Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Some properties of the coefficients of cyclotomic polynomials

An explicit formula is obtained for the coefficients of the cyclotomic polynomial F_n(x), where n is the product of two distinct odd primes. A recursion formula and a lower bound and an improvement of Bang’s upper bound for the coefficients of F_n(x) are also obtained, where n is the product of three distinct primes. The cyclotomic coefficients are also studied when n is the product of four distinct odd primes. A recursion formula and upper bounds for its coefficients are obtained. The last chapter includes a different approach to the cyclotomic coefficients. A connection is obtained between a certain partition function and the cyclotomic coefficients when n is the product of an arbitrary number of distinct odd primes. Finally, an upper bound for the coefficients is derived when n is the product of an arbitrary number of distinct and odd primes.

Invariants, Lie-Bäcklund operators and Bäcklund transformations

This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund groups of tangent transformations, particular cases of which are the Lie tangent and the Lie point groups, are extensively used.

In Chapter I we first review the classical results of Lie, Bäcklund and Bianchi as well as the more recent ones due mainly to Ovsjannikov. We then concentrate on the Lie-Bäcklund groups (or more precisely on the corresponding Lie-Bäcklund operators), as introduced by Ibragimov and Anderson, and prove some lemmas about them which are useful for the following chapters. Finally we introduce the concept of a conditionally admissible operator (as opposed to an admissible one) and show how this can be used to generate exact solutions.

In Chapter II we establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only Lie point groups is insufficient. For this purpose a special type of Lie-Bäcklund groups, those equivalent to Lie tangent groups, is used. It is also shown how these generalized groups induce Lie point groups on Hamilton's equations. The generalization of the above results to any first order equation, where the dependent variable does not appear explicitly, is obvious. In the second part of this chapter we investigate admissible operators (or equivalently constants of motion) of the Hamilton-Jacobi equation with polynornial dependence on the momenta. The form of the most general constant of motion linear, quadratic and cubic in the momenta is explicitly found. Emphasis is given to the quadratic case, where the particular case of a fixed (say zero) energy state is also considered; it is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered. These include potentials due to two centers and limiting cases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the corresponding invariant. Also some new cubic invariants are found.

In Chapter III we first establish the group nature of all separable solutions of any linear, homogeneous equation. We then concentrate on the Schrodinger equation and look for an algorithm which generates a quantum invariant from a classical one. The problem of an isomorphism between functions in classical observables and quantum observables is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl' s transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered; in this case Weyl's transform does not yield an isomorphism even for the quadratic case. However, for this case a correspondence mapping a classical invariant to a quantum orie is explicitly found.

Chapters IV and V are concerned with the general group structure of evolution equations. In Chapter IV we establish a one to one correspondence between admissible Lie-Bäcklund operators of evolution equations (derivable from a variational principle) and conservation laws of these equations. This correspondence takes the form of a simple algorithm.

In Chapter V we first establish the group nature of all Bäcklund transformations (BT) by proving that any solution generated by a BT is invariant under the action of some conditionally admissible operator. We then use an algorithm based on invariance criteria to rederive many known BT and to derive some new ones. Finally, we propose a generalization of BT which, among other advantages, clarifies the connection between the wave-train solution and a BT in the sense that, a BT may be thought of as a variation of parameters of some. special case of the wave-train solution (usually the solitary wave one). Some open problems are indicated.

Most of the material of Chapters II and III is contained in [I], [II], [III] and [IV] and the first part of Chapter V in [V].

Randomness-efficient curve sampling

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the sampled curve is still low-degree. This property is often used in combination with the sampling property and has found many applications, including PCP constructions, local decoding of codes, and algebraic PRG constructions.

The randomness complexity of curve samplers is a crucial parameter for its applications. It is known that (non-explicit) curve samplers using O(log N + log(1/δ)) random bits exist, where N is the domain size and δ is the confidence error. The question of explicitly constructing randomness-efficient curve samplers was first raised in [TU06] where they obtained curve samplers with near-optimal randomness complexity.

In this thesis, we present an explicit construction of low-degree curve samplers with optimal randomness complexity (up to a constant factor) that sample curves of degree (m log_q(1/δ))^O(1) in F_q^m. Our construction is a delicate combination of several components, including extractor machinery, limited independence, iterated sampling, and list-recoverable codes.

Velocity resolved - scalar modeled simulations of high Schmidt number turbulent transport

The objective of this thesis is to develop a framework to conduct velocity resolved - scalar modeled (VR-SM) simulations, which will enable accurate simulations at higher Reynolds and Schmidt (Sc) numbers than are currently feasible. The framework established will serve as a first step to enable future simulation studies for practical applications. To achieve this goal, in-depth analyses of the physical, numerical, and modeling aspects related to Sc>>1 are presented, specifically when modeling in the viscous-convective subrange. Transport characteristics are scrutinized by examining scalar-velocity Fourier mode interactions in Direct Numerical Simulation (DNS) datasets and suggest that scalar modes in the viscous-convective subrange do not directly affect large-scale transport for high Sc. Further observations confirm that discretization errors inherent in numerical schemes can be sufficiently large to wipe out any meaningful contribution from subfilter models. This provides strong incentive to develop more effective numerical schemes to support high Sc simulations. To lower numerical dissipation while maintaining physically and mathematically appropriate scalar bounds during the convection step, a novel method of enforcing bounds is formulated, specifically for use with cubic Hermite polynomials. Boundedness of the scalar being transported is effected by applying derivative limiting techniques, and physically plausible single sub-cell extrema are allowed to exist to help minimize numerical dissipation. The proposed bounding algorithm results in significant performance gain in DNS of turbulent mixing layers and of homogeneous isotropic turbulence. Next, the combined physical/mathematical behavior of the subfilter scalar-flux vector is analyzed in homogeneous isotropic turbulence, by examining vector orientation in the strain-rate eigenframe. The results indicate no discernible dependence on the modeled scalar field, and lead to the identification of the tensor-diffusivity model as a good representation of the subfilter flux. Velocity resolved - scalar modeled simulations of homogeneous isotropic turbulence are conducted to confirm the behavior theorized in these a priori analyses, and suggest that the tensor-diffusivity model is ideal for use in the viscous-convective subrange. Simulations of a turbulent mixing layer are also discussed, with the partial objective of analyzing Schmidt number dependence of a variety of scalar statistics. Large-scale statistics are confirmed to be relatively independent of the Schmidt number for Sc>>1, which is explained by the dominance of subfilter dissipation over resolved molecular dissipation in the simulations. Overall, the VR-SM framework presented is quite effective in predicting large-scale transport characteristics of high Schmidt number scalars, however, it is determined that prediction of subfilter quantities would entail additional modeling intended specifically for this purpose. The VR-SM simulations presented in this thesis provide us with the opportunity to overlap with experimental studies, while at the same time creating an assortment of baseline datasets for future validation of LES models, thereby satisfying the objectives outlined for this work.

Topological strings, double affine Hecke algebras, and exceptional knot homology

In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and various insights from topological string theory.

In particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.

In addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Geometrical effects on the resonance absorption of neutrons

An investigation was conducted to estimate the error when the flat-flux approximation is used to compute the resonance integral for a single absorber element embedded in a neutron source.

The investigation was initiated by assuming a parabolic flux distribution in computing the flux-averaged escape probability which occurs in the collision density equation. Furthermore, also assumed were both wide resonance and narrow resonance expressions for the resonance integral. The fact that this simple model demonstrated a decrease in the resonance integral motivated the more detailed investigation of the thesis.

An integral equation describing the collision density as a function of energy, position and angle is constructed and is subsequently specialized to the case of energy and spatial dependence. This equation is further simplified by expanding the spatial dependence in a series of Legendre polynomials (since a one-dimensional case is considered). In this form, the effects of slowing-down and flux depression may be accounted for to any degree of accuracy desired. The resulting integral equation for the energy dependence is thus solved numerically, considering the slowing down model and the infinite mass model as separate cases.

From the solution obtained by the above method, the error ascribable to the flat-flux approximation is obtained. In addition to this, the error introduced in the resonance integral in assuming no slowing down in the absorber is deduced. Results by Chernick for bismuth rods, and by Corngold for uranium slabs, are compared to the latter case, and these agree to within the approximations made.

Estimation of catalyst activity profiles and deactivation parameters from reactor operating data with applications to naphtha reforming

Techniques are developed for estimating activity profiles in fixed bed reactors and catalyst deactivation parameters from operating reactor data. These techniques are applicable, in general, to most industrial catalytic processes. The catalytic reforming of naphthas is taken as a broad example to illustrate the estimation schemes and to signify the physical meaning of the kinetic parameters of the estimation equations. The work is described in two parts. Part I deals with the modeling of kinetic rate expressions and the derivation of the working equations for estimation. Part II concentrates on developing various estimation techniques.

Part I: The reactions used to describe naphtha reforming are dehydrogenation and dehydroisomerization of cycloparaffins; isomerization, dehydrocyclization and hydrocracking of paraffins; and the catalyst deactivation reactions, namely coking on alumina sites and sintering of platinum crystallites. The rate expressions for the above reactions are formulated, and the effects of transport limitations on the overall reaction rates are discussed in the appendices. Moreover, various types of interaction between the metallic and acidic active centers of reforming catalysts are discussed as characterizing the different types of reforming reactions.

Part II: In catalytic reactor operation, the activity distribution along the reactor determines the kinetics of the main reaction and is needed for predicting the effect of changes in the feed state and the operating conditions on the reactor output. In the case of a monofunctional catalyst and of bifunctional catalysts in limiting conditions, the cumulative activity is sufficient for predicting steady reactor output. The estimation of this cumulative activity can be carried out easily from measurements at the reactor exit. For a general bifunctional catalytic system, the detailed activity distribution is needed for describing the reactor operation, and some approximation must be made to obtain practicable estimation schemes. This is accomplished by parametrization techniques using measurements at a few points along the reactor. Such parametrization techniques are illustrated numerically with a simplified model of naphtha reforming.

To determine long term catalyst utilization and regeneration policies, it is necessary to estimate catalyst deactivation parameters from the the current operating data. For a first order deactivation model with a monofunctional catalyst or with a bifunctional catalyst in special limiting circumstances, analytical techniques are presented to transform the partial differential equations to ordinary differential equations which admit more feasible estimation schemes. Numerical examples include the catalytic oxidation of butene to butadiene and a simplified model of naphtha reforming. For a general bifunctional system or in the case of a monofunctional catalyst subject to general power law deactivation, the estimation can only be accomplished approximately. The basic feature of an appropriate estimation scheme involves approximating the activity profile by certain polynomials and then estimating the deactivation parameters from the integrated form of the deactivation equation by regression techniques. Different bifunctional systems must be treated by different estimation algorithms, which are illustrated by several cases of naphtha reforming with different feed or catalyst composition.