Valaistusyhtälö

Tässä tietokonegrafiikan alan tutkielmassa tutkitaan annetun kohteen tai maailman valaistuksen määrittämistä laskennallisesti. Ilmiöt kuvataan fysikaalisesti ja valaistusta mallinnetaan ilmiöitä kuvaavilla yhtälöillä. Yhtälöiden ratkaisu perustuu valonsäteiden etenemisen ja sironnan laskennalliseen seurantaan. Näin johdettua matemaattista mallia tutkitaan ja yhtälöiden ratkaisut muotoillaan tietokoneella laskettavaan muotoon. Lopuksi ohjelmoidaan esiteltyä teoriaa hyödyntävä numeerinen ratkaisija ja esitellään käytetyt menetelmät ja tulokset. Valaistuksen matemaattista mallintamista lähestytään fysikaaliselta pohjalta. Aluksi esitellään valon luonnetta ja yhteyttä sähkömagneettiseen säteilyyn ja ihmisen näköjärjestelmään. Tämän jälkeen paneudutaan valonsäteiden heijastumiseen ja sirontaan ja yleistetään klassiset ideaaliset sirontamallit huomioimaan pinnan hienorakenne sirontajakaumilla. Tutkittu ongelma muotoillaan matemaattisiksi yhtälöiksi jotka ratkaistaan analyyttisesti. Ratkaisu osoittautuu intuitiiviseksi: valaistus saadaan laskemalla valonlähteistä emittoituvan valon kaikkien kertalukujen heijastukset. Analyyttinen ratkaisu muotoillaan rekursiiviseksi ja ratkaistaan todennäköisyyslaskentaan perustuvalla Monte Carlo -integroinnilla, jonka suppenemista nopeutetaan tärkeysotannalla. Numeerinen ratkaisu osoitetaan odotusarvoisesti harhattomaksi ja ratkaisun virheen osoitetaan puolittuvan laskentapisteiden määrän nelinkertaistuessa. Käytettävä todennäköisyyslaskenta esitellään pääpiirteittäin. Numeerinen ratkaisumenetelmä on stokastista säteenseurantaa yleistävä polunseuranta. Maailma määritellään kolmioverkkona ja pintojen normaalit annetaan kolmioiden kärkipisteissä. Kolmioista muodostetaan kuvaus tasopinnalle, josta voidaan tarvittaessa lukea esimerkiksi pinnan tarkemmat normaalit, sirontaominaisuudet tai absorptiospektri. Numeerisen ratkaisun eniten aikaa vievä osuus on valonsäteen seuraavan osumapisteen selvitys maailman pintojen välillä. Ratkaisua nopeutetaan tallentamalla maailman kolmiot tehokkaaseen tietorakenteeseen, kd-puuhun, joka mahdollistaa valonsäteen ja suurten kolmiojoukkojen nopeat leikkaustarkistukset. Kd-puun ajatus ja toteutus esitellään työssä lyhyesti. Lopuksi esitellään ratkaisun eri vaiheet ja teoria käytännössä ja nähdään konkreettisesti eri menetelmien merkitys numeerisen ratkaisijan tuottamaan kuvaan. Lisäksi esitellään tehokas prioriteettijonoon perustuva adaptiivinen menetelmä kuvaan jääneen kohinan pienentämiseksi tutkimalla näytteistyksen otoskeskihajonnan ja keskiarvon suhdetta kuvapisteittäin.

Vakuutustoiminnan sijoitusriskit

Riskiteorian klassisessa vararikkomallissa mallinnetaan ainoastaan yhtiön kassavirtaa, jossa tapahtuvaa tappiota kuvataan kunakin vuonna satunnaismuuttujilla Q, joiden oletetaan olevan toisistaan riipumattomia ja noudattavan samaa todennäköisyyslakia. Yhtiön tulkitaan olevan vararikossa mikäli Q:den summa ylittää jonain hetkenä yhtiön alkupääoman. Klassisessa vararikkomallissa rahan arvo on siis sama kaikkina ajanhetkinä. Todellisuudessa raha kasvaa korkoa ja vakuutusyhtiö harjoittaa myös riskillistä sijoitustoimintaa, jonka tulos vaikuttaa merkittävästi yhtiön kunkin hetkisen pääoman määrään. Tässä tutkielmassa vararikkomalliin lisätään yhtiön kunkin vuoden sijoitustuottoja kuvaavat satunnaismuuttujat M, jotka täytyy Q:den tapaan olettaa riippumattomiksi sekä samoin jakautuneiksi. Tämäkään malli ei kuvaa todellisen yhtiön tilannetta kovin hyvin, mutta on kuitenkin hieman realistisempi kuin klassinen malli. Yhtiön alkupääoman sekä tarkastelujakson pituuden täytyy lisäksi olettaa lähestyvän ääretöntä. Tällöin on mahdollista muodostaa ns. implisiittistä uusiutumisteoriaa hyödyntäen asymptoottisia tuloksia yhtiön vararikkotodennäköisyydelle.

Cliffordin analyysi ja sovelluksia

Cliffordin algebrat ovat äärellisulotteisia reaali- tai kompleksikertoimisia algebroja, jotka yleistävät kvaterneja ja kompleksilukuja. Näitä algebroja on kutsuttu myös geometrisiksi algebroiksi. Tässä tutkielmassa tarkastellaan analyysiä Cliffordin algebroilla ja sen sovelluksia. Analyysi tässä tarkoittaa sitä, että tarkastellaan Cliffordin algebraarvoisia funktioita, jotka omaavat erikseen määriteltyjä sileysominaisuuksia. Sovelluskohteina ovat osittaisdifferentiaaliyhtälöt ja reuna-arvo-ongelmat. Menetelmät ovat klassisia kompleksianalyysin menetelmiä. Tutkielmassa esitellään Cliffordin algebrat yleisille neliömuodollisille avaruuksille. Keskeisiä algebrallisia ominaisuuksia ovat Frobeniuksen teoreema ja perusoperaatiot. On yleisesti tunnettua, että kvaterneilla voidaan esittää kolmiulotteisen ja neljäulotteisen avaruuden rotaatiot. Tutkielmassa esitellään, miten Cliffordin ryhmiä, jotka ovat Cliffordin algebrojen osajoukkoja, käytetään useamman ulottuvuuden rotaatioiden esityksessä. Toinen sovelluskohde on Möbius-kuvausten esittäminen Vahlenin matriiseilla. Tutkielman toisessa osiossa määritellään monogeeniset funktiot erään Diracin operaattorin nollaratkaisuina. Monogeenisten funktioiden pääominaisuus on Cauchyn integraalikaava. Välittömiä seurauksia ovat esimerkiksi potenssisarjakehitelmät, analyyttisuus, Liuvillen teoreema ja muut klassisen kompleksianalyysin tuloksien yleistykset. Toisaalta monet kompleksianalyysin tulokset eivät yleisty. Esimerkiksi monogeenisten funktioiden tulo ei ole yleisesti ottaen monogeeninen. Potenssisarjat voidaan esittää monogeenisten polynomeiden avulla. Esitämme kannan monogeenisten polynomien avaruudelle käyttäen CK-laajennusta. Cauchyn ytimen ominaisuuksien avulla tarkastelemme Diracin operaattorin reuna-arvo-ongelmia ja nk. D-ongelmaa. Käyttäen Rungen lauseen yleistystä osoitamme D-ongelman yleisen ratkaistavuuden. Toisaalta reuna-arvo-ongelman ratkaistavuus karakterisoidaan käyttäen Cauchyn ytimen reuna-arvo-ominaisuuksia ja hyppyrelaatioita. Keskeinen sovellus tuloksille on aikaharmonisen Maxwellin yhtälön reuna-arvo-ongelmien tarkastelu. Mielenkiintoista on myös, miten Diracin operaattori linearisoi Laplacen operaattorin ja aalto-operaattorin. Toisaalta Diracin operaattorin avulla voidaan ilmaista Maxwellin yhtälöt tiiviissä muodossa. Muita tuloksia tutkielmassa ovat meromorfifunktioiden määritelmä ja Mittag-Lefflerin lause. Tutkielman lopuksi tarkastellaan lyhyesti harmonisten funktioiden ja monogeenisten funktioiden suhdetta. Jokainen harmoninen funktio on jonkin monogeenisen funktion reaaliosa. Tosin monogeeninen funktio ei ole yksikäsitteisesti määrätty sen reaaliosan avulla.

Embracing Integrability in Stellar and Planetary Dynamics

Hamiltonian systems in stellar and planetary dynamics are typically near integrable. For example, Solar System planets are almost in two-body orbits, and in simulations of the Galaxy, the orbits of stars seem regular. For such systems, sophisticated numerical methods can be developed through integrable approximations. Following this theme, we discuss three distinct problems. We start by considering numerical integration techniques for planetary systems. Perturbation methods (that utilize the integrability of the two-body motion) are preferred over conventional "blind" integration schemes. We introduce perturbation methods formulated with Cartesian variables. In our numerical comparisons, these are superior to their conventional counterparts, but, by definition, lack the energy-preserving properties of symplectic integrators. However, they are exceptionally well suited for relatively short-term integrations in which moderately high positional accuracy is required. The next exercise falls into the category of stability questions in solar systems. Traditionally, the interest has been on the orbital stability of planets, which have been quantified, e.g., by Liapunov exponents. We offer a complementary aspect by considering the protective effect that massive gas giants, like Jupiter, can offer to Earth-like planets inside the habitable zone of a planetary system. Our method produces a single quantity, called the escape rate, which characterizes the system of giant planets. We obtain some interesting results by computing escape rates for the Solar System. Galaxy modelling is our third and final topic. Because of the sheer number of stars (about 10^11 in Milky Way) galaxies are often modelled as smooth potentials hosting distributions of stars. Unfortunately, only a handful of suitable potentials are integrable (harmonic oscillator, isochrone and Stäckel potential). This severely limits the possibilities of finding an integrable approximation for an observed galaxy. A solution to this problem is torus construction; a method for numerically creating a foliation of invariant phase-space tori corresponding to a given target Hamiltonian. Canonically, the invariant tori are constructed by deforming the tori of some existing integrable toy Hamiltonian. Our contribution is to demonstrate how this can be accomplished by using a Stäckel toy Hamiltonian in ellipsoidal coordinates.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Geometric Characterizations for Patterson-Sullivan Measures of Geometrically Finite Kleinian Groups

The main results of this thesis show that a Patterson-Sullivan measure of a non-elementary geometrically finite Kleinian group can always be characterized using geometric covering and packing constructions. This means that if the standard covering and packing constructions are modified in a suitable way, one can use either one of them to construct a geometric measure which is identical to the Patterson-Sullivan measure. The main results generalize and modify results of D. Sullivan which show that one can sometimes use the standard covering construction to construct a suitable geometric measure and sometimes the standard packing construction. Sullivan has shown also that neither or both of the standard constructions can be used to construct the geometric measure in some situations. The main modifications of the standard constructions are based on certain geometric properties of limit sets of Kleinian groups studied first by P. Tukia. These geometric properties describe how closely the limit set of a given Kleinian group resembles euclidean planes or spheres of varying dimension on small scales. The main idea is to express these geometric properties in a quantitative form which can be incorporated into the gauge functions used in the modified covering and packing constructions. Certain estimation results for general conformal measures of Kleinian groups play a crucial role in the proofs of the main results. These estimation results are generalizations and modifications of similar results considered, among others, by B. Stratmann, D. Sullivan, P. Tukia and S. Velani. The modified constructions are in general defined without reference to Kleinian groups, so they or their variants may prove useful in some other contexts in addition to that of Kleinian groups.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Inverse problem for the wave equation : partial data and novel boundary sources

An inverse problem for the wave equation is a mathematical formulation of the problem to convert measurements of sound waves to information about the wave speed governing the propagation of the waves. This doctoral thesis extends the theory on the inverse problems for the wave equation in cases with partial measurement data and also considers detection of discontinuous interfaces in the wave speed. A possible application of the theory is obstetric sonography in which ultrasound measurements are transformed into an image of the fetus in its mother's uterus. The wave speed inside the body can not be directly observed but sound waves can be produced outside the body and their echoes from the body can be recorded. The present work contains five research articles. In the first and the fifth articles we show that it is possible to determine the wave speed uniquely by using far apart sound sources and receivers. This extends a previously known result which requires the sound waves to be produced and recorded in the same place. Our result is motivated by a possible application to reflection seismology which seeks to create an image of the Earth s crust from recording of echoes stimulated for example by explosions. For this purpose, the receivers can not typically lie near the powerful sound sources. In the second article we present a sound source that allows us to recover many essential features of the wave speed from the echo produced by the source. Moreover, these features are known to determine the wave speed under certain geometric assumptions. Previously known results permitted the same features to be recovered only by sequential measurement of echoes produced by multiple different sources. The reduced number of measurements could increase the number possible applications of acoustic probing. In the third and fourth articles we develop an acoustic probing method to locate discontinuous interfaces in the wave speed. These interfaces typically correspond to interfaces between different materials and their locations are of interest in many applications. There are many previous approaches to this problem but none of them exploits sound sources varying freely in time. Our use of more variable sources could allow more robust implementation of the probing.