Journal of Computational and Applied Mathematics最新文献

{"title":"Solving nonlinear neutral delay integro-differential equations via general linear methods","authors":"Yuexin Yu","doi":"10.1016/j.cam.2024.116342","DOIUrl":"10.1016/j.cam.2024.116342","url":null,"abstract":"<div><div>General linear methods are adapted for solving nonlinear neutral delay integro-differential equations. The sufficient conditions for the stability and asymptotic stability of <span><math><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>-algebraically stable general linear methods are derived. At last, a numerical test is given to validate the theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Semi-discrete Lagrangian-Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions","authors":"Eduardo Abreu ,&nbsp;Richard De la cruz ,&nbsp;Juan Juajibioy ,&nbsp;Wanderson Lambert","doi":"10.1016/j.cam.2024.116325","DOIUrl":"10.1016/j.cam.2024.116325","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this work, we have expanded upon the (local) semi-discrete Lagrangian-Eulerian method initially introduced in Abreu et al. (2022) to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux, referred to as the nonlocal model: &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;∂&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;∂&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in Abreu et al. (2016), with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian-Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution of Eq. &lt;span&gt;&lt;span&gt;(1)&lt;/span&gt;&lt;/span&gt;. Finally, we present a section of numerical examples to illustrate our results. Finally, we have examined examples discussed in Aggarwal et al. (2015) and Keime","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Collocation method for a functional equation arising in behavioral sciences","authors":"Josefa Caballero ,&nbsp;Hanna Okrasińska-Płociniczak ,&nbsp;Łukasz Płociniczak ,&nbsp;Kishin Sadarangani","doi":"10.1016/j.cam.2024.116343","DOIUrl":"10.1016/j.cam.2024.116343","url":null,"abstract":"<div><div>We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients.</div><div>As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm.</div><div>A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Least squares regression under weak moment conditions","authors":"Hongzhi Tong","doi":"10.1016/j.cam.2024.116336","DOIUrl":"10.1016/j.cam.2024.116336","url":null,"abstract":"<div><div>In this paper we consider the robust regression problem when the output variable may be heavy-tailed. In such scenarios, the traditional least squares regression paradigm is usually thought to be not a good choice as it lacks robustness to outliers. By projecting the outputs onto an adaptive interval, we show the regularized least squares regression can still work well when the conditional distribution satisfies a weak moment condition. Fast convergence rates in various norm are derived by tuning the projection scale parameter and regularization parameter in according with the sample size and the moment condition.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite difference schemes with non polynomial local conservation laws","authors":"Gianluca Frasca-Caccia","doi":"10.1016/j.cam.2024.116330","DOIUrl":"10.1016/j.cam.2024.116330","url":null,"abstract":"<div><div>A new technique has been recently introduced to define finite difference schemes that preserve local conservation laws. So far, this approach has been applied to find parametric families of numerical methods with polynomial conservation laws. This paper extends the existing approach to preserve non polynomial conservation laws. Although the approach is general, the treatment of the nonlinear terms depends on the problem at hand. New parameter depending families of conservative schemes are here introduced for the sine–Gordon equation and a magma equation. Optimal methods in each family are identified by finding values of the parameters that minimize a defect-based approximation of the local error in the time discretization.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A first order dynamical system and its discretization for a class of variational inequalities","authors":"Nguyen Buong","doi":"10.1016/j.cam.2024.116341","DOIUrl":"10.1016/j.cam.2024.116341","url":null,"abstract":"<div><div>In this paper, we study the variational inequality problem over the set of common fixed points of a Lipschitz continuous pseudo-contraction and a finite family of strictly pseudo-contractive operators on a real Hilbert space. We introduce a first order dynamical system in accordance with the Lavrentiev regularization method. The existence and strong convergence with a discretized variant of the trajectory of the dynamical system are proved under some mild conditions. Applications to solving the convex constrained monotone equations and to the LASSO problem with numerical experiments are given for validating our results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Statistical inference on the cumulative distribution function using judgment post stratification","authors":"Mina Azizi Kouhanestani ,&nbsp;Ehsan Zamanzade ,&nbsp;Sareh Goli","doi":"10.1016/j.cam.2024.116340","DOIUrl":"10.1016/j.cam.2024.116340","url":null,"abstract":"<div><div>In this work, we discuss a general class of estimators for the cumulative distribution function (CDF) based on judgment post stratification (JPS) sampling scheme, which includes both empirical and kernel distribution functions. Specifically, we obtain the expectation of the estimators in this class and show that they are asymptotically more efficient than their competitors in simple random sampling (SRS), as long as the rankings are better than random guessing. We find a mild condition that is necessary and sufficient for them to be asymptotically unbiased. We also prove that given the same condition, the estimators in this class are strongly uniformly consistent estimators of the true CDF, and converge in distribution to a normal distribution when the sample size approaches infinity. We then focus on the kernel distribution function (KDF) in the JPS design and obtain the optimal bandwidth. We next carry out a comprehensive Monte Carlo simulation to compare the performance of the KDF in the JPS design for different choices of sample size, set size, ranking quality, parent distribution, kernel function, as well as both perfect and imperfect rankings set-ups, with its counterpart in the SRS design. We find that the JPS estimator dramatically improves the efficiency of the KDF compared to its SRS competitor across a wide range of the settings. Finally, we apply the described procedure to a real dataset from a medical context to show its usefulness and applicability in practice.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An efficient modified conjugate gradient algorithm under Wolfe conditions with applications in compressive sensing","authors":"Zhibin Zhu ,&nbsp;Jiaqi Huang ,&nbsp;Ying Liu ,&nbsp;Yuehong Ding","doi":"10.1016/j.cam.2024.116335","DOIUrl":"10.1016/j.cam.2024.116335","url":null,"abstract":"<div><div>This paper presents a new modified conjugate gradient (NMCG) algorithm which satisfies the sufficient descent property under any line search for unconstrained optimization problems. We analyze that the algorithm is global convergence under the Wolfe line search. We use the proposed algorithm NMCG to unconstrained optimization problems to prove its effectiveness. Furthermore, we also extend it to solve image restoration and sparse signal recovery problems in compressive sensing, and the results indicate that our algorithm is effective and competitive.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Superconvergent results for fractional Volterra integro-differential equations with non-smooth solutions","authors":"Ruby,&nbsp;Moumita Mandal","doi":"10.1016/j.cam.2024.116337","DOIUrl":"10.1016/j.cam.2024.116337","url":null,"abstract":"<div><div>This article focuses on finding the approximate solutions of fractional Volterra integro-differential equations with non-smooth solutions using the shifted Jacobi spectral Galerkin method (SJSGM) and its iterated version. To deal with the singularity present in the kernel of the transformed weakly singular Volterra integral equation, we convert it into an equivalent weakly singular Fredholm integral equation. We first directly apply our proposed methods to this equivalent transformed equation and obtain improved convergence results by incorporating the singularity of the kernel function into the shifted Jacobi weight function. Further, we introduce a smoothing transformation and discuss the regularity of the transformed solution, and achieve superconvergence results for all <span><math><mrow><mi>γ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. Additionally, we obtain super-convergence results for classical first-order Volterra integro-differential equations. Finally, numerical examples with a comparative study are provided to validate our theoretical results and verify the efficiency of the proposed methods. We show that the convergence rates can be obtained to the desired degree by increasing the value of the smoothing index <span><math><mi>ϱ</mi></math></span> <span><math><mrow><mo>(</mo><mn>1</mn><mo>&lt;</mo><mi>ϱ</mi><mo>∈</mo><mi>N</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>N</mi></math></span> stands for the set of natural numbers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deterministic computation of quantiles in a Lipschitz framework","authors":"Yurun Gu ,&nbsp;Clément Rey","doi":"10.1016/j.cam.2024.116344","DOIUrl":"10.1016/j.cam.2024.116344","url":null,"abstract":"<div><div>In this article, we focus on computing the quantiles of a random variable <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>X</mi></math></span> is a <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span>-valued random variable, <span><math><mrow><mi>d</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, and <span><math><mrow><mi>f</mi><mo>:</mo><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>R</mi></mrow></math></span> is a deterministic Lipschitz function. We are particularly interested in scenarios where the cost of a single function evaluation is high, while the law of <span><math><mi>X</mi></math></span> is assumed to be known. In this context, we propose a deterministic algorithm to compute deterministic lower and upper bounds for the quantile of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> at a given level <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. With a fixed budget of <span><math><mi>N</mi></math></span> function calls, we demonstrate that our algorithm achieves an exponential deterministic convergence rate for <span><math><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span> (<span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>ρ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>) and a polynomial deterministic convergence rate for <span><math><mrow><mi>d</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span> (<span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></mrow></math></span>) and show the optimality of those rates. Furthermore, we design two algorithms, depending on whether the Lipschitz constant of <span><math><mi>f</mi></math></span> is known or unknown.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}