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At least 1,765 records · Page 98
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

SU-F-T-428: An Optimization-Based Commissioning Tool for Finite Size Pencil Beam Dose Calculations

Purpose: Finite size pencil beam (FSPB) algorithms are commonly used to pre-calculate the beamlet dose distribution for IMRT treatment planning. FSPB commissioning, which usually requires fine tuning of the FSPB kernel parameters, is crucial to the dose calculation accuracy and hence the plan quality. Yet due to the large number of beamlets, FSPB commissioning could be very tedious. This abstract reports an optimization-based FSPB commissioning tool we have developed in MatLab to facilitate the commissioning. Methods: A FSPB dose kernel generally contains two types of parameters: the profile parameters determining the dose kernel shape, and a 2D scaling factors accounting for the longitudinal and off-axis corrections. The former were fitted using the penumbra of a reference broad beam’s dose profile with Levenberg-Marquardt algorithm. Since the dose distribution of a broad beam is simply a linear superposition of the dose kernel of each beamlet calculated with the fitted profile parameters and scaled using the scaling factors, these factors could be determined by solving an optimization problem which minimizes the discrepancies between the calculated dose of broad beams and the reference dose. Results: We have commissioned a FSPB algorithm for three linac photon beams (6MV, 15MV and 6MVFFF). Dose of four field sizes (6*6cm2, 10*10cm2, 15*15cm2 and 20*20cm2) were calculated and compared with the reference dose exported from Eclipse TPS system. For depth dose curves, the differences are less than 1% of maximum dose after maximum dose depth for most cases. For lateral dose profiles, the differences are less than 2% of central dose at inner-beam regions. The differences of the output factors are within 1% for all the three beams. Conclusion: We have developed an optimization-based commissioning tool for FSPB algorithms to facilitate the commissioning, providing sufficient accuracy of beamlet dose calculation for IMRT optimization.

07 ISOTOPE AND RADIATION SOURCES↗
DOE OSTI2016

SU-G-BRA-03: PCA Based Imaging Angle Optimization for 2D Cine MRI Based Radiotherapy Guidance

Purpose: To develop an imaging angle optimization methodology for orthogonal 2D cine MRI based radiotherapy guidance using Principal Component Analysis (PCA) of target motion retrieved from 4DCT. Methods: We retrospectively analyzed 4DCT of 6 patients with lung tumor. A radiation oncologist manually contoured the target volume at the maximal inhalation phase of the respiratory cycle. An object constrained deformable image registration (DIR) method has been developed to track the target motion along the respiration at ten phases. The motion of the center of the target mass has been analyzed using the PCA to find out the principal motion components that were uncorrelated with each other. Two orthogonal image planes for cineMRI have been determined using this method to minimize the through plane motion during MRI based radiotherapy guidance. Results: 3D target respiratory motion for all 6 patients has been efficiently retrieved from 4DCT. In this process, the object constrained DIR demonstrated satisfactory accuracy and efficiency to enable the automatic motion tracking for clinical application. The average motion amplitude in the AP, lateral, and longitudinal directions were 3.6mm (min: 1.6mm, max: 5.6mm), 1.7mm (min: 0.6mm, max: 2.7mm), and 5.6mm (min: 1.8mm, max: 16.1mm), respectively. Based on PCA, the optimal orthogonal imaging planes were determined for cineMRI. The average angular difference between the PCA determined imaging planes and the traditional AP and lateral imaging planes were 47 and 31 degrees, respectively. After optimization, the average amplitude of through plane motion reduced from 3.6mm in AP images to 2.5mm (min:1.3mm, max:3.9mm); and from 1.7mm in lateral images to 0.6mm (min: 0.2mm, max:1.5mm), while the principal in plane motion amplitude increased from 5.6mm to 6.5mm (min: 2.8mm, max: 17mm). Conclusion: DIR and PCA can be used to optimize the orthogonal image planes of cineMRI to minimize the through plane motion during radiotherapy guidance.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

MO-FG-CAMPUS-TeP3-04: Deliverable Robust Optimization in IMPT Using Quadratic Objective Function

Purpose: To find and evaluate the way of applying deliverable MU constraints into robust spot intensity optimization in Intensity-Modulated- Proton-Therapy (IMPT) to prevent plan quality and robustness from degrading due to machine deliverable MU-constraints. Methods: Currently, the influence of the deliverable MU-constraints is retrospectively evaluated by post-processing immediately following optimization. In this study, we propose a new method based on the quasi-Newton-like L-BFGS-B algorithm with which we turn deliverable MU-constraints on and off alternatively during optimization. Seven patients with two different machine settings (small and large spot size) were planned with both conventional and new methods. For each patient, three kinds of plans were generated — conventional non-deliverable plan (plan A), conventional deliverable plan with post-processing (plan B), and new deliverable plan (plan C). We performed this study with both realistic (small) and artificial (large) deliverable MU-constraints. Results: With small minimum MU-constraints considered, new method achieved a slightly better plan quality than conventional method (D95% CTV normalized to the prescription dose: 0.994[0.992∼0.996] (Plan C) vs 0.992[0.986∼0.996] (Plan B)). With large minimum MU constraints considered, results show that the new method maintains plan quality while plan quality from the conventional method is degraded greatly (D95% CTV normalized to the prescription dose: 0.987[0.978∼0.994] (Plan C) vs 0.797[0.641∼1.000] (Plan B)). Meanwhile, plan robustness of these two method’s results is comparable. (For all 7 patients, CTV DVH band gap at D95% normalized to the prescription dose: 0.015[0.005∼0.043] (Plan C) vs 0.012[0.006∼0.038] (Plan B) with small MU-constraints and 0.019[0.009∼0.039] (Plan C) vs 0.030[0.015∼0.041] (Plan B) with large MU-constraints) Conclusion: Positive correlation has been found between plan quality degeneration and magnitude of deliverable minimal MU. Compared to conventional post-processing method, our new method of incorporating deliverable minimal MU-constraints directly into plan optimization, can produce machine-deliverable plans with better plan qualities and non-compromised plan robustness. This research was supported by the National Cancer Institute Career Developmental Award K25CA168984, by the Fraternal Order of Eagles Cancer Research Fund Career Development Award, by The Lawrence W. and Marilyn W. Matteson Fund for Cancer Research, by Mayo Arizona State University Seed Grant and by The Kemper Marley Foundation.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

WE-AB-207B-07: Dose Cloud: Generating “Big Data” for Radiation Therapy Treatment Plan Optimization Research

Purpose: To provide a tool to generate large sets of realistic virtual patient geometries and beamlet doses for treatment optimization research. This tool enables countless studies exploring the fundamental interplay between patient geometry, objective functions, weight selections, and achievable dose distributions for various algorithms and modalities. Methods: Generating realistic virtual patient geometries requires a small set of real patient data. We developed a normalized patient shape model (PSM) which captures organ and target contours in a correspondence-preserving manner. Using PSM-processed data, we perform principal component analysis (PCA) to extract major modes of variation from the population. These PCA modes can be shared without exposing patient information. The modes are re-combined with different weights to produce sets of realistic virtual patient contours. Because virtual patients lack imaging information, we developed a shape-based dose calculation (SBD) relying on the assumption that the region inside the body contour is water. SBD utilizes a 2D fluence-convolved scatter kernel, derived from Monte Carlo simulations, and can compute both full dose for a given set of fluence maps, or produce a dose matrix (dose per fluence pixel) for many modalities. Combining the shape model with SBD provides the data needed for treatment plan optimization research. Results: We used PSM to capture organ and target contours for 96 prostate cases, extracted the first 20 PCA modes, and generated 2048 virtual patient shapes by randomly sampling mode scores. Nearly half of the shapes were thrown out for failing anatomical checks, the remaining 1124 were used in computing dose matrices via SBD and a standard 7-beam protocol. As a proof of concept, and to generate data for later study, we performed fluence map optimization emphasizing PTV coverage. Conclusions: We successfully developed and tested a tool for creating customizable sets of virtual patients suitable for large-scale radiation therapy optimization research.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

WE-FG-BRB-03: Challenges and Opportunities for Implementing Biological Optimization in Particle Therapy

The physical pattern of energy deposition and the enhanced relative biological effectiveness (RBE) of protons and carbon ions compared to photons offer unique and not fully understood or exploited opportunities to improve the efficacy of radiation therapy. Variations in RBE within a pristine or spread out Bragg peak and between particle types may be exploited to enhance cell killing in target regions without a corresponding increase in damage to normal tissue structures. In addition, the decreased sensitivity of hypoxic tumors to photon-based therapies may be partially overcome through the use of more densely ionizing radiations. These and other differences between particle and photon beams may be used to generate biologically optimized treatments that reduce normal tissue complications. In this symposium, speakers will examine the impact of the RBE of charged particles on measurable biological endpoints, treatment plan optimization, and the prediction or retrospective assessment of treatment outcomes. In particular, an AAPM task group was formed to critically examine the evidence for a spatially-variant RBE in proton therapy. Current knowledge of proton RBE variation with respect to dose, biological endpoint, and physics parameters will be reviewed. Further, the clinical relevance of these variations will be discussed. Recent work focused on improving simulations of radiation physics and biological response in proton and carbon ion therapy will also be presented. Finally, relevant biology research and areas of research needs will be highlighted, including the dependence of RBE on genetic factors including status of DNA repair pathways, the sensitivity of cancer stem-like cells to charged particles, the role of charged particles in hypoxic tumors, and the importance of fractionation effects. In addition to the physical advantages of protons and more massive ions over photons, the future application of biologically optimized treatment plans and their potential to provide higher levels of local tumor control and improved normal tissue sparing will be discussed. Learning Objectives: To assess whether the current practice of a constant RBE of 1.1 should be revised or maintained in proton therapy and to evaluate the potential clinical consequences of delivering RBE-weighted dose distributions based on variable RBE To review current research on biological models used to predict the increased biological effectiveness of proton and carbon ions to help move towards a practical understanding and implementation of biological optimization in particle therapy To discuss potential differences in biological mechanisms between photons and charged particles (light and heavy ions) that could impact clinical cancer therapy H. Paganetti, NCI U19 CA21239D. Grosshans, Our research is supported by the NCIK. Held, Funding Support: National Cancer Institute of the National Institutes of Health, USA, under Award Number R21CA182259 and Federal Share of Program Income Earned by Massachusetts General Hospital on C06CA059267, Proton Therapy Research and Treatment Center.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TH-B-207B-00: Pediatric Image Quality Optimization

This imaging educational program will focus on solutions to common pediatric image quality optimization challenges. The speakers will present collective knowledge on best practices in pediatric imaging from their experience at dedicated children’s hospitals. One of the most commonly encountered pediatric imaging requirements for the non-specialist hospital is pediatric CT in the emergency room setting. Thus, this educational program will begin with optimization of pediatric CT in the emergency department. Though pediatric cardiovascular MRI may be less common in the non-specialist hospitals, low pediatric volumes and unique cardiovascular anatomy make optimization of these techniques difficult. Therefore, our second speaker will review best practices in pediatric cardiovascular MRI based on experiences from a children’s hospital with a large volume of cardiac patients. Learning Objectives: To learn techniques for optimizing radiation dose and image quality for CT of children in the emergency room setting. To learn solutions for consistently high quality cardiovascular MRI of children.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TH-B-207B-01: Optimizing Pediatric CT in the Emergency Department

This imaging educational program will focus on solutions to common pediatric image quality optimization challenges. The speakers will present collective knowledge on best practices in pediatric imaging from their experience at dedicated children’s hospitals. One of the most commonly encountered pediatric imaging requirements for the non-specialist hospital is pediatric CT in the emergency room setting. Thus, this educational program will begin with optimization of pediatric CT in the emergency department. Though pediatric cardiovascular MRI may be less common in the non-specialist hospitals, low pediatric volumes and unique cardiovascular anatomy make optimization of these techniques difficult. Therefore, our second speaker will review best practices in pediatric cardiovascular MRI based on experiences from a children’s hospital with a large volume of cardiac patients. Learning Objectives: To learn techniques for optimizing radiation dose and image quality for CT of children in the emergency room setting. To learn solutions for consistently high quality cardiovascular MRI of children.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TH-B-207B-02: Optimizing Pediatic Cardiovascular MRI

This imaging educational program will focus on solutions to common pediatric image quality optimization challenges. The speakers will present collective knowledge on best practices in pediatric imaging from their experience at dedicated children’s hospitals. One of the most commonly encountered pediatric imaging requirements for the non-specialist hospital is pediatric CT in the emergency room setting. Thus, this educational program will begin with optimization of pediatric CT in the emergency department. Though pediatric cardiovascular MRI may be less common in the non-specialist hospitals, low pediatric volumes and unique cardiovascular anatomy make optimization of these techniques difficult. Therefore, our second speaker will review best practices in pediatric cardiovascular MRI based on experiences from a children’s hospital with a large volume of cardiac patients. Learning Objectives: To learn techniques for optimizing radiation dose and image quality for CT of children in the emergency room setting. To learn solutions for consistently high quality cardiovascular MRI of children.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TH-CD-209-04: Fuzzy Robust Optimization in Intensity-Modulated Proton Therapy Planning to Account for Range and Patient Setup Uncertainties

Purpose: We propose to apply a robust optimization model based on fuzzy-logic constraints in the intensity-modulated proton therapy (IMPT) planning subject to range and patient setup uncertainties. The purpose is to ensure the plan robustness under uncertainty and obtain the best trade-off between tumor dose coverage and organ-at-risk(OAR) sparing. Methods: Two IMPT plans were generated for 3 head-and-neck cancer patients: one used the planning target volume(PTV) method; the other used the fuzzy robust optimization method. In the latter method, nine dose distributions were computed - the nominal one and one each for ±3mm setup uncertainties along three cardinal axes and for ±3.5% range uncertainty. For tumors, these nine dose distributions were explicitly controlled by adding hard constraints with adjustable parameters. For OARs, fuzzy constraints that allow the dose to vary within a certain range were used so that the tumor dose distribution was guaranteed by minimum compromise of that of OARs. We rendered this model tractable by converting the fuzzy constraints to linear constraints. The plan quality was evaluated using dose-volume histogram(DVH) indices such as tumor dose coverage(D95%), homogeneity(D5%-D95%), plan robustness(DVH band at D95%), and OAR sparing like D1% of brain and D1% of brainstem. Results: Our model could yield clinically acceptable plans. The fuzzy-logic robust optimization method produced IMPT plans with comparable target dose coverage and homogeneity compared to the PTV method(unit: Gy[RBE]; average[min, max])(CTV D95%: 59 [52.7, 63.5] vs 53.5[46.4, 60.1], CTV D5% - D95%: 11.1[5.3, 18.6] vs 14.4[9.2, 21.5]). It also generated more robust plans(CTV DVH band at D95%: 3.8[1.2, 5.6] vs 11.5[6.2, 16.7]). The parameters of tumor constraints could be adjusted to control the tradeoff between tumor coverage and OAR sparing. Conclusion: The fuzzy-logic robust optimization generates superior IMPT with minimum compromise of OAR sparing. This research was supported by the National Cancer Institute Career Developmental Award K25CA168984, by the Fraternal Order of Eagles Cancer Research Fund Career Development Award, by The Lawrence W. and Marilyn W. Matteson Fund for Cancer Research, by Mayo Arizona State University Seed Grant, and by The Kemper Marley Foundation. eRA Person ID(s) for the Principal Investigator: 11017970 (Research Supported by National Institutes of Health)

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TH-EF-BRB-02: Feasibility of Optimization for Dynamic Trajectory Radiotherapy

Purpose: Over the last years, volumetric modulated arc therapy (VMAT) has been widely introduced into clinical routine using a coplanar delivery technique. However, VMAT might be improved by including dynamic couch and collimator rotations, leading to dynamic trajectory radiotherapy (DTRT). In this work the feasibility and the potential benefit of DTRT was investigated. Methods: A general framework for the optimization was developed using the Eclipse Scripting Research Application Programming Interface (ESRAPI). Based on contoured target and organs at risk (OARs), the structures are extracted using the ESRAPI. Sampling potential beam directions, regularly distributed on a sphere using a Fibanocci-lattice, the fractional volume-overlap of each OAR and the target is determined and used to establish dynamic gantry-couch movements. Then, for each gantry-couch track the most suitable collimator angle is determined for each control point by optimizing the area between the MLC leaves and the target contour. The resulting dynamic trajectories are used as input to perform the optimization using a research version of the VMAT optimization algorithm and the ESRAPI. The feasibility of this procedure was tested for a clinically motivated head and neck case. Resulting dose distributions for the VMAT plan and for the dynamic trajectory treatment plan were compared based on DVH-parameters. Results: While the DVH for the target is virtually preserved, improvements in maximum dose for the DTRT plan were achieved for all OARs except for the inner-ear, where maximum dose remains the same. The major improvements in maximum dose were 6.5% of the prescribed dose (66 Gy) for the parotid and 5.5% for the myelon and the eye. Conclusion: The result of this work suggests that DTRT has a great potential to reduce dose to OARs with similar target coverage when compared to conventional VMAT treatment plans. This work was supported by Varian Medical Systems. This work was supported by Varian Medical Systems.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Poster - 19: Investigation of Electron Reference Dosimetry Based on Optimal Chamber Shift

An addendum/revision to AAPM TG-51 electron reference dosimetry is highly expected to meet the clinical requirement with the increasing usage of new ion chambers not covered in TG-51. A recent study, Med. Phys. 41, 111701, proposed a new fitting equation for the beam quality conversion factor k’{sub Q} to a wide spectrum of chambers. In the study, an optimal Effective Point of Measurement (EPOM) from Monte Carlo calculations was recommended and the fitting parameters to k’{sub Q} was based on it. We investigated the absolute dose obtained based on the optimal EPOM method and the original TG-51 method with k’{sub R50} determined differently. The results showed that using the Markus curve is a better choice than the well-guarded chamber fitting for an IBA PPC-05 parallel plate chamber if we need to strictly follow the AAPM TG-51 protocol. We also examined the usage of the new fitting equation with measurement performed at the physical EPOM, instead of the optimal EPOM. The former is more readily determined and more practical in clinics. Our study indicated that the k’{sub Q} fitting based on the optimal EPOM can be used to measurement at the physical EPOM with no significant clinical impact. The inclusion of Farmer chamber gradient correction P{sub gr} in k’{sub Q}, as in the mentioned study, asks for the precise positioning of chamber center at dref. It is not recommended in clinics to avoid over-correction for low electron energies, especially for an institute having matching Linacs implemented.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2011

On time-optimal NMR control of states of qutrits represented by quadrupole nuclei with the spin I = 1

Elementary logical operators (selective rotation, Fourier transform, controllable phase shift, and SUM gate) are considered for a quantum computer based on three-level systems (qutrits) represented by nuclear spins I = 1 under nuclear magnetic resonance conditions. The computer simulation of the realization of these operators by means of simple and composite selective radiofrequency (RF) pulses and optimized RF pulses is performed. The time dependence of the amplitude of last pulses is found by numerical optimization at different durations. Two variants are proposed for realization of a two-qutrit SUM gate by using one-qutrit or two-qutrit optimized RF pulses. The calculated time dependences of realization errors were used to study the time optimality of different methods for obtaining gates, proposed earlier and in this paper. The advantages and disadvantages of each of the methods are evaluated for different values of physical parameters.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Configuration analysis and optimization on multipolar Galatea trap

Multipolar Galatea magnetic trap simulation model was established with the finite element simulation software COMSOL Multiphysics. Analyses about the magnetic section configuration show that better magnetic configuration should make more plasma stay in the weak magnetic field rather than the annular magnetic shell field. Then an optimization model was established with axial electromagnetic force, weak magnetic field area and average magnetic mirror ratio as the optimization goals and with the currents of myxines as design variables. Select appropriate weight coefficients and get optimization results by applying genetic algorithm. Results show that the superiority of the target value of typical application parameters, including the average magnetic mirror can reduce more than 5%, the weak magnetic field area can increase at least 65%, at the same time, axial electromagnetic force acting on the outer myxines can be reduced to less than 50 N. Finally, the results were proved by COMSOL Multiphysics and the results proved the optimized magnetic trap configuration with more plasma in the weak magnetic field can reduce the plasma diffusion velocity and is more conducive for the constraint of plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2014

An Optimization-based Atomistic-to-Continuum Coupling Method

In this paper, we present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the latter as a constrained optimization problem with virtual Dirichlet controls on the interfaces between the atomistic and continuum subdomains. The optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains, while the atomistic and continuum force balance equations provide the constraints. Separation, rather then blending of the atomistic and continuum problems, and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing AtC formulations. Finally, we present and analyze the method in the context of a one-dimensional chain of atoms modeled using a linearized two-body potential with next-nearest neighbor interactions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Optimization Methods for Large-Scale Machine Learning

Here this paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Error Estimates for the Optimal Control of a Parabolic Fractional PDE

In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration

This paper proposes novel noise-free Bayesian optimization strategies that rely on a random exploration step to enhance the accuracy of Gaussian process surrogate models. The new algorithms retain the ease of implementation of the classical GP-UCB algorithm, but the additional random exploration step accelerates their convergence, nearly achieving the optimal convergence rate. Furthermore, to facilitate Bayesian inference with intractable likelihoods, we propose to utilize optimization iterates for maximum a posteriori estimation to build a Gaussian process surrogate model for the unnormalized log-posterior density. We provide bounds for the Hellinger distance between the true and the approximate posterior distributions in terms of the number of design points. We demonstrate the effectiveness of our Bayesian optimization algorithms in nonconvex benchmark objective functions, in a machine learning hyperparameter tuning problem, and in a black-box engineering design problem. The effectiveness of our posterior approximation approach is demonstrated in two Bayesian inference problems for parameters of dynamical systems.

Bayesian inference↗
DOE OSTI2026

Optimal Transfer Operators in Algebraic Two-Level Methods for Nonsymmetric and Indefinite Problems

Consider an algebraic two-level method applied to the 𝑛-dimensional linear system 𝐴⁢𝒙 = 𝒃 using fine-space preconditioner (i.e., “relaxation” or “smoother”) 𝑀, with 𝑀 ≈ 𝐴, restriction and interpolation 𝑅 and 𝑃, and algebraic coarse-space operator 𝐴 𝑐 : = 𝑅 ∗ ⁢𝐴⁢𝑃. Then, what are the best possible transfer operators 𝑅 and 𝑃 of a given dimension 𝑛 𝑐 < 𝑛? Brannick et al. [12] showed that when 𝐴 and 𝑀 are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the 𝑛 𝑐 smallest generalized eigenvectors of the matrix pencil (𝐴, 𝑀). Recently, in Ali et al. [5] we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of (𝐴, 𝑀) and defining corresponding nonsymmetric transfer operators {𝑅#, 𝑃#}. Tight convergence bounds for {𝑅#, 𝑃#} are derived in spectral radius, as well as a proof of pseudo-optimality. Note, {𝑅#, 𝑃#} are typically complex valued, which is not practical for real-valued problems. Here, in this work, we build on [5], first characterizing all inner products in which the coarse-space correction defined by {𝑅#, 𝑃#} is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators {𝑅#, 𝑃#} are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from [12]. Finally, we show how to construct optimal, real-valued transfer operators in the case of that 𝐴 and 𝑀 are real valued, but are not HPD. Numerical examples arising from a discretized advection-reaction equation, wave-equation, and Stokes equations are used to verify and illustrate the theory.

97 MATHEMATICS AND COMPUTING↗