Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Stochastic approximation methods in stochastic optimization

Stochastic approximation methods for stochastic optimization are considered. Reviewed the main methods of stochastic approximation: stochastic quasi-gradient algorithm, Kiefer-Wolfowitz algorithm and adaptive rules for them, simultaneous perturbation stochastic approximation (SPSA) algorithm. Suggested the model and the solution of the retailer's profit optimization problem and considered an application of the SQG-algorithm for the optimization problems with objective functions given in the form of ordinary differential equation.

Transfer of MPEG-2 Transport Stream over MPLS

This study examines how MPEG-2 Transport Stream, used in DVB-T video transmission, can be reliably and efficiently transferred to remote locations over an MPLS network. All the relevant technologies used in this scenario are also discussed in the study. This study was done for Digita Oy, which is a major radio and television content distributor in Finland. The theoretical part of the study begins with the introduction to MPLS technology and continues with explanation of IP Multicast and its components. The fourth section discusses MPEG-2 and the formation and content of MPEG-2 Transport Stream. These technologies were studied in relevant literature and RFC documentation. After the theoretical part of the study, the test setup and the test cases are presented. The results of the test cases, and the conclusions that can be drawn based on them, are discussed in the last section of the study. The tests showed that it is possible to transfer digital video quite reliably over an MPLS network using IP Multicast. By configuring the equipment correctly, the recovery time of the network in case of a failure can be shortened remarkably. Also, the unwanted effect of other traffic on the critical video traffic can be eliminated by defining the Quality of Service parameters correctly. There are, however, some issues that need to be tested further before this setup can be used in broadcast networks. Reliable operation of IP Multicast and proper error correction are the main subjects for future testing.

Agricultural phosphorus and water quality : sources, transport and management

Selostus: Maatalous, fosfori ja veden laatu: alkuperä, kulkeutuminen ja vesistökuormituksen hallinta

Harvesting and transport of root biomass from fast-growing poplar plantations

Abstract

New directions in Stochastic Multicriteria Acceptability Analysis

Decisions taken in modern organizations are often multi-dimensional, involving multiple decision makers and several criteria measured on different scales. Multiple Criteria Decision Making (MCDM) methods are designed to analyze and to give recommendations in this kind of situations. Among the numerous MCDM methods, two large families of methods are the multi-attribute utility theory based methods and the outranking methods. Traditionally both method families require exact values for technical parameters and criteria measurements, as well as for preferences expressed as weights. Often it is hard, if not impossible, to obtain exact values. Stochastic Multicriteria Acceptability Analysis (SMAA) is a family of methods designed to help in this type of situations where exact values are not available. Different variants of SMAA allow handling all types of MCDM problems. They support defining the model through uncertain, imprecise, or completely missing values. The methods are based on simulation that is applied to obtain descriptive indices characterizing the problem. In this thesis we present new advances in the SMAA methodology. We present and analyze algorithms for the SMAA-2 method and its extension to handle ordinal preferences. We then present an application of SMAA-2 to an area where MCDM models have not been applied before: planning elevator groups for high-rise buildings. Following this, we introduce two new methods to the family: SMAA-TRI that extends ELECTRE TRI for sorting problems with uncertain parameter values, and SMAA-III that extends ELECTRE III in a similar way. An efficient software implementing these two methods has been developed in conjunction with this work, and is briefly presented in this thesis. The thesis is closed with a comprehensive survey of SMAA methodology including a definition of a unified framework.

Quantum transport phenomena and shallow impurity states in CdSb

This work is dedicated to investigation of the energy spectrum of one of the most anisotropic narrow-gap semiconductors, CdSb. At the beginning of the present studies even the model of its energy band structure was not clear. Measurements of galvanomagnetic effects in wide temperature range (1.6 - 300 K) and in magnetic fields up to 30 T were chosen for clarifying of the energy spectrum in the intentionally undoped CdSb single crystals and doped with shallow impurities (In, Ag). Detection of the Shubnikov - de Haas oscillations allowed estimating the fundamental energy spectrum parameters. The shapes of the Fermi surfaces of electrons (sphere) and holes (ellipsoid), the number of the equivalent extremums for valence band (2) and their positions in the Brillouin zone were determined for the first time in this work. Also anisotropy coefficients, components of the tensor of effective masses of carriers, effective masses of density of states, nonparabolicity of the conduction and valence bands, g-factor and its anisotropy for n- and p-CdSb were estimated for the first time during these studies. All the results obtained are compared with the cyclotron resonance data and the corresponding theoretical calculations for p-CdSb. This is basic information for the analyses of the complex transport properties of CdSb and for working out the energy spectrum model of the shallow energy levels of defects and impurities in this semiconductor. It was found out existence of different mechanisms of hopping conductivity in the presence of metal - insulator transition induced by magnetic field in n- and p-CdSb. Quite unusual feature opened in CdSb is that different types of hopping conductivity may take place in the same crystal depending on temperature, magnetic field or even orientation of crystal in magnetic field. Transport properties of undoped p-CdSb samples show that the anisotropy of the resistivity in weak and strong magnetic fields is determined completely by the anisotropy of the effective mass of the holes. Temperature and magnetic field dependence of the Hall coefficient and magnetoresistance is attributed to presence of two groups of holes with different concentrations and mobilities. The analysis demonstrates that below Tcr ~ 20 K and down to ~ 6 - 7 K the low-mobile carriers are itinerant holes with energy E2 ≈ 6 meV. The high-mobile carriers, at all temperatures T < Tcr, are holes activated thermally from a deeper acceptor band to itinerant states of a shallower acceptor band with energy E1 ≈ 3 meV. Analysis of temperature dependences of mobilities confirms the existence of the heavy-hole band or a non-equivalent maximum and two equivalent maxima of the light-hole valence band. Galvanomagnetic effects in n-CdSb reveal the existence of two groups of carriers. These are the electrons of a single minimum in isotropic conduction band and the itinerant electrons of the narrow impurity band, having at low temperatures the energies above the bottom of the conduction band. It is found that above this impurity band exists second impurity band of only localized states and the energy of both impurity bands depend on temperature so that they sink into the band gap when temperature is increased. The bands are splitted by the spin, and in strong magnetic fields the energy difference between them decreases and redistribution of the electrons between the two impurity bands takes place. Mobility of the conduction band carriers demonstrates that scattering in n-CdSb at low temperatures is strongly anisotropic. This is because of domination from scattering on the neutral impurity centers and increasing of the contribution to mobility from scattering by acoustic phonons when temperature increases. Metallic conductivity in zero or weak magnetic field is changed to activated conductivity with increasing of magnetic field. This exhibits a metal-insulator transition (MIT) induced by the magnetic field due to shift of the Fermi level from the interval of extended states to that of the localized states of the electron spectrum near the edge of the conduction band. The Mott variablerange hopping conductivity is observed in the low- and high-field intervals on the insulating side of the MIT. The results yield information about the density of states, the localization radius of the resonant impurity band with completely localized states and about the donor band. In high magnetic fields this band is separated from the conduction band and lies below the resonant impurity bands.

Syttyminen ja palamisen eteneminen partikkelikerroksessa

Syttymistä ja palamisen etenemistä partikkelikerroksessa tutkitaan paloturvallisuuden parantamista sekä kiinteitä polttoaineita käyttävien polttolaitteiden toiminnan tuntemista ja kehittämistä varten. Tässä tutkimuksessa on tavoitteena kerätä yhteen syttymiseen ja liekkirintaman etenemiseen liittyviä kokeellisia ja teoreettisia tutkimustuloksia, jotka auttavat kiinteäkerrospoltto- ja -kaasutus-laitteiden kehittämisessä ja suunnittelussa. Työ on esitutkimus sitä seuraavalle kokeelliselle ja teoreettiselle osalle. Käsittelyssä keskitytään erityisesti puuperäisiin polttoaineisiin. Hiilidioksidipäästöjen vähentämistavoitteet sekä kiinteiden jätteiden energiakäytön lisääminen ja kaatopaikalle viennin vähentäminen aiheuttavat lähitulevaisuudessa kerrospolton lisääntymistä. Kuljetusmatkojen optimoinnin takia joudutaan rakentamaan melko pieniä polttolaitoksia, joissa kerrospolttotekniikka on edullisin vaihtoehto. Syttymispisteellä tarkoitetaan Semenovin määritelmän mukaan tilaa ja ajankohtaa, jolloin polttoaineen ja hapen reaktioissa muodostuva nettoenergia aikayksikössä on yhtäsuuri kuin ympäristöön siirtyvä nettoenergiavirta. Itsesyttyminen tarkoittaa syttymistä ympäristön lämpötilan tai paineen suurenemisen seurauksena. Pakotettu syttyminen tapahtuu, kun syttymispisteen läheisyydessä on esimerkiksi liekki tai hehkuva kiinteä kappale, joka aiheuttaa paikallisen syttymisen ja syttymisrintaman leviämisen muualle polttoaineeseen. Kokeellinen tutkimus on osoittanut tärkeimmiksi syttymiseen ja syttymisrintaman etenemiseen vaikuttaviksi tekijöiksi polttoaineen kosteuden, haihtuvien aineiden pitoisuuden ja lämpöarvon, partikkelikerroksen huokoisuuden, partikkelien koon ja muodon, polttoaineen pinnalle tulevan säteilylämpövirran tiheyden, kaasun virtausnopeuden kerroksessa, hapen osuuden ympäristössä sekä palamisilman esilämmityksen. Kosteuden lisääntyminen suurentaa syttymisenergiaa ja -lämpötilaa sekä pidentää syttymisaikaa. Mitä enemmän polttoaine sisältää haihtuvia aineita sitä pienemmässä lämpötilassa se syttyy. Syttyminen ja syttymisrintaman eteneminen ovat sitä nopeampia mitä suurempi on polttoaineen lämpöarvo. Kerroksen huokoisuuden kasvun on havaittu suurentavan palamisen etenemisnopeutta. Pienet partikkelit syttyvät yleensä nopeammin ja pienemmässä lämpötilassa kuin suuret. Syttymisrintaman eteneminen nopeutuu partikkelien pinta-ala - tilavuussuhteen kasvaessa. Säteilylämpövirran tiheys on useissa polttosovellutuksissa merkittävin lämmönsiirtotekijä, jonka kasvu luonnollisesti nopeuttaa syttymistä. Ilman ja palamiskaasujen virtausnopeus kerroksessa vaikuttaa konvektiiviseen lämmönsiirtoon ja hapen pitoisuuteen syttymisvyöhykkeellä. Ilmavirtaus voi jäähdyttää ja kuumankaasun virtaus lämmittää kerrosta. Hapen osuuden kasvaminen nopeuttaa syttymistä ja liekkirintaman etenemistä kunnes saavutetaan tila, jota suuremmilla virtauksilla ilma jäähdyttää ja laimentaa reaktiovyöhykettä. Palamisilman esilämmitys nopeuttaa syttymisrintaman etenemistä. Syttymistä ja liekkirintaman etenemistä kuvataan yleensä empiirisillä tai säilyvyysyhtälöihin perustuvilla malleilla. Empiiriset mallit perustuvat mittaustuloksista tehtyihin korrelaatioihin sekä joihinkin tunnettuihin fysikaalisiin lainalaisuuksiin. Säilyvyysyhtälöihin perustuvissa malleissa systeemille määritetään massan, energian, liikemäärän ja alkuaineiden säilymisyhtälöt, joiden nopeutta kuvaavien siirtoyhtälöiden muodostamiseen käytetään teoreettisella ja kokeellisella tutkimuksella saatuja yhtälöitä. Nämä mallinnusluokat ovat osittain päällekkäisiä. Pintojen syttymistä kuvataan usein säilyvyysyhtälöihin perustuvilla malleilla. Partikkelikerrosten mallinnuksessa tukeudutaan enimmäkseen empiirisiin yhtälöihin. Partikkelikerroksia kuvaavista malleista Xien ja Liangin hiilipartikkelikerroksen syttymiseen liittyvä tutkimus ja Gortin puun ja jätteen polttoon liittyvä reaktiorintaman etenemistutkimus ovat lähimpänä säilyvyysyhtälöihin perustuvaa mallintamista. Kaikissa malleissa joudutaan kuitenkin yksinkertaistamaan todellista tapausta esimerkiksi vähentämällä dimensioita, reaktioita ja yhdisteitä sekä eliminoimalla vähemmän merkittävät siirtomekanismit. Suoraan kerrospolttoa ja -kaasutusta palvelevia syttymisen ja palamisen etenemisen tutkimuksia on vähän. Muita tarkoituksia varten tehtyjen tutkimusten polttoaineet, kerrokset ja ympäristöolosuhteet poikkeavat yleensä selvästi polttolaitteiden vastaavista olosuhteista. Erikokoisten polttoainepartikkelien ja ominaisuuksiltaan erilaisten polttoaineiden seospolttoa ei ole tutkittu juuri ollenkaan. Polttoainepartikkelien muodon vaikutuksesta on vain vähän tutkimusta.Ilman kanavoitumisen vaikutuksista ei löytynyt tutkimuksia.