Nonlinear dimension reduction for surrogate modeling using gradient information

We introduce a method for the nonlinear dimension reduction of a high-dimensional function $u:\mathbb{R}^d\rightarrow\mathbb{R}$, $d\gg1$. Our objective is to identify a nonlinear feature map $g:\mathbb{R}^d\rightarrow\mathbb{R}^m$, with a prescribed intermediate dimension $m\ll d$, so that $u$ can be well approximated by $f\circ g$ for some profile function $f:\mathbb{R}^m\rightarrow\mathbb{R}$. We propose to build the feature map by aligning the Jacobian $\nabla g$ with the gradient $\nabla u$, and we theoretically analyze the properties of the resulting $g$. Once $g$ is built, we construct $f$ by solving a gradient-enhanced least squares problem. Our practical algorithm makes use of a sample $\{x^{(i)},u(x^{(i)}),\nabla u(x^{(i)})\}_{i=1}^N$ and builds both $g$ and $f$ on adaptive downward-closed polynomial spaces, using cross validation to avoid overfitting. We numerically evaluate the performance of our algorithm across different benchmarks, and explore the impact of the intermediate dimension $m$. We show that building a nonlinear feature map $g$ can permit more accurate approximation of $u$ than a linear $g$, for the same input data set

Nonlinear dimension reduction for surrogate modeling using gradient information

We introduce a method for the nonlinear dimension reduction of a high-dimensional function $u:\mathbb{R}^d\rightarrow\mathbb{R}$, $d\gg1$. Our objective is to identify a nonlinear feature map $g:\mathbb{R}^d\rightarrow\mathbb{R}^m$, with a prescribed intermediate dimension $m\ll d$, so that $u$ can be well approximated by $f\circ g$ for some profile function $f:\mathbb{R}^m\rightarrow\mathbb{R}$. We propose to build the feature map by aligning the Jacobian $\nabla g$ with the gradient $\nabla u$, and we theoretically analyze the properties of the resulting $g$. Once $g$ is built, we construct $f$ by solving a gradient-enhanced least squares problem. Our practical algorithm makes use of a sample $\{x^{(i)},u(x^{(i)}),\nabla u(x^{(i)})\}_{i=1}^N$ and builds both $g$ and $f$ on adaptive downward-closed polynomial spaces, using cross validation to avoid overfitting. We numerically evaluate the performance of our algorithm across different benchmarks, and explore the impact of the intermediate dimension $m$. We show that building a nonlinear feature map $g$ can permit more accurate approximation of $u$ than a linear $g$, for the same input data set

Scalable electronic structure methods to solve the Kohn-Sham equation

From the single hydrogen to proteins in the hundreds of thousands of kilodaltons, scientists can use the electronic structure of interacting atoms to predict their material properties. Knowing the material properties through solving the electronic structure problem, would allow for the controlled prediction and corresponding design of materials. The Kohn-Sham equations, based on density functional theory, transform a many-body problem impossible to solve for anything but the smallest molecules, into a practical problem which can be used to predict material properties. Although KSDFT scales as the cube of the number of electrons in the system, there are additional well documented approximations to further reduce the number of electrons, such as the pseudopotential method. The incoming exascale era will lead to unavoidable challenges in solving the Kohn-Sham equations. These challenges include communication and hardware considerations. Old paradigms, epitomized by repeated series of globally forced synchronization points, will give way to new breeds of algorithms to maximizing scaling performance while maintaining portability. This thesis focuses on the solution to Kohn-Sham DFT in real space at scale. Key to this effort is a parallel treatment of numerical elements involving the Rayleigh-Ritz method. At minimum, the Rayleigh-Ritz projection requires a number of distributed matrix vector operations equal to the number of electrons solved for in a system. Furthermore, the projection requires that number, squared and then halved, of dot products. The memory cost for such an algorithm also grows very large quickly, and explicit intelligent management is not an option. I demonstrate the computational requirements for the various steps in solving for the electronic structure problem for both large and small molecular systems. This thesis also discusses opportunities in real space Kohn-Sham DFT to further utilize floating point optimized hardware the with higher order stencils.Chemical Engineerin

Advanced Algebraic Concepts for Efficient Multi-Channel Signal Processing

Unsere moderne Gesellschaft ist Zeuge eines fundamentalen Wandels in der Art und Weise wie wir mit Technologie interagieren. Geräte werden zunehmend intelligenter - sie verfügen über mehr und mehr Rechenleistung und häufiger über eigene Kommunikationsschnittstellen. Das beginnt bei einfachen Haushaltsgeräten und reicht über Transportmittel bis zu großen überregionalen Systemen wie etwa dem Stromnetz. Die Erfassung, die Verarbeitung und der Austausch digitaler Informationen gewinnt daher immer mehr an Bedeutung. Die Tatsache, dass ein wachsender Anteil der Geräte heutzutage mobil und deshalb batteriebetrieben ist, begründet den Anspruch, digitale Signalverarbeitungsalgorithmen besonders effizient zu gestalten. Dies kommt auch dem Wunsch nach einer Echtzeitverarbeitung der großen anfallenden Datenmengen zugute. Die vorliegende Arbeit demonstriert Methoden zum Finden effizienter algebraischer Lösungen für eine Vielzahl von Anwendungen mehrkanaliger digitaler Signalverarbeitung. Solche Ansätze liefern nicht immer unbedingt die bestmögliche Lösung, kommen dieser jedoch häufig recht nahe und sind gleichzeitig bedeutend einfacher zu beschreiben und umzusetzen. Die einfache Beschreibungsform ermöglicht eine tiefgehende Analyse ihrer Leistungsfähigkeit, was für den Entwurf eines robusten und zuverlässigen Systems unabdingbar ist. Die Tatsache, dass sie nur gebräuchliche algebraische Hilfsmittel benötigen, erlaubt ihre direkte und zügige Umsetzung und den Test unter realen Bedingungen. Diese Grundidee wird anhand von drei verschiedenen Anwendungsgebieten demonstriert. Zunächst wird ein semi-algebraisches Framework zur Berechnung der kanonisch polyadischen (CP) Zerlegung mehrdimensionaler Signale vorgestellt. Dabei handelt es sich um ein sehr grundlegendes Werkzeug der multilinearen Algebra mit einem breiten Anwendungsspektrum von Mobilkommunikation über Chemie bis zur Bildverarbeitung. Verglichen mit existierenden iterativen Lösungsverfahren bietet das neue Framework die Möglichkeit, den Rechenaufwand und damit die Güte der erzielten Lösung zu steuern. Es ist außerdem weniger anfällig gegen eine schlechte Konditionierung der Ausgangsdaten. Das zweite Gebiet, das in der Arbeit besprochen wird, ist die unterraumbasierte hochauflösende Parameterschätzung für mehrdimensionale Signale, mit Anwendungsgebieten im RADAR, der Modellierung von Wellenausbreitung, oder bildgebenden Verfahren in der Medizin. Es wird gezeigt, dass sich derartige mehrdimensionale Signale mit Tensoren darstellen lassen. Dies erlaubt eine natürlichere Beschreibung und eine bessere Ausnutzung ihrer Struktur als das mit Matrizen möglich ist. Basierend auf dieser Idee entwickeln wir eine tensor-basierte Schätzung des Signalraums, welche genutzt werden kann um beliebige existierende Matrix-basierte Verfahren zu verbessern. Dies wird im Anschluss exemplarisch am Beispiel der ESPRIT-artigen Verfahren gezeigt, für die verbesserte Versionen vorgeschlagen werden, die die mehrdimensionale Struktur der Daten (Tensor-ESPRIT), nichzirkuläre Quellsymbole (NC ESPRIT), sowie beides gleichzeitig (NC Tensor-ESPRIT) ausnutzen. Um die endgültige Schätzgenauigkeit objektiv einschätzen zu können wird dann ein Framework für die analytische Beschreibung der Leistungsfähigkeit beliebiger ESPRIT-artiger Algorithmen diskutiert. Verglichen mit existierenden analytischen Ausdrücken ist unser Ansatz allgemeiner, da keine Annahmen über die statistische Verteilung von Nutzsignal und Rauschen benötigt werden und die Anzahl der zur Verfügung stehenden Schnappschüsse beliebig klein sein kann. Dies führt auf vereinfachte Ausdrücke für den mittleren quadratischen Schätzfehler, die Schlussfolgerungen über die Effizienz der Verfahren unter verschiedenen Bedingungen zulassen. Das dritte Anwendungsgebiet ist der bidirektionale Datenaustausch mit Hilfe von Relay-Stationen. Insbesondere liegt hier der Fokus auf Zwei-Wege-Relaying mit Hilfe von Amplify-and-Forward-Relays mit mehreren Antennen, da dieser Ansatz ein besonders gutes Kosten-Nutzen-Verhältnis verspricht. Es wird gezeigt, dass sich die nötige Kanalkenntnis mit einem einfachen algebraischen Tensor-basierten Schätzverfahren gewinnen lässt. Außerdem werden Verfahren zum Finden einer günstigen Relay-Verstärkungs-Strategie diskutiert. Bestehende Ansätze basieren entweder auf komplexen numerischen Optimierungsverfahren oder auf Ad-Hoc-Ansätzen die keine zufriedenstellende Bitfehlerrate oder Summenrate liefern. Deshalb schlagen wir algebraische Ansätze zum Finden der Relayverstärkungsmatrix vor, die von relevanten Systemmetriken inspiriert sind und doch einfach zu berechnen sind. Wir zeigen das algebraische ANOMAX-Verfahren zum Erreichen einer niedrigen Bitfehlerrate und seine Modifikation RR-ANOMAX zum Erreichen einer hohen Summenrate. Für den Spezialfall, in dem die Endgeräte nur eine Antenne verwenden, leiten wir eine semi-algebraische Lösung zum Finden der Summenraten-optimalen Strategie (RAGES) her. Anhand von numerischen Simulationen wird die Leistungsfähigkeit dieser Verfahren bezüglich Bitfehlerrate und erreichbarer Datenrate bewertet und ihre Effektivität gezeigt.Modern society is undergoing a fundamental change in the way we interact with technology. More and more devices are becoming "smart" by gaining advanced computation capabilities and communication interfaces, from household appliances over transportation systems to large-scale networks like the power grid. Recording, processing, and exchanging digital information is thus becoming increasingly important. As a growing share of devices is nowadays mobile and hence battery-powered, a particular interest in efficient digital signal processing techniques emerges. This thesis contributes to this goal by demonstrating methods for finding efficient algebraic solutions to various applications of multi-channel digital signal processing. These may not always result in the best possible system performance. However, they often come close while being significantly simpler to describe and to implement. The simpler description facilitates a thorough analysis of their performance which is crucial to design robust and reliable systems. The fact that they rely on standard algebraic methods only allows their rapid implementation and test under real-world conditions. We demonstrate this concept in three different application areas. First, we present a semi-algebraic framework to compute the Canonical Polyadic (CP) decompositions of multidimensional signals, a very fundamental tool in multilinear algebra with applications ranging from chemistry over communications to image compression. Compared to state-of-the art iterative solutions, our framework offers a flexible control of the complexity-accuracy trade-off and is less sensitive to badly conditioned data. The second application area is multidimensional subspace-based high-resolution parameter estimation with applications in RADAR, wave propagation modeling, or biomedical imaging. We demonstrate that multidimensional signals can be represented by tensors, providing a convenient description and allowing to exploit the multidimensional structure in a better way than using matrices only. Based on this idea, we introduce the tensor-based subspace estimate which can be applied to enhance existing matrix-based parameter estimation schemes significantly. We demonstrate the enhancements by choosing the family of ESPRIT-type algorithms as an example and introducing enhanced versions that exploit the multidimensional structure (Tensor-ESPRIT), non-circular source amplitudes (NC ESPRIT), and both jointly (NC Tensor-ESPRIT). To objectively judge the resulting estimation accuracy, we derive a framework for the analytical performance assessment of arbitrary ESPRIT-type algorithms by virtue of an asymptotical first order perturbation expansion. Our results are more general than existing analytical results since we do not need any assumptions about the distribution of the desired signal and the noise and we do not require the number of samples to be large. At the end, we obtain simplified expressions for the mean square estimation error that provide insights into efficiency of the methods under various conditions. The third application area is bidirectional relay-assisted communications. Due to its particularly low complexity and its efficient use of the radio resources we choose two-way relaying with a MIMO amplify and forward relay. We demonstrate that the required channel knowledge can be obtained by a simple algebraic tensor-based channel estimation scheme. We also discuss the design of the relay amplification matrix in such a setting. Existing approaches are either based on complicated numerical optimization procedures or on ad-hoc solutions that to not perform well in terms of the bit error rate or the sum-rate. Therefore, we propose algebraic solutions that are inspired by these performance metrics and therefore perform well while being easy to compute. For the MIMO case, we introduce the algebraic norm maximizing (ANOMAX) scheme, which achieves a very low bit error rate, and its extension Rank-Restored ANOMAX (RR-ANOMAX) that achieves a sum-rate close to an upper bound. Moreover, for the special case of single antenna terminals we derive the semi-algebraic RAGES scheme which finds the sum-rate optimal relay amplification matrix based on generalized eigenvectors. Numerical simulations evaluate the resulting system performance in terms of bit error rate and system sum rate which demonstrates the effectiveness of the proposed algebraic solutions

Joint Sensing and Reception Design of SIMO Hybrid Cognitive Radio Systems

In this paper, the problem of joint design of Spectrum Sensing (SS) and receive beamforming (BF), with reference to a Cognitive Radio (CR) system, is considered. The aim of the proposed design is the maximization of the achievable average uplink rate of a Secondary User (SU), subject to an outage-based Quality-of-Service (QoS) constraint for primary communication. A hybrid CR system approach is studied, according to which, the system either operates as an interweave (i.e., opportunistic) or as an underlay (i.e., spectrum sharing) CR system, based on SS results. A realistic Channel State Information (CSI) framework is assumed, according to which, the direct channel links are known by the multiple antenna receivers (RXs), while, merely statistical (covariance) information is available for the interference links. A new, closed form approximation is derived for the outage probability of primary communication, and the problem of rate-optimal selection of SS parameters and receive beamformers is addressed for hybrid, interweave and underlay CR systems. It is proven that our proposed system design outperforms both underlay and interweave CR systems for a range of system scenarios

Aperture-Level Simultaneous Transmit and Receive (STAR) with Digital Phased Arrays

In the signal processing community, it has long been assumed that transmitting and receiving useful signals at the same time in the same frequency band at the same physical location was impossible. A number of insights in antenna design, analog hardware, and digital signal processing have allowed researchers to achieve simultaneous transmit and receive (STAR) capability, sometimes also referred to as in-band full-duplex (IBFD). All STAR systems must mitigate the interference in the receive channel caused by the signals emitted by the system. This poses a significant challenge because of the immense disparity in the power of the transmitted and received signals. As an analogy, imagine a person that wanted to be able to hear a whisper from across the room while screaming at the top of their lungs. The sound of their own voice would completely drown out the whisper. Approaches to increasing the isolation between the transmit and receive channels of a system attempt to successively reduce the magnitude of the transmitted interference at various points in the received signal processing chain. Many researchers believe that STAR cannot be achieved practically without some combination of modified antennas, analog self-interference cancellation hardware, digital adaptive beamforming, and digital self-interference cancellation. The aperture-level simultaneous transmit and receive (ALSTAR) paradigm confronts that assumption by creating isolation between transmit and receive subarrays in a phased array using only digital adaptive transmit and receive beamforming and digital self-interference cancellation. This dissertation explores the boundaries of performance for the ALSTAR architecture both in terms of isolation and in terms of spatial imaging resolution. It also makes significant strides towards practical ALSTAR implementation by determining the performance capabilities and computational costs of an adaptive beamforming and self-interference cancellation implementation inspired by the mathematical structure of the isolation performance limits and designed for real-time operation

A parallel multigrid method for band structure computation of 3D photonic crystals with higher order finite elements

The band structure computation turns into solving a family of Maxwell eigenvalue problems on the periodicity domain. The discretization is done by the finite element method with special higher order H(curl)- and H1-conforming modified elements. The eigenvalue problem is solved by a preconditioned iterative eigenvalue solver with a projection onto the divergence-free vector fields. As a preconditioner we use the parallel multigrid method with a special Hiptmair smoother

Source and propagation effects of Rayleigh waves from Central Asian earthquakes.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Earth and Planetary Science.Microfiche copy available in Archives and Science.Bibliography: leaves 279-290.Ph.D