• 제목/요약/키워드: $L_p error estimates

검색결과 11건 처리시간 0.023초

Lp and W1,p Error Estimates for First Order GDM on One-Dimensional Elliptic and Parabolic Problems

  • Gong, Jing;Li, Qian
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제4권2호
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    • pp.41-57
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    • 2000
  • In this paper, we consider first order generalized difference scheme for the two-point boundary value problem and one-dimensional second order parabolic type problem. The optimal error estimates in $L_p$ and $W^{1,p}$ ($2{\leq}p{\leq}{\infty}$) as well as some superconvergence estimates in $W^{1,p}$ ($2{\leq}p{\leq}{\infty}$) are obtained. The main results in this paper perfect the theory of GDM.

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Lp error estimates and superconvergence for finite element approximations for nonlinear parabolic problems

  • LI, QIAN;DU, HONGWEI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제4권1호
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    • pp.67-77
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    • 2000
  • In this paper we consider finite element mathods for nonlinear parabolic problems defined in ${\Omega}{\subset}R^d$ ($d{\leq}4$). A new initial approximation is taken. Optimal order error estimates in $L_p$ for $2{\leq}p{\leq}{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2{\leq}q{\leq}{\infty}$ are demonstrated as well.

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[ $L_p$ ] ERROR ESTIMATES AND SUPERCONVERGENCE FOR FINITE ELEMENT APPROXIMATIONS FOR NONLINEAR HYPERBOLIC INTEGRO-DIFFERENTIAL PROBLEMS

  • Li, Qian;Jian, Jinfeng;Shen, Wanfang
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제9권1호
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    • pp.17-29
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    • 2005
  • In this paper we consider finite element methods for nonlinear hyperbolic integro-differential problems defined in ${\Omega}\;{\subset}\;R^d(d\;{\leq}\;4)$. A new initial approximation of $u_t(0)$ is taken. Optimal order error estimates in $L_p$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are demonstrated as well.

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GENERALIZED DIFFERENCE METHODS FOR ONE-DIMENSIONAL VISCOELASTIC PROBLEMS

  • Li, Huanrong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제9권2호
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    • pp.55-64
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    • 2005
  • In this paper, generalized difference methods(GDM) for one-dimensional viscoelastic problems are proposed and analyzed. The new initial values are given in the generalized difference scheme, so we obtain optimal error estimates in $L^p$ and $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ between the GDM solution and the generalized Ritz-Volterra projection of the exact solution.

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FINITE VOLUME ELEMENT METHODS FOR NONLINEAR PARABOLIC INTEGRODIFFERENTIAL PROBLEMS

  • Li, Huanrong;Li, Qian
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제7권2호
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    • pp.35-49
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    • 2003
  • In this paper, finite volume element methods for nonlinear parabolic integrodifferential problems are proposed and analyzed. The optimal error estimates in $L^p\;and\;W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ are obtained. The main results in this paper perfect the theory of FVE methods.

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FINITE VOLUME ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

  • LI, QIAN;LIU, ZHONGYAN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제6권2호
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    • pp.85-97
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    • 2002
  • In this paper, finite volume element methods for nonlinear parabolic problems are proposed and analyzed. Optimal order error estimates in $W^{1,p}$ and $L_p$ are derived for $2{\leq}p{\leq}{\infty}$. In addition, superconvergence for the error between the approximation solution and the generalized elliptic projection of the exact solution (or and the finite element solution) is also obtained.

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건강한 한국인에서 미다졸람 집단약동학 분석: CYP3A 매개 약물상호작용 평가 (Population Pharmacokinetics of Midazolam in Healthy Koreans: Effect of Cytochrome P450 3A-mediated Drug-drug Interaction)

  • 신광희
    • 한국임상약학회지
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    • 제26권4호
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    • pp.312-317
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    • 2016
  • Objective: Midazolam is mainly metabolized by cytochrome P450 (CYP) 3A. Inhibition or induction of CYP3A can affect the pharmacological activity of midazolam. The aims of this study were to develop a population pharmacokinetic (PK) model and evaluate the effect of CYP3A-mediated interactions among ketoconazole, rifampicin, and midazolam. Methods: Three-treatment, three-period, crossover study was conducted in 24 healthy male subjects. Each subject received 1 mg midazolam (control), 1 mg midazolam after pretreatment with 400 mg ketoconazole once daily for 4 days (CYP3A inhibition phase), and 2.5 mg midazolam after pretreatment with 600 mg rifampicin once daily for 10 days (CYP3A induction phase). The population PK analysis was performed using a nonlinear mixed effect model ($NONMEM^{(R)}$ 7.2) based on plasma midazolam concentrations. The PK model was developed, and the first-order conditional estimation with interaction was applied for the model run. A three-compartment model with first-order elimination described the PK. The influence of ketoconazole and rifampicin, CYP3A5 genotype, and demographic characteristics on PK parameters was examined. Goodness-of-fit (GOF) diagnostics and visual predictive checks, as well as bootstrap were used to evaluate the adequacy of the model fit and predictions. Results: Twenty-four subjects contributed to 900 midazolam concentrations. The final parameter estimates (% relative standard error, RSE) were as follows; clearance (CL), 31.8 L/h (6.0%); inter-compartmental clearance (Q) 2, 36.4 L/h (9.7%); Q3, 7.37 L/h (12.0%), volume of distribution (V) 1, 70.7 L (3.6%), V2, 32.9 L (8.8%); and V3, 44.4 L (6.7%). The midazolam CL decreased and increased to 32.5 and 199.9% in the inhibition and induction phases, respectively, compared to that in control phase. Conclusion: A PK model for midazolam co-treatment with ketoconazole and rifampicin was developed using data of healthy volunteers, and the subject's CYP3A status influenced the midazolam PK parameters. Therefore, a population PK model with enzyme-mediated drug interactions may be useful for quantitatively predicting PK alterations.

Evaluation of the Degenerative Changes of the Distal Intervertebral Discs after Internal Fixation Surgery in Adolescent Idiopathic Scoliosis

  • Dehnokhalaji, Morteza;Golbakhsh, Mohammad Reza;Siavashi, Babak;Talebian, Parham;Javidmehr, Sina;Bozorgmanesh, Mohammadreza
    • Asian Spine Journal
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    • 제12권6호
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    • pp.1060-1068
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    • 2018
  • Study Design: Retrospective study. Purpose: Lumbar intervertebral disc degeneration is an important cause of low back pain. Overview of Literature: Spinal fusion is often reported to have a good course for adolescent idiopathic scoliosis (AIS). However, many studies have reported that adjacent segment degeneration is accelerated after lumbar spinal fusion. Radiography is a simple method used to evaluate the orientation of the vertebral column. magnetic resonance imaging (MRI) is the method most often used to specifically evaluate intervertebral disc degeneration. The Pfirrmann classification is a well-known method used to evaluate degenerative lumbar disease. After spinal fusion, an increase in stress, excess mobility, increased intra-disc pressure, and posterior displacement of the axis of motion have been observed in the adjacent segments. Methods: we retrospectively secured and analyzed the data of 15 patients (four boys and 11 girls) with AIS who underwent a spinal fusion surgery. We studied the full-length view of the spine (anterior-posterior and lateral) from the X-ray and MRI obtained from all patients before surgery. Postoperatively, another full-length spine X-ray and lumbosacral MRI were obtained from all participants. Then, pelvic tilt, sacral slope, curve correction, and fused and free segments before and after surgery were calculated based on X-ray studies. MRI images were used to estimate the degree to which intervertebral discs were degenerated using Pfirrmann grading system. Pfirrmann grade before and after surgery were compared with Wilcoxon signed rank test. While analyzing the contribution of potential risk factors for the post-spinal fusion Pfirrmann grade of disc degeneration, we used generalized linear models with robust standard error estimates to account for intraclass correlation that may have been present between discs of the same patient. Results: The mean age of the participant was 14 years, and the mean curvature before and after surgery were 67.8 and 23.8, respectively (p<0.05). During the median follow-up of 5 years, the mean degree of the disc degeneration significantly increased in all patients after surgery (p<0.05) with a Pfirrmann grade of 1 and 2.8 in the L2-L3 before and after surgery, respectively. The corresponding figures at L3-L4, L4-L5, and L5-S1 levels were 1.28 and 2.43, 1.07 and 2.35, and 1 and 2.33, respectively. The lower was the number of free discs below the fusion level, the higher was the Pfirrmann grade of degeneration (p<0.001). Conversely, the higher was the number of the discs fused together, the higher was the Pfirrmann grade. Conclusions: we observed that the disc degeneration aggravated after spinal fusion for scoliosis. While the degree of degeneration as measured by Pfirrmann grade was directly correlated by the number of fused segments, it was negatively correlated with the number of discs that remained free below the lowermost level of the fusion.

Genetic Evaluation of First Lactation Traits in Sahiwal Cattle Using Restricted Maximum Likelihood Technique

  • Choudhary, V.;Kothekar, M.D.;Raheja, K.L.;Kasturiwale, N.N.;Khire, D.W.;Kumar, P.
    • Asian-Australasian Journal of Animal Sciences
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    • 제16권5호
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    • pp.639-643
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    • 2003
  • The data on 283 Sahiwal cows, sired by 16 bulls, maintained at Cattle Breeding Farm of Nagpur Veterinary College and Dairy Farm of Agricultural College, Nagpur, were considered for the estimation of genetic parameters. Variance and covariance estimates of first lactation traits were obtained using restricted maximum likelihood technique (REML). When first lactation milk yield (FLMY), first lactation length (FLL) and average daily yield (ADY) traits were considered for REML analysis, the heritabilities were $0.184{\pm}0.146$, $0.132{\pm}0.131$ and $0.141{\pm}0.133$, respectively. While, genetic and phenotypic correlations between them were medium to high except phenotypic correlations between FLL and ADY (-0.025). REML procedure considering FLMY, age at first calving (AFC) and first service period (FSP) combination exhibits heritabilities as $0.274{\pm}0.173$, $0.506{\pm}0.233$ and $0.274{\pm}0.172$, respectively. Genetic correlations were $-0.120{\pm}0.376$, $0.225{\pm}0.423$ and $0.365{\pm}0.331$ between FLMY and AFC, FLMY and FSP, AFC and FSP, respectively. Phenotypic correlations were 0.057, 0.289 and 0.123, respectively. Considering all five traits REML combination heritabilities estimated were $0.238{\pm}0.162$, $0.160{\pm}0.139$, $0.136{\pm}0.132$, $0.409{\pm}0.209$ and $0.259{\pm}0.168$ for FLMY, FLL, ADY, AFC and FSP, respectively. The genetic correlations were positive except FLMY and AFC. The phenotypic correlations were also positive except FLL and ADY, ADY and FSP. Almost all estimates were associated with high standard error.

상엽 수확고 측정에 관한 연구 - 제1보 엽면적에 의한 상엽량의 순서 - (Studies on the Estimation of Leaf Production in Mulberry Trees 1. Estimation of the leaf production by leaf area determination)

  • 한경수;장권열;안정준
    • 한국잠사곤충학회지
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    • 제8권
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    • pp.11-25
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    • 1968
  • 상엽의 수엽량을 견적하여 잠아의 사육량을 정하는데 이용하기 위하여 개량서반, 일지뢰, 노상, 수원상4호의 4품종의 상수를 재료로 지조장, 지조직경, 절수, 지조종중량, 지조중, 엽중 그리고 엽면적의 7개형질에 대한 조사 실험을 하였다. 조사항목별 평균치, 분산, 표준편차 등을 구하고 지조장과 수엽량, 지조직경과 수엽량, 절수와 수엽량 그리고 엽면적과 이들 형질과의 관계를 보고 수엽량을 견적할 수 있도록 수식을 유도하여 보았다. 실험결과를 요약하면 다음과 같다. 1. 지조장, 지조직경, 절수의 평균치, 분산, 표준편차 그리고 표준오차는 제1표와 같고 이들 평균치와 지조총중량 등을 보면 일지뢰가 제1 크고 개량서반이 다음이며 수원상4호, 노상의 순위로 되어 있으나 단위지당 엽면적은 일지뢰, 노상, 수원제4호, 개량서반의 순이고 1엽당 엽면적은 수원제4호, 노상, 일지뢰,개량서반의 순으로 되어있다. 2. 지조수에 의한 수엽량의 견적을 위하여는 y$_1$=a$_1$X$_1$$\times$p$_1$.......식\circled1이 유도되었다. 이 때 a$_1$은 지당엽량, X$_1$은 주당지조수이고 p$_1$은 반당주수를 나타낸다. 3. 지조장에 의한 엽량의 견적에는 y$_2$=(a$_2$$\pm$S. E.$\times$X$_2$)+p$_1$........식\circled2이 유도되었다. 이 때 a$_2$는 지조장 1m당 엽량, S. E.는 동표준오차. X$_2$는 주당지조장이고 p$_1$은 전기 반당주수를 나타낸다. 4. 조지직경에 의한 수엽량의 견적에는 y$_3$=(a$_3$$\pm$S. E$\times$X$_3$)$\times$p$_1$......식\circled3이 유도된다. 이 때 a$_3$는 지조직경, lcm당 엽중, X$_3$는 주당 지조총직경을 나타낸다. 5. 절수에 의한 수엽량의 견적에는 y$_4$=(a$_4$$\pm$S. E.$\times$X$_4$)$\times$p$_1$........식\circled4가 유도된다. 이 때 a$_4$는절당엽중, X$_4$는 주당총절수를 나타낸다. 6. 지조장과 엽면적에 의한 수엽량의 견적에는 y$_{5}$= {(a$_{5}$$\pm$S. E.$\times$X$_2$)Kv}$\times$p$_1$......식\circled5 이 때 a$_{5}$는 지조 1m조 당엽면적, S. E.는 동 표준오차, X$_2$는 주당 지조장, Kv는엽면적 l00$\textrm{cm}^2$당 중량을 나타낸다. 7. 엽면적과 지조장에 의한 수엽량의 견적에는 y$_{6}$={(X$_2$$\div$a$_{6}$$\pm$S. E.)}$\times$K$_{v}$ $\times$p$_1$......식\circled6이 유도된다. 이 때 X$_2$는 주당 지조장, a$_{6}$는 엽면적 l00$\textrm{cm}^2$당 지조장이다. 8. 지조직경과 엽면적에 의한 수엽량의 견적에는 y$_{7}$= {(a$_{7}$$\pm$S. E. $\times$X$_3$)}$\times$K$_{v}$ $\times$p$_1$......식\circled7이 유도되고 이 때 a$_{7}$는 지조직경 lcm당 엽면적이고 X$_3$는주당 총직경을 나타낸다. 9. 엽면적과 지조직경에 의한 수엽량의 견적에는 y$_{8}$= {(X$_3$$\div$a$_{8}$$\pm$S. E.)}$\times$K$_{v}$ $\times$p$_1$......식\circled8이 유도되고 이 때 a$_{8}$는 엽면적 l00$\textrm{cm}^2$당 지조직경을 나타내며 X$_3$, $K_{v}$ , p$_1$는전항과 같다. 10. 절수와 엽면적에 의한 수엽량의 견적에는 y$_{9}$= {(a$_{9}$$\pm$S. E.$\times$X$_4$)$\times$K$_{v}$ }$\times$p$_1$.........식\circled9이 유도되고 a$_{9}$는 절당 엽면적, X$_4$는 주당총절수를 나타낸다. 11. 엽면적과 절수에 의한 수엽량의 견적에는 y$_{10}$= {(X$_4$$\div$a$_{10}$$\div$S. E.)$\times$K$_{v}$ }$\times$p$_1$.........식\circled10이 유도되고 a$_{10}$은 엽면적 l00$\textrm{cm}^2$당절수를 나타내며 X$_4$, $K_{v}$ , p$_1$ 등은 전항에 기술한 바와 같다. 이상 여러가지 식에 의해서 수엽량의 예상량을 견적하여 본 바 지조장에 의하는 견적치가 절수에 의하는 것보다 절수에 의해서 수엽량을 견적하는 것이 지조직경에 의하여 수엽량을 견적한 것보다 편차가 적고 이들 방법에 의하는 것보다 엽면적을 측정하여 지조장과 엽면적에 의하여 수엽량을 견적하는 것이 편차가 적고 다음이 절수와 엽면적에 의하여 견적하는 것이 나았다. 지조직경을 측정하는 것은 오차가 많고 여기에 엽면적을 측정하는 것을 가미한다고 하여도 지조직경에 의한 수엽량을 견적하는 것은 편차가 심하고 적정한 방법이라고는 할 수없음을 일았다. 비교적 측정하기 쉬운 지조장과 지당 중앙부의 엽면적을 간이측정법에 의하여 측정하여 상지조를 벌채하지 아니하여도 반당수엽량을 견적하는 것은 잠아의 사육량을 정하는데 또는 노력면에 있어서 도움이 되는 의의 깊은 일이라사료된다.

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