COLLOQUIUM– UL OVERVIEW

Bruce A. Wade

Department of Mathematics

bruce.wade@louisiana.edu

UNIVERSIDAD EAFIT

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Bruce A. Wade, Ph.D.Professor & Head

C.B.I.T. TC/LEQSF Regents ProfessorDepartment of Mathematics

University of Louisiana at Lafayette

SpecialtyNumerical analysis (algorithms & analysis for partial differential equations),computational mathematics (simulations in application, PDE models,optimization, machine learning), interdisciplinary mathematics.

Ph.D., Mathematics, University of Wisconsin-Madison, 1987Adviser: Professor J.C. Strikwerda.

M.A., Mathematics, University of Wisconsin-Madison, 1984

B.S., Mathematics, University of Wisconsin-Madison, 1982

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Chair, Department of Mathematical Sciences, University ofWisconsin–Milwaukee, 2017– 2018

Professor, Department of Mathematical Sciences, University ofWisconsin–Milwaukee, 2006– 2018

Post-doctoral Fellow, Mathematical Sciences Institute, CornellUniversity, Adviser: L. B. Wahlbin, 1987–89

Professor Emeritus, University of Wisconsin– Milwaukee, 2018–

Founder & Director, Center for Industrial Mathematics, University ofWisconsin–Milwaukee, 1998–2012

Profesor Visitante, Departamento de Matemática Aplicada,Universidad de Salamanca, Spain, 2005

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The University of Louisiana

We pride ourselves on the quality of our programs and theinnovativeness of our students and faculty.

We are proud to rank among the nation’s leading programs in:

Biology & Ecology

Computing, informatics, and smart systems development

Nursing and health care systems and support

Programs in environment, energy, and economics

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Academics– The University of Louisiana

Fall 2019 enrollment: 19,403Includes 2,700 graduate students

Over 80 majors and more than 30 graduate programs.Faculty scholars bring real problems and pressing research questionsto the classroom.

College of the ArtsCollege of Business AdministrationCollege of EducationCollege of EngineeringCollege of ScienceGraduate School

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College of Sciences

BiologyChemistryMathematicsPhysicsSchool of Computing and InformaticsSchool of Geosciences

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College of Engineering

Chemical EngineeringCivil EngineeringElectrical & Computer EngineeringIndustrial TechnologyMechanical EngineeringPetroleum EngineeringSystems Engineering

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Department of Mathematics

Applied Math, Pure Math, Statistics, B.S., M.S., Ph.D.Research Active Faculty: 21We have Two unfilled Endowed Chair positions (Math & Statistics);& Two unfilled research faculty positionsInstructors & Teaching-Track Professors: 17Ph.D. students: 46Annual Average # Research Articles Published: 40Annual Math Dept. (UL Overall) Budget : $5.5 million ($250 million)Average Annual # Ph.D. Graduates: 42018 # of Student-Classes Taught in Dept. of Math: 10,935

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Department of Mathematics

Areas of specialty:

Applied Math: PDE (Deng, Vatsala)Applied Math: Computational Mathematics & Numerical Analysis(Ackleh, Li, Wade, Wang)Applied Math: Bio-Mathematics (Ackleh, Browne, Gulbudak,Salceanu, Veprauskas, Wang)Pure Math: Algebra (Birkenmeier, Lynd, Magidin)Pure Math: Topology (Davis, Hackney, Koytcheff)Pure Math: Analysis & Functional Analysis (Ng, Robert-Gonzalez)Statistics: (Kim, Krishnamoorthy, Pal, Sang)

Math Dept. has 4 Endowed Professors, 1 Endowed Chair, 2 openpositions for Endowed Chair

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Department of Mathematics

UL MATH PROFESSIONAL ACTIVITIES REPORT

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Research Groups School of Computing and Informatics

Research Groups

Applied Discrete Geometry LaboratoryBioinformatics LaboratoryBiological Artificial Intelligence Laboratory (BAIL)Computer Architecture and Networks (CAN) LaboratoryHigh Performance Cloud Computing (HPCC) LaboratoryHuman Computer Interaction (HCI) LaboratoryLaboratory for Internet Computing (LINC)Intelligent Systems, Modeling and Simulation (ISMS) LabNetwork Science Research Group (NSRG)Software Research Laboratory (SRL)Virtual Reality LaboratoryVLSI Laboratory

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Research Groups Department of Biology

Systematics, Biogeography & EvolutionMarine BiologyFunctional Morphology & PhysiologyConservation & EcologyMolecular & Cell Biology and MicrobiologyResearch CollaborationsCoastal Water ConsortiumNational Institute for Mathematical and Biological Synthesis(NIMBioS)The National Wetland Research CenterAustralian Institute of Marine SciencesGlobal Change Institute, University of QueenslandSmithsonian Tropical Research InstituteThe Water Institute of the Gulf

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Research Groups Department of Biology

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Research Groups Department of Biology

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Research Groups Department of Biology

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Research Groups UL Applied Math & Stats People

Mathematical Biology

Azmy Ackleh (Ph.D. Univ of Tennessee): Development ofdeterministic and stochastic population models with particularemphasis on age/size structured populations, invasive species,amphibians, phenotypic selection-mutation, and epidemics.Computational methods for PDE arising in biology.Cameron Browne (Ph.D. Univ of Florida): Mathematical Modeling,Differential Equations and Dynamical Systems. Application ofdifferential equations and dynamical systems to modeling populationdynamics and evolution of infectious diseases.Hayriye Gulbudak (Ph.D. Univ of Florida): Interface of DynamicalSystems, Differential Equations, Numerical Analysis, and theirapplication to modeling biological systems. Formulation and analysisof structured population models in infectious diseases to study theecology/evolution of pathogen-host systems.

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Research Groups UL Applied Math & Stats People

Mathematical Biology

Paul Salceanu (Arizona State Univ.): Difference and DifferentialEquations and Dynamical Systems. Focused on the study ofpersistence in discrete and continuous time dynamical systems, withapplications in population biology and epidemiology.Amy Veprauskas (Univ. of Arizona): Mathematical models ofpopulation and evolutionary dynamics. Dynamical systems theory,structured population models.Xiang Sheng Wang (Ph.D. Hong Kong City Univ.): Asymptoticanalysis and computational mathematics, dynamical systems andmathematical biology.

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Research Groups UL Applied Math & Stats People

Computational Mathematics

Azmy Ackleh (Ph.D. Univ of Tennessee): Computational methods forPDE arising in biology.Longfei Li (Univ. of Delaware): High performance scientificcomputation and numerical analysis. Mathematical modeling and highfidelity simulations. Software: Development team ofOvertureFramework.Baker Kearfott, Emeritus, (Univ. of Utah): Interval Computations &Global Optimization.Bruce Wade, Endowed Professor (Univ. of Wisconsin): Numericalanalysis for PDE. Computational mathematics, mathematicalmodeling, and industrial mathematics.Xiang-Sheng Wang (Ph.D. Hong Kong City Univ.): Asymptoticanalysis and computational mathematics, dynamical systems andmathematical biology.

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Research Groups UL Applied Math & Stats People

Statistics

Sungsu Kim (Univ. of California, Riverside): Circular statistics &applications.

Kalimuthu Krishnamoorthy, Endowed Professor & Fellow of ASA,(Indian Institute of Technology): Multivariate data analysis,inferences with missing data, statistical tolerance region andcalibration, & meta analysis. .

Nabendu Pal (Univ. of Maryland Baltimore County): DecisionTheory & Bayesian analysis reliability & life testing Biostatistics.

Yongli Sang (Univ. of Mississippi): Nonparametric statistics, robuststatistics and correlated data analysis.

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Research Groups UL Applied Math & Stats People

Applied & Applicable Mathematics

Keng Deng, Endowed Professor (Iowa State Univ.): PartialDifferential Equations, applicatioins of PDE in mathematical Biology.

Aghalaya Vatsala, Endowed Professor (Indian Institute ofTechnology): PDE.

Endowed Professor (4): Donors support travel and summer research.$200,000 donation.Endowed Chair (3): Donors support travel, post-doc and summerresearch. $1million donation.

UL Math emphasis on research quality from research trackprofessors & many Ph.D. holding teaching track faculty to aid inthe overal effort

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Research Groups UL Applied Math & Stats People

Nonlinear Reaction-Diffusion Systems

Nonlinear parabolic initial-boundary value problem:

ut + Au = f (t ,u) in Ω, t ∈ (0,T ) ,

u(·,0) = u0

Ω : bounded domain in Rd ;A = −D∆, D is diagonal & positive definite;Boundary conditions are homogeneous Dirichlet, homogeneousNeumann, or periodic;f : nonlinear reaction.

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Research Groups UL Applied Math & Stats People

Exact dynamics:

u(t) = E(t)u0 +

∫ t

0E(t − s) f (s,u(s)) ds

0 < k ≤ k0, tn = nk ,0 ≤ n ≤ N.

Normalize over one time step tn to tn+1 & use E(t) = e−tA:

Capture the single step exact dynamics

u(tn+1) = e−kAu(tn) + k∫ 1

0e−kA(1−τ)f (tn + τk ,u(tn + τk)) dτ

Options: Approximate the whole integrand (quadrature) orapproximate f & integrate exactly.

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Research Groups UL Applied Math & Stats People

Single Step Exact Dynamics Exploited

For e−kA we have choices: Rational approximation (Padé or othertype), Krylov subspace methods...None appears easy due to e−kA(1−τ)

inside the integral. However, we have found a way to deal with theproblem.

Two interesting rational approximations are:(1,1) - Padé Scheme (Crank-Nicolson):

R1,1(−kA) = (I +12

kA)−1(I − 12

kA)

Real-Distinct Poles (RDP) Scheme (L-stable, Khaliq-Voss):

r(−kA) =

(I − 5

12Ak)(

I +14

Ak)−1(

I +13

Ak)−1

Equivalent: = 9(I +13

kA)−1 − 8(I +14

kA)−1

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L-Stability to damp spurious oscillations

R(z) ≈ e−z is A-acceptable if |R(z)| < 1 whenever Re(z) > 0 andL-acceptable if also |R(z)| → 0 as Re(z)→∞.

RDP is L-acceptable & has a simple form: 9(1+13

z)−1−8(1+14

z)−1 ≈ e−z

0 50 100 150−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z

exp(−z)Pade(0,2)Pade(1,1)RDP

Figure: L-Stability of RDP in comparison with Pade(0,2), Pade(1,1)

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ETD-RDP Scheme

Semi-discrete ETD using RDP

Linear appoximation for f , integrate exactly:

u(tn+1) = e−Aku(tn) + A−1(I − e−Ak )f (tn,u(tn))

+ k−1A−2(e−kA − I + kA)(f (tn,u(tn+1)− f (tn,u(tn)))

& use —

e−kA ≈(

I − 512

kA)(

I +14

kA)−1(

I +13

kA)−1

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ETD-RDP Scheme

Semi-discrete⇒ fully discrete

Intermediate form —

Un+1 =

(I − 5

12Ak)(

I +14

Ak)−1(

I +13

Ak)−1

Un

+k2

(I +

Ak4

)−1(I +

Ak3

)−1

f (tn,Un)

+k2

(I +

16

kA)(

I +14

kA)−1(

I +13

kA)−1

f (tn+1,U∗n+1)

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ETD-RDP Scheme

Partial fraction forms simplify the computation

(I − 5

12Ak)(

I +14

Ak)−1(

I +13

Ak)−1

= 9(

I +13

Ak)−1

− 8(

I +14

Ak)−1

(I +

Ak4

)−1(I +

Ak3

)−1

= 4(

I +13

Ak)− 3

(I +

14

Ak)−1

(I +

Ak6

)(I +

Ak4

)−1(I +

Ak3

)−1

= 2(

I +13

Ak)−1

−(

I +14

Ak)−1

RE Efficiency with linear algebra solvers

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ETD-RDP Scheme

ETD-RDP: Efficient implementation

1 Solve for the estimator U∗

(I + Ak)U∗ = Un + kf (Un)

2 Solve for Un+1 (serial or parallel)(I +

13

Ak)

Ua = 9Un + 2kf (Un) + kf (U∗)(I +

14

Ak)

Ub = −8Un −32

kf (Un)− k2

f (U∗)

Un+1 = Ua + Ub

Suggests parallel implementation & splitting.Allows highly efficient linear solvers.

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ETD-RDP Scheme

2D Efficient ETD-RDP-IF Algorithm

(I + A2k)u∗ = (un + kf (un))

(I + A1k)u∗ = u∗

(I +13

A1k)a1 = un & (I +14

A1k)b1 = un

(I +13

A1k)a2 = f (un)

(I +14

A1k)b2 = f (un)

c1 = 9a1 − 8b1 c2 = 9a2 − 8b2

(I +13

A2k)d1 = 9c1 + 2kc2 + kf (u∗)

(I +14

A2k)d2 = −8c1 −32

kc2 −12

kf (u∗)

Un+1 = d1 + d2.

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ETD-RDP Scheme

Schematic: 3D, parallel, multi-threaded PC

Figure: 3D with Three threads.30 / 32

Exp2: 2D Brusselator

2D Brusselator

∂u1

∂t= ε1∆u1 + u2

1u2 − (A + 1)u1 + B

∂u2

∂t= ε2∆u2 − u2

1u2 + Au1

ε1 = ε2 = 2.10−3,A = 1.0,B = 3.4.

At the boundary of the domain homogeneous Neumann conditions areimposed. Initial conditions:

u(x , y ,0) =12

+ y , v(x , y ,0) = 1 + 5x

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Exp2: 2D Brusselator

Snapshot of convergence

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