Monotone Version Research Articles

Iterative image reconstruction algorithm analysis for optical CT radiochromic gel dosimetry

Background. Modern radiation therapy technologies aim to enhance radiation dose precision to the tumor and utilize hypofractionated treatment regimens. Verifying the dose distributions associated with these advanced radiation therapy treatments remains an active research area due to the complexity of delivery systems and the lack of suitable three-dimensional dosimetry tools. Gel dosimeters are a potential tool for measuring these complex dose distributions. A prototype tabletop solid-tank fan-beam optical CT scanner for readout of gel dosimeters was recently developed. This scanner does not have a straight raypath from source to detector, thus images cannot be reconstructed using filtered backprojection (FBP) and iterative techniques are required. Purpose. To compare a subset of the top performing algorithms in terms of image quality and quantitatively determine the optimal algorithm while accounting for refraction within the optical CT system. The following algorithms were compared: Landweber, superiorized Landweber with the fast gradient projection perturbation routine (S-LAND-FGP), the fast iterative shrinkage/thresholding algorithm with total variation penalty term (FISTA-TV), a monotone version of FISTA-TV (MFISTA-TV), superiorized conjugate gradient with the nonascending perturbation routine (S-CG-NA), superiorized conjugate gradient with the fast gradient projection perturbation routine (S-CG-FGP), superiorized conjugate gradient with with two iterations of CG performed on the current iterate and the nonascending perturbation routine (S-CG-2-NA). Methods. A ray tracing simulator was developed to track the path of light rays as they traverse the different mediums of the optical CT scanner. Two clinical phantoms and several synthetic phantoms were produced and used to evaluate the reconstruction techniques under known conditions. Reconstructed images were analyzed in terms of spatial resolution, signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), signal non-uniformity (SNU), mean relative difference (MRD) and reconstruction time. We developed an image quality based method to find the optimal stopping iteration window for each algorithm. Imaging data from the prototype optical CT scanner was reconstructed and analysed to determine the optimal algorithm for this application. Results. The optimal algorithms found through the quantitative scoring metric were FISTA-TV and S-CG-2-NA. MFISTA-TV was found to behave almost identically to FISTA-TV however MFISTA-TV was unable to resolve some of the synthetic phantoms. S-CG-NA showed extreme fluctuations in the SNR and CNR values. S-CG-FGP had large fluctuations in the SNR and CNR values and the algorithm has less noise reduction than FISTA-TV and worse spatial resolution than S-CG-2-NA. S-LAND-FGP had many of the same characteristics as FISTA-TV; high noise reduction and stability from over iterating. However, S-LAND-FGP has worse SNR, CNR and SNU values as well as longer reconstruction time. S-CG-2-NA has superior spatial resolution to all algorithms while still maintaining good noise reduction and is uniquely stable from over iterating. Conclusions. Both optimal algorithms (FISTA-TV and S-CG-2-NA) are stable from over iterating and have excellent edge detection with ESF MTF 50% values of 1.266 mm−1 and 0.992 mm−1. FISTA-TV had the greatest noise reduction with SNR, CNR and SNU values of 424, 434 and 0.91 × 10−4, respectively. However, low spatial resolution makes FISTA-TV only viable for large field dosimetry. S-CG-2-NA has better spatial resolution than FISTA-TV with PSF and LSF MTF 50% values of 1.581 mm−1 and 0.738 mm−1, but less noise reduction. S-CG-2-NA still maintains good SNR, CNR, and SNU values of 168, 158 and 1.13 × 10−4, respectively. Thus, S-CG-2-NA is a well rounded reconstruction algorithm that would be the preferable choice for small field dosimetry.

Effects of Adjusting for Instrumental Variables on the Bias and Precision of Propensity Score Weighted Estimators: Analysis Under Complete, Near, and No Positivity Violations.

To demonstrate that using an instrumental variable (IV) with monotonicity reduces the accuracy of propensity score (PS) weighted estimators for the average treatment effect (ATE). Monotonicity in the relationship between a binary IV and a binary treatment variable is an important assumption to identify the ATE for compliers who would only take treatment when encouraged by the IV. We perform theoretical and numerical investigations to study the impact of using the IV that satisfies monotonicity on the PS of treatment in terms of the positivity assumption, which requires that the PS be strictly between 0 and 1, and the accuracy of PS weighted estimators. Two versions of monotonicity that result in one-sided or two-sided noncompliance are considered. The PS adjusting for the IV always violates the positivity assumption when noncompliance occurs in one direction (one-sided noncompliance) and is more extreme than without the IV under two-sided noncompliance. These results are valid if the probability of being encouraged to get treatment and the compliance score, the probability of being a complier, are strictly between 0 and 1. Using a binary IV with monotonicity as a covariate for the PS model makes the estimated PSs unnecessarily extreme, reducing the accuracy of the PS weighted estimators.

On monotone pseudocontractive operators and Krasnoselskij iterations in an ordered Hilbert space

The aim of this work is to establish fixed point results in ordered Hilbert spaces for monotone operators with a pseudocontractive property. We state monotone versions of Theorem 12 in [F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 2.1 in [Berinde, Vasile. Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators, Annals of West University of Timisoara-Mathematics and Computer Science, vol. 56, no. 2, 2018, pp. 13–27], as well as, several related results. Further results, in Hilbert spaces without a partial order, are stated too.

Refined ratio monotonicity of the coordinator polynomials of the root lattice of type B n

Abstract Ratio monotonicity, a property stronger than both log-concavity and the spiral property, describes the behavior of the coefficients of many classical polynomials. It is known that the coordinator polynomials of the root lattice of type B n {B}_{n} possess log-concavity. In this paper, we show that the coordinator polynomials of type B n {B}_{n} have a refined version of ratio monotonicity except for the first and the last terms. To our knowledge, this refined version is novel.

Insertion of lattice-valued functions for some topological spaces

In [14] and [8], insertion theorems for normal spaces by lattice-valued functions were obtained which extend the well known Katětov-Tong insertion theorem. As a monotone version, some analogous results for monotonically normal spaces were presented in [16]. In this paper, we consider insertion of lattice-valued functions for some other spaces such as stratifiable spaces and MCP-spaces. The results obtained extend some corresponding insertion theorems formulated with real-valued functions in literature.

Optimal reinsurance and investment strategies for an insurer under monotone mean-variance criterion

This paper considers the optimal investment-reinsurance problem under the monotone mean-variance preference. The monotone mean-variance preference is a monotone version of the classical mean-variance preference. First of all, we reformulate the original problem as a zero-sum stochastic differential game. Secondly, the optimal strategy and the optimal value function for the monotone mean-variance problem are derived by the approach of dynamic programming and the Hamilton-Jacobi-Bellman-Isaacs equation. Thirdly, the efficient frontier is obtained and it is proved that the optimal strategy is an efficient strategy. Finally, the continuous-time monotone capital asset pricing model is derived.

Sharpness of Lenglart’s domination inequality and a sharp monotone version

We prove that the best so far known constant cp=p−p1−p,p∈(0,1) of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor [12]. Motivated by the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy inequality, we also study the domination inequality under an additional monotonicity assumption. In this special case, a constant which stays bounded for p near 1 was proven by Pratelli and Lenglart. We provide the sharp constant for this case.

Equilibrium points and equilibrium sets of some $$\textit{GI}/M/1$$ queues

Queues can be seen as a service facility where quality of service (QoS) is an important measure for the performance of the system. In many cases, the queue implements the optimal admission control (either discounted or average) policy in the presence of holding/congestion cost and revenue collected from admitted customers. In this paper, users offer an arrival rate at stationarity that depends on the QoS they experience. We study the interaction between arriving customers and such a queue under two different QoS measures—the asymptotic rate of the customers lost and the fraction of customers lost in the long run. In particular, we investigate the behaviour of equilibrium points and equilibrium sets associated with this interaction and their interpretations in terms of business cycles. We provide sufficient conditions for existence of equilibrium sets for M/M/1 queue. These conditions further help us to identify the relationship among system parameters for which equilibrium sets exist. Next, we consider $$\textit{GI}/M/1$$ queues and provide a sufficient condition for existence of multiple optimal revenue policies. We then specialize these results to study the equilibrium sets of (i) a D/M/1 queue and (ii) a queue where the arrival rate is locally continuous. The equilibrium behaviour in the latter case is more interesting as there may be multiple equilibrium points or sets. Motivated by such queues, we introduce a weaker version of monotonicity and investigate the existence of generalized equilibrium sets.

Monotone versions of some selection principles II

A space X is monotonically Lindelöf if there is an operator r that assigns to every open cover U a countable open cover r(U) so that r(U) refines U, and r(U) refines r(V) whenever U refines V. We continue the study begun in [5] of monotone versions of Menger, Hurewicz and Rothberger properties.

Expander construction in VNC1

We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system VNC1 (corresponding to the “NC1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in VNC1. This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlák [9].

<b>p</b>-dominant Implementation

We introduce and study a partial implementation notion, which we term p-dominant implementation, where p ∈ [0, 1]^N and N is the number of agents. When agents have interdependent valuation, this notion captures robustness to strategic uncertainty, contrasting with ex-post implementation. The corresponding solution concept generalizes p-dominant equilibrium (Morris et al., 1995) to games with incomplete information so as to be employed in the mechanism design problem. The incentive compatibility (p-DIC) condition requires each agent i find it optimal to truthfully report his/her type as long as the opponents are believed to do so with probability at least pi. In the quasilinear environment with one-dimensional payoff type spaces, we completely characterize p-dominant implementable allocation rules for any p. In the special case of agents with private valuation, we find that p-dominant implementable allocation rule is completely characterized by p-monotonicity of allocation rule. On the other hand, when agents’ valuation is interdependent, the condition is necessary but not sufficient. The complete characterization involves a stronger version of monotonicity of marginal benefit of reporting a higher type) with respect to agent’s true type for any report-indistinguishable reporting strategies of the opponents. Especially, with continuous types and interdependent valuation, the condition requires that any p-dominant implementable allocation rule to be 0-dominant implementable; we show that an allocation rule is p-dominant implementable for p ≪ 1 if and only if it satisfies ex-post monotonicity and additive separability. As an application of our result, we also study constrained efficient allocation that is 0-dominant implementable. We find that when there are two agents, the constrained efficient auction is a discriminatory posted-price mechanism, where the planner determines one and only one agent to whom the planner tries to sell the item.

Combinatorics for General Kinetically Constrained Spin Models

We study the set of possible configurations for a general kinetically constrained model (KCM), a non monotone version of the $\mathcal{U}$-bootstrap percolation cellular automata. We solve a combinatorial question that is a generalization of a problem addressed by Chung, Diaconis and Graham in 2001 for a specific one-dimensional KCM, the East model. Since the general models we consider are in any dimension and lack the oriented character of the East dynamics, we have to follow a completely different route than the one taken by Chung, Diaconis and Graham. Our combinatorial result is used by Mar\^ech\'e, Martinelli and Toninelli to complete the proof of a conjecture put forward by Morris.

Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set

We present new axiomatic characterizations of five classes of TU-values, the classes of the weighted, positively weighted, and multiweighted Shapley values, random order values, and the Harsanyi set. The axiomatizations are given in parallel, i.e., they differ only in one axiom. In conjunction with marginality, a new property, called coalitional differential dependence, is the key that allows us to dispense with additivity. In addition, we propose new axiomatizations of the above five classes, in which, in part new, different versions of monotonicity, associated with the strong monotonicity in Young (1985), are decisive.

Combined DG scheme that maintains increased accuracy in areas of shock waves

A combined scheme of the discontinuous Galerkin method is proposed. This scheme monotonously localizes the fronts of shock waves and simultaneously maintains increased accuracy in the regions of smoothness of the calculated solutions. In this scheme, a non-monotonic version of the third-order DG method is used as the baseline and a monotonic version of this method is used as the internal one, in which a nonlinear correction of numerical flows is used. Tests demonstrating the advantages of the new scheme compared to the standard monotonized variants of the DG method are provided.

Combined DG Scheme That Maintains Increased Accuracy in Shock Wave Areas

A combined scheme for the discontinuous Galerkin (DG) method is proposed. This scheme monotonically localizes the fronts of shock waves and simultaneously maintains increased accuracy in the regions of smoothness of the computed weak solutions. In this scheme, a nonmonotone version of the third-order DG method is used as a baseline scheme and a monotone version of this method is used as an internal scheme, in which a nonlinear correction of numerical fluxes is used. Tests demonstrate the advantages of the new scheme as compared to standard monotonized variants of the DG method.

Non-monotone inexact restoration method for nonlinear programming

This paper deals with a new variant of the inexact restoration method of Fischer and Friedlander (Comput Optim Appl 46:333–346, 2010) for nonlinear programming. We propose an algorithm that replaces the monotone line search performed in the tangent phase by a non-monotone one, using the sharp Lagrangian as merit function. Convergence to feasible points satisfying the convex approximate gradient projection condition is proved under mild assumptions. Numerical results on representative test problems show that the proposed approach outperforms the monotone version when a suitable non-monotone parameter is chosen and is also competitive against other globalization strategies for inexact restoration.

Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

The Swendsen‐Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is conjectured to converge to equilibrium in O(|V|1/4) steps at any (inverse) temperature β, yet there are few results providing o(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, when β < βc(d) where βc(d) denotes the uniqueness/nonuniqueness threshold on infinite d‐regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a monotone version of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW dynamics has mixing time and relaxation time Θ(1) on any graph of maximum degree d for all β < βc(d). Our proof technology can be applied to general monotone Markov chains, including for example the heat‐bath block dynamics, for which we obtain new tight mixing time bounds.

Monotone versions of some selection principles

A space X is monotonically Lindelöf if there is an operator r that assigns to every open cover U a countable open cover r(U) so that r(U) refines U, and r(U) refines r(V) whenever U refines V. We consider the monotone versions of the multicover forms of the Lindelöf property: Menger, Hurewicz, Rothberger.

Repeated implementation: A practical characterization

We characterize the social choice functions that are repeatedly implementable. The necessary and sufficient condition is formulated in terms of the equilibrium payoff set of an associated repeated game. It follows that the implementability of a function can be tested numerically by approximating the equilibrium payoff set. Additionally, with the help of our characterization, we demonstrate that an efficient function is implementable if and only if it satisfies a weaker version of Maskin monotonicity. As an application, we prove that utilitarian social choice functions are implementable by showing that continuation payoff promises effectively play the role of side-payments, which are needed for implementation in static setups.

Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids

We study the gap of discrete spectra of the Laplace operator in 1d for non-uniform meshes by analyzing the corresponding spectral symbol, which allows to show how to design the discretization grid for improving the gap behavior. The main tool is the study of a univariate monotonic version of the spectral symbol, obtained by employing a proper rearrangement via the GLT theory. We treat in detail the case of basic finite-difference approximations. In a second step, we pass to precise approximation schemes, coming from the celebrated Galerkin isogeometric analysis based on B-splines of degree p and global regularity $$C^{p-1}$$ , and finally we address the case of finite-elements with global regularity $$C^0$$ and local polynomial degree p. The surprising result is that the GLT approach allows a unified spectral treatment of the various schemes also in terms of the preservation of the average gap property, which is necessary for the uniform gap property. The analytical results are illustrated by a number of numerical experiments. We conclude by discussing some open problems.