of CearÃ, the socioeconomic profile of the municipalities that have benefited, as well as the effects of the implemented policy. For doing this, teconometric models were performed in order to estimate beta convergence, to identify the variables that affect the probability of a municipality have been chosen as beneficiary of the quota share criteria changes, and to evaluate the effect of such changes for policy purposes. The empirical results indicate that in the period studied there was convergence of per capita quota transferred to municipalities, which also was identified between socioeconomic characteristics, municipalities that have reduced small population, formal labor market less developed, higher education indicators and proportion of poor are more likely to benefit. Despite the observed convergence and the characteristics that indicate benefits from the transfer of quota share, the effect of implementing the policy established by Law 14.023/2007 was not significant when comparing municipalities of Cearà with the municipalities of Pernambuco.

Neste trabalho, uma abordagem micromecânica com aproximação da geometria dada por funções de Bézier triangulares com continuidade geométrica G1 é inserida ao Método dos Elementos de Contorno, o qual é aplicado em problemas da elastostática tridimensional. Para consideração do efeito microestrutural, foi utilizado a teoria gradiente elástica simplificada de Aifantis, a qual é uma particularização da teoria geral de Mindlin. Nesta teoria, um argumento variacional é estabelecido para determinar todas as possíveis condições de contorno, clássica e não-clássica, para o problema de valor de contorno geral. A partir deste argumento, a solução fundamental da elasticidade gradiente é explicitada e com o auxílio da identidade integral recíproca é construído a representação integral de contorno. Para tornar o problema de valor de contorno bem-posto, em adição à representação integral de contorno para deslocamento, uma segunda representação integral para derivada normal do deslocamento foi utilizada. Expressões integrais para deslocamento e tensão em pontos internos são apresentadas. Todos os núcleos das equações integrais são explicitamente desenvolvidos. Para a discretização do MEC foram utilizados elementos triangulares curvos, aproximados tanto para a geometria quanto para os parâmetros físicos por funções de Proriol (com características espectrais) e por funções aqui chamadas de Polinomiais, onde esta última é construída a partir de uma base nodal equidistante e pela imposição da partição da unidade. Entretanto estas funções aproximadoras garantem apenas continuidade C0 entre os elementos triangulares, ou seja, a garantia da continuidade do plano tangente não necessariamente é satisfeita. Com o objetivo de anular o termo de integral de linha presente na formulação microestrutural, a hipótese de superfície suave se faz necessária e assim funções de Bézier com continuidade geométrica G1, a qual depende apenas da posição e das normais dos nós nos vértices da malha triangular é utilizada. Para auxiliar na obtenção das coordenadas e das normais nodais para geometrias complexas foi utilizado o software de computação gráfica BlenderTM 2.7, o qual foi acoplado ao programa do MEC elastostático gradiente. Na sequência foi verificada, por meio de exemplos, a suavidade na intersecção entre os elementos triangulares G1 e estes foram comparados com as aproximações de Proriol e Polinomial. Em seguida, as singularidades presentes nas soluções fundamentais foram tratadas através da expansão em série de Laurent aplicada à técnica de subtração de singularidade. Condições necessárias e suficientes para a convergência das expansões em série das soluções fundamentais, estimador do erro para estas expansões, assim como, a correlação matemática entre o tamanho da malha e o parâmetro micromecânico g foram estabelecidos. Expressões explicitas da série de Laurent dos núcleos das integrais singulares e hipersingulares do MEC clássico e não clássico foram apresentadas. A verificação do tratamento da singularidade aplicado a elementos triangulares curvos foi realizada, tanto na direção radial quanto na direção angular. E pôde ser observado que ocorre uma perda de eficiência no tratamento da singularidade na direção angular, devida a presença do efeito de camada limite para elementos curvos distorcidos. Entretanto, este efeito de quase singularidade pode ser amenizado por meio da abordagem micromecânica, uma vez que foi observado menor presença do efeito da camada limite à medida que o parâmetro g é diminuído. Por último, foi desenvolvido um programa na linguagem FORTRAN 11.0, o qual contempla as abordagens clássica e micromecânica com continuidade geométrica G1. Sua validação foi feita por meio de exemplos considerados Benchmarks. / In this work, a micromechanical approach with approximation of geometry solved by Bézier triangular functions that guaranty continuity G1 is inserted to the Boundary element Method (BEM). This formulation is applied in three-dimensional elastostatic problems. The simplified elastic gradient theory proposed by Aifantis, which is a particularization of the general theory of Mindlin is used to consider the microstructural effect. In this theory a variational argument is established to determine all possible boundary conditions, classical and non-classical, for the general boundary value problem. From this argument, the fundamental solution of the gradient elasticity is explicited and by the reciprocal integral identity the boundary integral representation is achieved. In addition to the boundary integral representation for dispacement, a second integral representation regarding its normal derivative is used to make the well-posed boundary value problem. Integral expressions for displacement and stress on internal points are also presented. All kernels in the integral equations are explicitly developed. Curved triangular elements are used for the discretization of the BEM. The approximation of both the geometry and physical parameters is performed by Proriol functions (with spectral characteristics) and by Polynomial functions. The last is built from an equidistant nodal basis enforcing the partition of unity. However these approximating functions ensure only C0 continuity between the triangular elements, that is, the tangent plane continuity assurance is not necessarily satisfied. In order to cancel line integral terms in the microstructural approach, the hypothesis of smooth surface is required and thus Bézier function with geometric continuity G1, which depends only on the position and the normal of the nodes at the vertices of the triangular mesh is used. In this study the computer graphics software called BlenderTM 2.7 is used to assist in obtaining coordinates and normal vectors at nodes when complex geometries are analyzed. BlenderTM 2.7 is coupled to the gradient elastic BEM program. The smoothness of the resulting mesh using G1 elements is compared to Proriol and Polynomial approximations by means of simple examples. The singularities present in the fundamental solutions are treated by employing the expansion in Laurent series and the singularity subtraction technique. Necessary and sufficient conditions for the convergence of expansions in series of fundamental solutions, error estimator for these expansions, as well as the mathematical correlation between the size of the mesh and the micromechanical parameter, g, are established. Explicit expressions of Laurent series of the classical and micromechanical kernels forthe singular and hipersingular BEM integrals are presented. Treatment of singularity, both in the radial direction and in the angular direction, applied to curved triangular elements is verified. It can be observed that there is a loss of efficiency in the treatment of singularity in the angular direction, due to the presence of the boundary layer effect for distorted curved boundary elements. However, this nearly singularity effect could be alleviated by micromechanics approach, since minor boundary layer effect was observed as the parameter g is decreased. Finally, using FORTRAN 11.0 language, a computational code is developed, which includes the classic and micromechanics approach with geometric continuity G1, and its results are validated by means of Benchmark examples.

Philosophiae Doctor - PhD / In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis. / South Africa

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature. / South Africa

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Yes / This paper investigates the interaction phenomena of the coupled axons while the mutual coupling factor is presented as a pairwise description. Based on the Hodgkin-Huxley model and the coupling factor matrix, the membrane potentials of the coupled myelinated/unmyelinated axons are quantified which implies that the neural coupling can be characterised by the presented coupling factor. Meanwhile the equivalent electric circuit is supplied to illustrate the physical meaning of this extended model. In order to estimate the coupling factor, a data-based iterative learning identification algorithm is presented where the Rényi entropy of the estimation error has been minimised. The convergence of the presented algorithm is analysed and the learning rate is designed. To verified the presented model and the algorithm, the numerical simulation results indicate the correctness and the effectiveness. Furthermore, the statistical description of the neural coupling, the approximation using ordinary differential equation, the measurement and the conduction of the nerve signals are discussed respectively as advanced topics. The novelties can be summarised as follows: 1) the Hodgkin-Huxley model has been extended considering the mutual interaction between the neural axon membranes, 2) the iterative learning approach has been developed for factor identification using entropy criterion, and 3) the theoretical framework has been established for this class of system identification problems with convergence analysis. / This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 51807010, and in part by the Natural Science Foundation of Hunan under Grant 1541 and Grant 1734. / Research Development Fund Publication Prize Award winner, Nov 2020.

Abstract This thesis focuses on two important challenges in wireless downlink transmission: multi-user (MU) precoder design and scheduling of users over time, frequency, and spatial resources at any given instant. Data streams intended for different users are transmitted by a multiple-input multiple-output (MIMO) multi-antenna orthogonal frequency division multiplexing (OFDM) system. The transmit precoders are designed jointly across space-frequency resources to minimize the number of backlogged packets waiting at the coordinating base stations (BSs), thereby implicitly performing user scheduling. Then the problem of multicast beamformer design is considered wherein a subset of users belonging to a multicasting group are served by a common group-specific data. The design objective is to either minimize the transmit power for a guaranteed quality-of-service, or to maximize the minimum achievable rate among users for a given transmit power. Unlike existing techniques, the proposed design utilizes both the spatial and frequency resources jointly while designing multi-group beamformers. As an extension to coordinated precoding, the problem of beamformer design for cloud radio access network is considered wherein beamformers are designed centrally, quantized and sent along with data to the respective BSs via backhaul. Since the users can be served by multiple BSs, beamformer design becomes a nonconvex combinatorial problem. Unlike existing solutions, beamformer overhead is also included in the backhaul utilization along with the associated data. As the number of antennas increases, backhaul utilization is dominated by the beamformers. Thus, to reduce the overhead, two techniques are proposed: varying the quantization precision, and reducing the number of active antennas used for transmission. Finally, to reduce the complexity involved in the design of joint space- frequency approach, a two-step procedure is proposed, where a MU-MIMO scheduling algorithm is employed to find a subset of users for each scheduling block. The precoders are then designed only for the chosen users, thus reducing the complexity without compromising much on the throughput. In contrast to the null-space-based existing techniques, a low-complexity scheduling algorithm is proposed based on vector projections. The real-time performance of all the schedulers are evaluated by implementing them on both Xilinx ZYNQ-ZC702 system-on-chip (SoC) and TI TCI6636K2H multi-core SoC. / Tiivistelmä Tässä väitöskirjassa keskitytään kahteen tärkeään langattoman tiedonsiirron haasteeseen alalinkkilähetyksissä: usean käyttäjän (MU) esikooderisuunnitteluun ja käyttäjien skedulointiin aika-, taajuus- ja tilaresurssien yli. Eri käyttäjille tarkoitettuja datavirtoja lähetetään käyttämällä monitulo-monilähtötekniikkaa (MIMO) yhdistettynä monikantoaaltomodulointiin (OFDM). Lähettimien esikooderit suunnitellaan yhteisesti tila- ja taajuusresurssien yli, jotta keskenään yhteistoiminnallisten tukiasemien jonossa olevien pakettien määrää voitaisiin minimoida samalla kun tehdään epäsuorasti käyttäjien skedulointia. Tämän jälkeen työssä paneudutaan monilähetysten (multicast) keilanmuodostussuunnitteluun, jossa monilähetysryhmään kuuluvien käyttäjien alijoukolle lähetetään yhteistä ryhmäspesifistä dataa. Suunnittelun päämääränä on joko minimoida kokonaislähetysteho tietyllä palvelunlaatuvaatimuksella tai maksimoida pienin saavutettavissa oleva siirtonopeus käyttäjien joukossa tietyllä lähetysteholla. Toisin kuin olemassa olevat menetelmät, ehdotetussa mallissa käytetään yhteisesti sekä aika- että taajuusresursseja usean ryhmän keilanmuodostusta suunniteltaessa. Laajennuksena yhteistoiminnalliselle esikoodaukselle, väitöskirjassa käsitellään myös keilanmuodostusta pilvipohjaisessa radioliityntäverkkoarkkitehtuurissa. Keilanmuodostajat suunnitellaan keskitetysti, kvantisoidaan ja lähetetään datan mukana tukiasemille käyttäen runkoverkkoyhteyttä. Koska käyttäjiä voidaan palvella usealta tukiasemalta, keilanmuodostussuunnittelu muuttuu ei-konveksiksi kombinatoriseksi ongelmaksi. Toisin kuin olemassa olevissa ratkaisuissa, ehdotettu malli sisällyttää käyttäjien datan lisäksi keilanmuodostajien resursoinnin tarpeen runkoverkkoon. Tukiaseman antennien määrän lisääntyessä, keilanmuodostajien osuus runkoverkon käyttöasteesta kasvaa suureksi. Jotta keilanmuodostajien aiheuttamaa ylimääräistä tiedonsiirtotarvetta voitaisiin minimoida, esitellään kaksi tekniikkaa: kvantisointitarkkuuden muunteleminen sekä lähetykseen käytettävien aktiivisten antennien määrän vähentäminen. Lopuksi, jotta yhdistetyn tila-taajuussuunnittelun aiheuttamaa kompleksisuutta saataisiin vähennettyä, ehdotetaan kaksivaiheista menetelmää. MU-MIMO skedulointialgoritmin avulla etsitään ensin alijoukko käyttäjiä jokaiselle skedulointilohkolle. Esikooderit suunnitellaan vain valituille käyttäjille, mikä vähentää kompleksisuutta, heikentämättä suorituskykyä kuitenkaan olennaisesti. Poiketen nolla-avaruuteen perustuvista tekniikoista, esitetään yksinkertainen vektoriprojektioihin perustuva skeduleri. Kaikkien skedulerien reaaliaikasuorituskykyä on arvioitu toteuttamalla ne ohjelmoitavilla Xilinx ZYNQ-ZC702 system-on-chip (SoC) ja TI TCI6636K2H moniydinalustoilla.